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Research article

Preliminary assessment of solid waste in Philippine Fabrication Laboratories

  • Received: 08 March 2021 Accepted: 10 June 2021 Published: 30 June 2021
  • Solid waste management is seen as a response to the increase in waste generation due to the rising number of industrial facilities. This includes digital manufacturing facilities such as Fabrication Laboratories (FAB LAB) which acts as innovation centers that generates prototypes using a common set of digital fabrication equipment. Previous studies have tackled with the environmental impacts of FAB LABs in a macro-level scale; however, there has been a lack of research specifically assessing the solid waste of laboratories, more so on Philippine FAB LABs. A baseline assessment study on FAB LABs of the Philippines could be applicable in future implementations of solid waste management systems through the crafting of institutional policies and guidelines for environmental sustainability. Using data gathered from 11 respondent FAB LABs, this study quantified percentage compositions of the waste according to waste type as well as the relative waste generated by each respondent FAB LAB. Machine availability was seen as a factor in waste generation resulting in the high generation of wood and plastic waste. Moreover, it was observed that earlier established laboratories generally had more active makers than recently established ones, hence the older FAB LABs statistically produced more waste. Approximately 53% of the overall waste produced was considered recyclable by Philippine standards but the actual recyclability of the waste was still undetermined due to the ambiguous criteria for recyclables and the lack of feedback data from recycling facilities. The initial findings suggest that an implementation of continuous waste monitoring, sufficient in-laboratory protocols, and coordination between FAB LABs and recycling facilities could improve actual waste recyclability and—by extension—the environmental sustainability of Philippine FAB LABs.

    Citation: Lemuel Clark Velasco, Mary Jane Burden, Marie Joy Satiniaman, Rachelle Bea Uy, Luchin Valrian Pueblos, Reynald Gimena. Preliminary assessment of solid waste in Philippine Fabrication Laboratories[J]. AIMS Environmental Science, 2021, 8(3): 255-267. doi: 10.3934/environsci.2021017

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  • Solid waste management is seen as a response to the increase in waste generation due to the rising number of industrial facilities. This includes digital manufacturing facilities such as Fabrication Laboratories (FAB LAB) which acts as innovation centers that generates prototypes using a common set of digital fabrication equipment. Previous studies have tackled with the environmental impacts of FAB LABs in a macro-level scale; however, there has been a lack of research specifically assessing the solid waste of laboratories, more so on Philippine FAB LABs. A baseline assessment study on FAB LABs of the Philippines could be applicable in future implementations of solid waste management systems through the crafting of institutional policies and guidelines for environmental sustainability. Using data gathered from 11 respondent FAB LABs, this study quantified percentage compositions of the waste according to waste type as well as the relative waste generated by each respondent FAB LAB. Machine availability was seen as a factor in waste generation resulting in the high generation of wood and plastic waste. Moreover, it was observed that earlier established laboratories generally had more active makers than recently established ones, hence the older FAB LABs statistically produced more waste. Approximately 53% of the overall waste produced was considered recyclable by Philippine standards but the actual recyclability of the waste was still undetermined due to the ambiguous criteria for recyclables and the lack of feedback data from recycling facilities. The initial findings suggest that an implementation of continuous waste monitoring, sufficient in-laboratory protocols, and coordination between FAB LABs and recycling facilities could improve actual waste recyclability and—by extension—the environmental sustainability of Philippine FAB LABs.



    Smooth varieties have certain nice properties, and both algebraic and analytic methods can be applied to them. However, when studying problems in birational geometry, particularly those related to the minimal model program, it becomes necessary to investigate varieties with singularities. Fortunately, as Hironaka's famous theorem states, every variety in characteristic zero has a resolution of singularity. The existence of resolutions of singularities provides a way to study singular varieties. For instance, by comparing a variety to its resolution of singularities, one can measure the complexity of a singularity. This is a fundamental technique in higher-dimensional birational geometry.

    Since we want to understand a singularity through its resolution, it is natural to inquire about the difference between two distinct resolutions of singularities. For an algebraic surface S, there exists a smooth surface known as the minimal resolution of S. This is a resolution of singularities ˉSS such that ρ(ˉS/S) is minimal. The minimal resolution ˉS is unique, and any birational morphism SS from a smooth surface S to S factors through ˉSS.

    In this paper, we want to find a higher-dimensional analog of the minimal resolution for surfaces. It is not reasonable to assume the existence of a unique minimal resolution for higher-dimensional singularities. For instance, if XX is a smooth flop over W, then X and X are two different resolutions of singularities for W. Since flops are symmetric (at least in dimension three), it appears that X and X are both minimal. Thus, we need to consider the following issues:

    (1) To define "minimal resolutions", which ideally should be resolutions of singularities with some minimal geometric invariants.

    (2) To compare two different minimal resolutions. We need some symmetry between them, so that even if minimal resolutions are not unique, it is not necessary to distinguish them.

    (3) To compare a minimal resolution with an arbitrary resolution of singularities.

    Inspired by the two-dimensional case, it is natural to consider resolutions of singularities with minimal Picard number (we will call such resolutions P-minimal resolutions, see Section 7 for a more precise definition). We know that a fixed singularity may have more than one P-minimal resolution, and two different P-minimal resolutions can differ by a smooth flop. It is also possible that two different P-minimal resolutions "differ by a singular flop": consider XX as a possibly singular flop over W. Let ˜XX and ~XX be P-minimal resolutions of X and X, respectively. Then, because of the symmetry between flops, one may expect that ˜X and ~X are two different P-minimal resolutions of W. We call the birational map ˜X~X a P-desingularization of the flop XX (a precise definition can be found in Section 7). If we consider P-desingularizations of flops as elementary birational maps, then in dimension three, P-minimal resolutions have nice properties.

    Theorem 1.1. Assume that X is a projective threefold over the complex numbers and ˜X1, ˜X2 are two different P-minimal resolutions of X. Then ˜X1 and ˜X2 are connected by P-desingularizations of terminal and Q-factorial flops.

    Moreover, if X has terminal and Q-factorial singularities, then the birational map ˜X1˜X2 has an Ω-type factorization.

    Please see Section 6 for the definition of Ω-type factorizations.

    Theorem 1.2. Assume that X is a projective threefold over the complex numbers and WX is a birational morphism from a smooth threefold W to X. Then, for any P-minimal resolution ˜X of X, one has a factorization

    W=˜Xk...˜X1˜X0=˜X

    such that ˜Xi+1˜Xi is either a smooth blow-down or a P-desingularization of a terminal Q-factorial flop.

    Since three-dimensional terminal flops are topologically symmetric, some topological invariants like Betti numbers will not change after P-desingularizations of terminal flops. Hence, it is easy to see that P-minimal resolutions are the resolution of singularities with minimal Betti numbers.

    Corollary 1.3. Assume that X is a projective threefold over the complex numbers and WX is a birational morphism from a smooth threefold W to X. Then, for any P-minimal resolution ˜X of X, one has that bi(˜X)bi(W) for all i=0, ..., 6.

    Although in dimension three P-minimal resolutions behave well, for singularities of dimension greater than three, P-minimal resolutions may not be truly "minimal". A simple example is a smooth flip. If XX is a smooth flip over W, then both X and X are P-minimal resolutions of W, but X is better than X. Notice that the only known smooth flips are standard flips [1, Section 11.3], and if XX is a standard flip, then it is easy to see that bi(X)bi(X) for all i and the inequality is strict for some i. Thus, the resolution of singularities with minimal Betti numbers may be the right minimal resolution for higher-dimensional singularities. Because of Corollary 1.3, in dimension three P-minimal resolutions are exactly those smooth resolutions which have minimal Betti numbers. Therefore, this new definition of minimal resolutions is compatible with our three-dimensional theorems.

    We now return to the proof of our main theorems. Let X be a threefold and WX be a resolution of singularities. One can run KW-MMP over X as

    W=X0X1...Xk=X.

    Let ˜Xi be a P-minimal resolution of Xi. Then ˜X0=W and it is easy to see that ˜Xk is also a P-minimal resolution of X. Thus, our main theorems can be easily proved if we know the relation between ˜Xi and ˜Xi+1. Since Xi has only terminal and Q-factorial singularities, studying P-minimal resolutions of Xi becomes simpler.

    In [2], Chen introduced feasible resolutions for terminal threefolds, which is a resolution of singularities consisting of a sequence of divisorial contractions to points with minimal discrepancies (see Section 2.3.3 for more detail). Given a terminal threefold X and a feasible resolution ˉX of X, one can define the generalized depth of X to be the integer ρ(ˉX/X). The generalized depth is a very useful geometric invariant of a terminal threefold. In our application, the crucial factor is that one can test whether a resolution of singularities WX is a feasible resolution or not by comparing ρ(W/X) and the generalized depth of X. We need to understand how generalized depths change after steps of the minimal model program. After that, we can prove that for terminal and Q-factorial threefolds, P-minimal resolutions and feasible resolutions coincide.

    Now we only need to figure out the following two things: how generalized depths change after a step of minimal model program (MMP), and how P-minimal resolutions change after a step of MMP. To answer those questions, we have to factorize a step of MMP into more simpler birational maps. In [3], Chen and Hacon proved that three-dimensional terminal flips and divisorial contractions to curves can be factorized into a composition of (inverses of) divisorial contractions and flops. In this paper, we construct a similar factorization for divisorial contractions to points. After knowing the factorization, we are able to answer the two questions above and prove our main theorems.

    In addition to the above, we introduce the notion of Gorenstein depth for terminal threefolds. The basic idea is as follows: given a sequence of steps of MMP of terminal threefolds

    X0X1...Xk,

    one can show that the generalized depth of Xk is bounded above by the integer k and the generalized depth of X0. That is to say, the number of steps of MMP bounds the singularities on the minimal model. One may ask whether there is an opposite bound. Specifically, if we know the singularities of the minimal model Xk, can we bound singularities of X0? In this paper, we define the Gorenstein depth of terminal threefolds, which roughly speaking measures only Gorenstein singularities. One can show that the Gorenstein depth is always non-decreasing when running three-dimensional terminal MMP. Our result on Gorenstein depth will have important applications in [4].

    This paper is structured as follows: Section 2 is a preliminary section. In Section 3, we develop some useful tools to construct relations between divisorial contractions to points. Those tools, as well as the explicit classification of divisorial contractions, will be used in Section 4 to construct links of different divisorial contractions to points. In Section 5, we prove the property of the generalized depth. The construction of diagrams in Theorem 1.1 will be given in Section 6. All our main theorems will be proved in Section 7. In the last section, we discuss possible higher-dimensional generalizations of the notion of minimal resolutions, and possible applications of our main theorems.

    In this paper we only consider varieties over complex numbers.

    Let X and Y be two algebraic varieties. We say that X and Y are birational if there exists Zariski open sets UX and VY such that U and V are isomorphic. If X and Y are birational, we say that ϕ:XY is a birational map, and we will denote ϕ|U to be the isomorphism UV. If ϕ:XY is a morphism between X and Y and there exists a Zariski open set UX such that ϕ|U is an isomorphism, then we say that ϕ is a birational morphism.

    For a divisorial contraction, we mean a birational morphism YX which contracts an irreducible divisor E to a locus of codimension at least two, such that KY is Q-Cartier and is anti-ample over X. We will denote by vE the valuation that corresponds to E.

    Let G be a cyclic group of order r generated by τ. For any Z-valued n-tuple (a1,...,an), one can define a G-action on An(x1,...,xn) by τ(xi)=ξaixi, where ξ=e2πir. We will denote the quotient space An/G by An(x1,...,xn)/1r(a1,...,an).

    We say that w is a weight on W/G=An(x1,...,xn)/G defined by w(x1,...,xn)=1r(b1,...,bn) if w is a map OW1rZ0 such that

    w((i1,...,in)Zn0c(i1,...,in)xi11...xinn)=min{1r(b1i1+...+bnin) |c(i1,...,in)0}.

    Assume that ϕ:XY is a birational map. Let UX be the largest open set such that ϕ|U is an isomorphism and ZX be an irreducible subset such that Z intersects U non-trivially. We will denote by ZY the closure of ϕ|U(Z|U).

    Let W=An and G be a finite cyclic group, such that ˉW=W/GAn(x1,...,xn)/1r(a1,...,an). There is an elementary way to construct a birational morphism WˉW, so called the weighted blow-up, defined as follows.

    We write everything in the language of toric varieties. Let N be the lattice e1,...,en,vZ, where e1, ..., en is the standard basis of Rn and v=1r(a1,...,an). Let σ=e1,...,enR0. We have ˉWSpecC[Nσ].

    Let w=1r(b1,...,bn) be a vector such that bi=λai+kir for λN and kiZ with bi0. We define a weighted blow-up of ˉW with weight w to be the toric variety defined by the fan consisting of the cones

    σi=e1,...,ei1,w,ei+1,...,en.

    Let Ui be the toric variety defined by the cone σi and lattice N, namely

    Ui=SpecC[Nσi].

    Lemma 2.1. One has that

    UiAn/τ,τ

    where τ is the action given by

    xiξrbixi,xjξbjbixj,ji

    and τ is the action given by

    xiξaibixi,xjξajbiaibjrbixj,ji.

    Here, ξk denotes a k-th roots of unity for any positive integer k.

    In particular, the exceptional divisor of WˉW is P(b1,...,bn)/G where G is a cyclic group of order m where m is an integer that divides λ.

    Proof. Let Ti be a linear transformation such that Tiej=ej if ji and Tiw=ei. One can see that

    Tiei=rbi(eijibjrej)

    and

    Tiv=jiajrej+airrbi(eijibjrej)=aibiei+jiajbiaibjrbjej.

    Under this linear transformation, σi becomes the standard cone e1,...,enR0. Note that

    kiTiei+λTiv=kir+λaibiei+jiλ(ajbiaibj)kibjrrbiej=ei+jiλajbibibjrbiej=eijikjej.

    Hence, eiTiN and TiN=e1,...,en,Tiei,TivZ. Now Tiei corresponds to the action τ and Tiv corresponds to the action τ. This means that UiAn/τ,τ.

    The computation above shows that τki=τλ. If we glue (xi=0)An/τ together, then we get a weight projective space P(b1,...,bn). The relation τki=τλ implies that (xi=0)Ui can be viewed as P(b1,...,bn)/G where G is a cyclic group of order m for some factor m of λ.

    Corollary 2.2. Let x1, ..., xn be the local coordinates of W and let y1, ..., yn be the local coordinates of Ui. The change of coordinates of the morphism UiˉW are given by xj=yjybjri and xi=ybiri.

    Proof. The change of coordinates is defined by Tti, where Ti is defined as in Lemma 2.1.

    Corollary 2.3. Assume that

    S=(f1(x1,...,xn)=...=fk(x1,...,xn)=0)ˉW

    is a complete intersection and S is the proper transform of S on W. Assume that the exceptional locus E of SS is irreducible and reduced. Then

    a(E,S)=b1+...+bnrki=1w(fk)1.

    Proof. Assume first that k=0. Denote ϕ:WˉW. Then, on Ui, we have

    ϕdx1...dxn=birybir1i(jiybjri)dy1...dyn,

    hence KW=ϕKˉW+(b1+...+bnr1)F where F=exc(WˉW).

    Now the statement follows from the adjunction formula.

    Corollary 2.4. Let F=exc(WˉW). Then

    Fn=(1)n1rn1b1...bnm.

    Here, m is the integer in Lemma 2.1.

    Proof. From the change of coordinate formula in Corollary 2.2, one can see that F|F=OP(b1,...,bn)/G(r). It follows that

    Fn=(F|F)n1=(1)n1rn1b1...bnm.

    Definition 2.5. Let ϕi:UiˉW be the morphism in Corollary 2.2. For any G-semi-invariant function uOW, we can define the strict transform of u on Ui by (ϕ1i)(u)=ϕ(u)/yw(u)i.

    In this paper, we will consider terminal threefolds which are embedded into a cyclic quotient of A4 or A5

    XA4(x,y,z,u)/1r(a,b,c,d)orXA5(x,y,z,u,t)/1r(a,b,c,d,e).

    We say that YX is a weighted blow-up with weight w if Y is the proper transform of X inside the weighted blow-up of A4(x,y,z,u)/1r(a,b,c,d) or A5(x,y,z,u,t)/1r(a,b,c,d,e) with weight w.

    Notation 2.6. Assume that X is of the above form and let YX be a weighted blow-up. The notation Ux, Uy, Uz, Uu and Ut will stand for U1, ..., U5 in Lemma 2.1.

    Notation 2.7. Assume that w is a weight on An(x1,...,xn) determined by w(x1,...,xn)=(a1,...,an) and

    f(x1,...,xn)=(i1,...,in)Zn0λi1,...,inxi11...xinn

    is a regular function on An. We denote

    fw=a1i1+...+anin=w(f)λi1,...,inxi11...xinn.

    The local classification of terminal threefolds was done by Reid [5] for Gorenstein cases and Mori [6] for non-Gorenstein cases.

    Definition 2.8. A compound Du Val point PX is a hypersurface singularity which is defined by f(x,y,z)+tg(x,y,z,t)=0, where f(x,y,z) is an analytic function which defines a Du Val singularity.

    Theorem 2.9 ([5,Theorem 1.1]). Let PX be a point of threefold. Then PX is an isolated compound Du Val point if and only if PX is terminal and KX is Cartier near P.

    Theorem 2.10 ([6, cf. [7,Theorem 6.1]). Let PX be a germ of three-dimensional terminal singularity such that KX has Cartier index r>1. Then

    X(f(x,y,z,u)=0)A4(x,y,z,u)/1r(a1,...,a4)

    such that f, r and ai are given by Table 1.

    Table 1.  Classification of terminal threefolds.
    Type f(x,y,z,u) r ai condition
    cA/r xy+g(zr,u) any (α,α,1,r) gm2P α and r are coprime
    cAx/4 xy+z2+g(u)x2+z2+g(y,u) 4 (1,1,3,2) gm3P
    cAx/2 xy+g(z,u) 2 (0,1,1,1) gm4P
    cD/3 x2+y3+z3+u3x2+y3+z2u+yg(z,u)+h(z,u)x2+y3+z3+yg(z,u)+h(z,u) 3 (0,2,1,1) gm4Phm6P
    cD/2 x2+y3+yzu+g(z,u)x2+yzu+yn+g(z,u)x2+yz2+yn+g(z,u) 2 (1,0,1,1) gm4P,n4n3
    cE/2 x2+y3+yg(z,u)+h(z,u) 2 (1,0,1,1) g,hm4Ph40

     | Show Table
    DownLoad: CSV

    Assume that PX is a three-dimensional terminal singularity. Then there exists a section H|KX| which has Du Val singularities (referred to as a general elephant). Please see [7,(6.4)] for details.

    Divisorial contractions to points between terminal threefolds are well-classified by Kawamata [8], Hayakawa [9,10,11], Kawakita [12,13,14] and Yamamoto [15].

    Theorem 2.11. Assume that YX is a divisorial contraction to a point between terminal threefolds. Then there exists an embedding XW with W=A4(x,y,z,u) or A5(x,y,z,u,t) and a weight w(x,y,z,u)=1r(a1,...,a4) or w(x,y,z,u,t)=1r(a1,...,a5), respectively, such that YX is a weighted blow-up with respect to w.

    The defining equation of XW and the weight are given in Table 2 to Table 11.

    Table 2.  Divisorial contractions to cAx/r points.
    No. defining equations (r;ai)weight typea(X,E) condition
    A1 xy+zrk+gka(z,u) (r;β,β,1,r)1r(b,c,a,r) cA/ra/r baβ(modr),b+c=rka
    A2 x2y2+z3+xu2+g6(x,y,z,u) (1;)(4,3,2,1) cA23 xzg(x,y,z,u)

     | Show Table
    DownLoad: CSV
    Table 3.  Divisorial contractions to cAx/r points.
    No. defining equations (r;ai)weight typea(X,E) condition
    Ax1 x2+y2+g2k+12(z,u) (4;1,3,1,2)14(b,c,1,2) cAx/41/4 (b,c)=(2k+1,2k+3) or (2k+3,2k+1)
    Ax2 x2+y2+(λx+μy)p2k+14(z,u)+g2k+32(z,u) (4;1,3,1,2)14(b,c,1,2) cAx/41/4 (b,c,λ,μ)=(2k+5,2k+3,1,0) or (2k+3,2k+5,0,1)
    Ax3 x2+y2+gk(z,u) (2;0,1,1,1)12(b,c,1,1) cAx/21/2 (b,c)=(k,k+1) or (k+1,k)
    Ax4 x2+y2+(λx+μy)pk2(z,u)+gk+1(z,u) (2;0,1,1,1)12(b,c,1,1) cAx/21/2 (b,c,λ,μ)=(k+2,k+1,1,0) or (k+1,k+2,0,1)

     | Show Table
    DownLoad: CSV
    Table 4.  Divisorial contractions to cD points with discrepancy one.
    No. defining equations weight typea(X,E) condition
    D1 x2+y2u+λyzk+gl(z,u) (b,b1,1,2) cD1 b=min{k1,l2}
    D2 x2+y2u+λyzk+g2l(z,u) (b,b,1,1) cD1 b=min{k,l}
    D3 {x2+ut+λyzk+g2b+2(z,u)y2+p2b(x,z,u)+t (b+1,b,1,1,2b+1) cD1 kb+2
    D4 x2+y2u+yhk(z,u)+g2b+1(x,z,u) (b+1,b,1,1) cD1 kb+1
    D5 {x2+yt+g2b(z,u)yu+pb(z,u)+t (b,b1,1,1,b+1) cD1 zbp(z,u)

     | Show Table
    DownLoad: CSV
    Table 5.  Divisorial contractions to cD points with discrepancies greater than one.
    No. defining equations weight typea(X,E) condition
    D6 x2+y2u+zk+g2b+1(x,y,z,u) (b+1,b,a,1) cDa ak=2b+1
    D7 {x2+yt+g2b+2(y,z,u)yu+zk+pb+1(z,u)+t (b+1,b,a,1,b+2) cDa ak=b+1
    D8 {x2+ut+λzb+14+gb+1(y,z,u)y2+μzb14+pb1(x,z,u)+t (b+12,b12,4,1,b) cD4 b+14N,λ=1,μ=0, or b14N,μ=1,λ=0.
    D9 {x2+ut+zb+12+gb+1(y,z,u)y2+pb1(x,z,u)+t (b+12,b12,2,1,b) cD2
    D10 x2+y2u+zb+g2b(y,z,u) (b,b,2,1) cD2
    D11 x2+y2u+yp3(z,u)+u3+g6(z,u) (3,3,1,2) cD42 z3p(z,u)
    D12 x2+y2u+z3+yu2+g6(y,z,u) (3,4,2,1) cD43

     | Show Table
    DownLoad: CSV
    Table 6.  Divisorial contractions to cD/r points with discrepancy one.
    No. defining equations (r;ai)weight typea(X,E) condition
    D13 x2+y3+gk(y,z,u) (3;0,2,1,1)13(3,2,4,1) cD/31/3 k=2 and zu2 or z3g, or k=3 and z2ug
    D14 x2+y3+z3+g4(y,z,u) (3;0,2,1,1)13(6,5,4,1) cD/31/3
    D15 x2+yzu+g2(y,z,u) (2;1,1,1,0)12(3,1,1,2) cD/21/2
    D16 x2+yzu+g3(y,z,u) (2;1,1,1,0)12(3,b,c,d) cD/21/2 (b,c,d)=(3,1,2)(1,1,4)
    D17 {x2+yt+g3(z,u)zu+y3+t (2;1,1,1,0,1)12(3,1,1,2,5) cD/21/2
    D18 x2+y2u+λyzk+gl(z,u) (2;1,1,1,0)12(b,b2,1,4) cD/21/2 b=min{k2,l21}
    D19 x2+y2u+λyzk+gl(z,u) (2;1,1,1,0)12(b,b,1,2) cD/21/2 b=min{k,l}
    D20 {x2+ut+λyzk+gb+2(z,u)y2+pb(x,z,u)+t (2;1,1,1,0,0)12(b+2,b,1,2,2b+2) cD/21/2 kb+4
    D21 x2+y2u+yhk(z,u)+gb+1(x,z,u) (2;1,1,1,0)12(b+2,b,1,2) cD/21/2 kb+2
    D22 {x2+yt+g2b(z,u)yu+zb+t (2;1,1,1,0,1)12(b,b2,1,2,b+2) cD/21/2

     | Show Table
    DownLoad: CSV
    Table 7.  Divisorial contractions to cD/r points with large disprepancies.
    No. defining equations (r;ai)weight typea(X,E) condition
    D23 x2+y2u+zm+gb+1(x,y,z,u) (2;1,1,1,0)12(b+2,b,a,2) cD/2a/2 ma=2b+2,aandbareodd
    D24 {x2+yt+gb+2(z,u)yu+zm+pb2+1(z,u)+t (2;1,1,1,0,1)12(b+2,b,a,2,b+4) cD/2a/2 ma=b+2ab(mod2)
    D25 x2+y2u+z4b+g4b(y,z,u) (2;1,1,1,0)(2b,2b,1,1) cD/21
    D26 x2+yzu+y4+zb+uc (2;1,1,1,0)(2,1,2,1) cD/21 b,c4 b is even
    D27 {x2+ut+y4+z4yz+u2+t (2;1,1,1,0,0)(2,1,1,1,3) cD/21
    D28 {x2+ut+g2b+2(y,z,u)y2+p2b(x,z,u)+t (2;1,1,1,0,0)(b+1,b,1,1,2b+1) cD/21 Either b is odd, or b is even and xzb1 or z2bp
    D29 {x2+ut+g2b+2(y,z,u)y2+p2b(x,z,u)+t (2;1,1,1,0,0)(b+1,b,2,1,2b+1) cD/22 xzb12 or zbp

     | Show Table
    DownLoad: CSV
    Table 8.  Divisorial contractions to cE points with discrepancy one.
    No. defining equations weight typea(X,E) condition
    E1 x2+y3+g4(y,z,u) (2,2,1,1) cE61 2y2g(y,z,u)=0
    E2 x2+xp2(z,u)+y3+g5(y,z,u) (3,2,1,1) cE6,71
    E3 x2+y3+gq6(y,z,u) (3,2,2,1) cE1
    E4 x2+y3+y2p2(z,u)+gq8(y,z,u) (4,3,2,1) cE1
    E5 x2+xp4(y,z,u)+y3+g9(y,z,u) (5,3,2,1) cE1
    E6 x2+y3+y2p3(z,u)+g10(y,z,u) (5,4,2,1) cE7,81
    E7 x2+y3+g12(y,z,u) (6,4,3,1) cE1
    E8 x2+y3+y2p4(z,u)+g14(y,z,u) (7,5,3,1) cE7,81
    E9 x2+xp7(y,z,u)+y3+g15(y,z,u) (8,5,3,1) cE7,81
    E10 x2+y3+g18(y,z,u) (9,6,4,1) cE7,81
    E11 x2+y3+y2p6(z,u)+g20(y,z,u) (10,7,4,1) cE81
    E12 x2+y3+g24(y,z,u) (12,8,5,1) cE81
    E13 x2+y3+g30(y,z,u) (15,10,6,1) cE81

     | Show Table
    DownLoad: CSV
    Table 9.  Divisorial contractions to cE points with discrepancy one, continued.
    No. defining equations weight typea(X,E) condition
    E14 {x2+y3+tz+g6(y,z,u)p4(x,y,z,u)+t (3,2,1,1,5) cE6,71 p(x,y,z,u) is irreducible
    E15 x2+xp2(z,u)+y3+g6(x,y,z,u) (4,2,1,1) cE61
    E16 {x2+y3+tp2(z,u)+g6(y,z,u)q3(y,z,u)+t (3,2,1,1,4) cE71 q(y,z,u) is irreducible
    E17 x2+y3+yz3+g6(y,z,u) (3,3,1,1) cE71 y2u2g
    E18 {x2+yt+g10(y,z,u)y2+p6(y,z,u)+t (5,3,2,1,7) cE7,81 y2+p(y,z,u) is irreducible

     | Show Table
    DownLoad: CSV
    Table 10.  Divisorial contractions to cE points with large disprepancies.
    No. defining equations weight typea(X,E) condition
    E19 x2+(y+p2(z,u))3+yu3+g6(z,u) (3,3,2,1) cE62 zp(z,u)
    E20 {x2+yt+g10(y,z,u)y2+p6(z,u)+t (5,3,2,2,7) cE72 gcd(p6,g10)=1
    E21 x2+y3+u7+g14(z,u) (7,5,3,2) cE7,82 yz3,z5 or z4ug(z,u)

     | Show Table
    DownLoad: CSV
    Table 11.  Divisorial contractions to cE/2 points.
    No. defining equations (r;ai)weight typea(X,E)
    E22 x2+y3+g3(y,z,u) (2;1,0,1,1)12(3,2,3,1) cE/21/2
    E23 x2+y3+g5(y,z,u) (2;1,0,1,1)12(5,4,3,1) cE/21/2
    E24 x2+xp52(y,z,u)+y3+g6(y,z,u) (2;1,0,1,1)12(7,4,3,1) cE/21/2
    E25 x2+y3+g9(y,z,u) (2;1,0,1,1)12(9,6,5,1) cE/21/2
    E26 x2+y3+z4+u8+g8(y,z,u) (2;1,0,1,1)(4,3,2,1) cE/21

     | Show Table
    DownLoad: CSV

    For the reader's convenience, we put these tables in Section 4. In those tables, we use the following notation: For a non-negative integer m, the notation gm represents a function gOW such that w(g)=m. The notation pm represents a function pOW which is homogeneous of weight m with respect to the weight w.

    The reference of each of the cases in Table 2, Table 3, Table 5, ..., Table 7, Table 9, ..., Table 11 is as follows:

    ● Case A1 is [13, Theorem 1.2 (ⅰ)]. Case A2 is [15, Theorem 2.6].

    ● Case Ax1–Ax4 are [9 Theorems 7.4, 7.9, 8.4 and 8.8 respectively.

    ● Cases D6 and D7 are [13, Theorem 1.2 (ⅱ)]. Case D8–D11 is [15] Theorem 2.1–2.4. Case D12 is [15, Theorem 2.7].

    ● Case D13 is [9, Theorems 9.9, 9.14, 9.20]. Case D14 is [9, Theorem 9.25].

    ● Case D16 is [10, Proposition 4.4]. Case D17 is [10, Proposition 4.7, 4.12]. Case D18 is [10, Proposition 4.9]. Case D18 is [10, Proposition 5.4]. Case D19 is [10, Propositions 5.8, 5.13, 5.22, 5.28, and 5.35]. Case D20 is [10, Propositions 5.18 and 5.25]. Case D21 is [10, Propositions 5.16 and 5.32]. Case D22 is [10, Propositions 5.9 and 5.36].

    ● Cases D23 and D24 is [13, Theorem 1.2(ⅱ)] and [11, Theorem 1.1 (ⅲ)]. Case D25–D28 is [11 Theorem 1.1 (ⅰ), (ⅰ'), (ⅱ'), (ⅲ), (ⅱ) respectively. Case D29 is [14, Theorem 2].

    ● Case E19–E21 is [15] Theorems 2.5, 2.9 and 2.10 respectively.

    ● Case E22 is [9, Theorems 10.11, 10.17, 10.22, 10.28, 10.33 and 10.41]. Case E23 is [9, Theorems 10.33 and 10.47]. Case E24 is [9, Theorems 10.54 and 10.61]. Case E25 is [9], Theorem 10.67]. Case E26 is [11, Theorem 1.2].

    Divisorial contractions to cD points of discrepancy one (Case D1–D5 in Table 4) and divisorial contractions to cE points of discrepancy one (Case E1–E18 in Table 8) was completely classified by Hayakawa in his two unpublished papers "Divisorial contractions to cD points" and "Divisorial contractions to cE points". We will briefly introduce how to derive this classification. For more detail, please contact the author or Professor Takayuki Hayakawa in Kanazawa University.

    Let (oX) be a germ of three-dimensional Gorenstein terminal singular point with type cD or cE.

    Step1: Construct a divisorial contraction X1X which contracts an exceptional divisor of discrepancy one to o. We refer to [2, Section 4, Section 6] for the explicit construction. X1X can be viewed as a weighted blow-up with respect to an explicit embedding and a explicit weight.

    Step2: Find all exceptional divisors E over X such that a(E,X)=1 and CenterXE=o. We know that exc(X1X) is an exceptional divisor of discrepancy one. Assume that Eexc(X1X). Then an easy computation on discrepancies shows that a(E,X1)<1. In particular, CenterX1E is a non-Gorenstein point. Since X1X is an explicit weighted blow-up, all non-Gorenstein points on X1 can be explicitly computed, and all exceptional divisors of discrepancy less than one can be explicitly write down. Say S is the set of exceptional divisors over X1 with discrepancy less than one. One can compute a(E,X) for ES. If a(E,X)>1, then we remove E form S. After that, S is a set consisting exceptional divisors over X of discrepancy one.

    Step3: For any exceptional divisor ES, the valuation of E on X can be calculated. One can construct a weighted blow-up YEX with respect to this valuation. If YE do not have terminal singularities, then we remove E form S. Now, YEX for all ES, together with X1X, are all divisorial contractions to o with discrepancy one.

    Definition 2.12. Let YX be a divisorial contraction which contracts a divisor E to a point P. We say that YX is a w-morphism if a(X,E)=1rP, where rP is the Cartier index of KX near P.

    Definition 2.13. The depth of a terminal singularity PX, dep(PX), is the minimal length of the sequence

    XmXm1X1X0=X,

    such that Xm is Gorenstein and XiXi1 is a w-morphism for all 1im.

    The generalized depth of a terminal singularity PX, gdep(PX), is the minimal length of the sequence

    XnXn1X1X0=X,

    such that Xn is smooth and XiXi1 is a w-morphism for all 1in. The variety Xn is called a feasible resolution of PX.

    The Gorenstein depth of a terminal singularity PX, depGor(PX), is defined by gdep(PX)dep(PX).

    For a terminal threefold we can define

    dep(X)=Pdep(PX),
    gdep(X)=Pgdep(PX)

    and

    depGor(X)=PdepGor(PX).

    Remark 2.14. In the above definition, the existence of a sequence

    XmXm1X1X0=X,

    such that Xm is Gorenstein follows from [10, Theorem 1.2]. The existence of a sequence

    XnXn1X1X0=X,

    such that Xn is smooth follows from [2, Theorem 2].

    Definition 2.15. Assume that YX is a w-morphism such that gdep(Y)=gdep(X)1. Then we say that YX is a strict w-morphism.

    Lemma 2.16. Assume that YX is a divisorial contraction which is a weighted blow-up with the weight w(x1,...,xn)=1r(a1,...,an) with respect to an embedding XAn(x1,...,xn)/G where G is a cyclic group of index r. Assume that E is an exceptional divisor over X and vE(x1,...,xn)=1r(b1,...,bn). Then CenterYEUi non-trivially if and only if biaibjaj for all 1jn. Here, U1, ..., Un denotes the canonical affine chart of the weighted blow-up on Y.

    Proof. Let y1, ..., yn be the local coordinates of Ui. Then we have the following change of coordinates formula:

    xi=yairi,xj=yajriyj if ij.

    One can see that

    vE(yi)=biai,vE(yj)=birajbirai if ij.

    We know that CenterEY intersects Ui non-trivially if and only if birajbirai0 for all ji, or, equivalently, bjajbiai for all ji.

    Corollary 2.17. Assume that YX and Y1X are two different w-morphisms over the same point. Let E and F be the exceptional divisors of YX and Y1X, respectively. Then there exists uOX such that vE(u)<vF(u).

    Proof. Let XAn(x1,...,xn)/G be the embedding so that Y1X can be obtained by the weighted blow-up with respect to the embedding. We may assume that (xn=0) defines a Du Val section. Then

    vE(xn)=a(E,X)=a(F,X)=vF(xn).

    It follows that an=bn=1 where (a1,...,an) and (b1,...,bn) are integers in Lemma 2.16. Now, since a(F,X)=a(E,X), one has that CenterY1E is a non-Gorenstein point (if Y1 is generically Gorenstein along CenterY1E, then an easy computation shows that a(E,X)>a(F,X)). It follows that CenterY1EUn is empty since an=1 implies that Un is Gorenstein. Thus, by Lemma 2.16 we know that there exists j so that bjaj<1. Hence, vE(u)>vF(u) if u=xj.

    We have the following factorization of steps of three-dimensional terminal MMP by Chen and Hacon [3].

    Theorem 2.18 ([3, Theorem 3.3]). Assume that either XX is a flip over V, or XV is a divisorial contraction to a curve such that the exceptional locus contains a non-Gorenstein singular point. Then there exists a diagram

    such that Y1X is a w-morphism, YkX is a divisorial contraction, Y1Y2 is a flip or a flop, and YiYi+1 is a flip for i>1. If XV is divisorial, then YkX is a divisorial contraction to a curve and XV is a divisorial contraction to a point.

    Remark 2.19. Notation as in the above theorem. From the construction of the diagram, we can state the following:

    (1) Let CY1 be a flipping/flopping curve of Y1Y2. Then CX is a flipping curve of XX. Here, we use the notation introduced in Section 2.1, so CX is the image of CY1 on X.

    (2) Assume that the exceptional locus of XV contains a non-Gorenstein point P which is not a cA/r or a cAx/r point. Then Y1X can be chosen to be any w-morphism over P. This statement follows from the proof of [3, Theorem 3.1].

    We have the following properties of the depth [3, Propositions 2.15, 3.8 and 3.9]:

    Lemma 2.20. Let X be a terminal threefold.

    1. If YX is a divisorial contraction to a point, then dep(Y)dep(X)1.

    2. If YX is a divisorial contraction to a curve, then dep(Y)dep(X).

    3. If XX is a flip, then dep(X)>dep(X).

    We have the following negativity lemma for flips.

    Lemma 2.21. Assume that XX is a (KX+D)-flip. Then, for all exceptional divisors E, one has that a(E,X,D)a(E,X,DX). The inequality is strict if CenterXE is contained in the flipping locus.

    Proof. It is a special case of [16, Lemma 3.38].

    What we really need is the following corollary of the negativity lemma.

    Corollary 2.22. Assume that XX is a (KX+D)-flip and CX is an irreducible curve which is not a flipping curve. Then (KX+D).C(KX+DX).CX. The inequality is strict if C intersects the flipping locus non-trivially.

    Proof. Let XϕWϕX be a common resolution such that C is not contained in the indeterminacy locus of ϕ. Then Lemma 2.21 implies that F=ϕ(KX+D)ϕ(KX+DX) is an effective divisor and is supported on exactly those exceptional divisors whose centers on X are contained in the flipping locus. Hence,

    (KX+D).C(KX+DX).CX=(ϕ(KX+D)ϕ(KX+DX)).CW=F.CW0.

    The last inequality is strict if and only if CW intersects F non-trivially, or, equivalently, C intersects the flipping locus non-trivially.

    Let YX be a divisorial contraction between Q-factorial terminal threefolds that contracts a divisor E to a point. We construct the diagram

    as follows: Let Z1Y be a w-morphism and let H|KX| be a Du Val section. According to [3, Lemma 2.7 (ⅱ)], we have a(E,X,H)=0. We run the (KZ1+HZ1+ϵEZ1)-MMP over X for some ϵ>0 such that (Z1,HZ1+ϵEZ1) is klt. Notice that a general curve inside EZ1 intersects the pair negatively, and a general curve in F intersects the pair positively where F=exc(Z1Y). Thus, after finitely many (KZ1+HZ1+ϵEZ1)-flips Z1...Zk, the MMP ends with a divisorial contraction ZkY1 which contracts EZk, and Y1X is a divisorial contraction which contracts FY1.

    Lemma 3.1. Keeping the above notation, assume that KZ1 is anti-nef over X and EZ1 is not covered by KZ1-trivial curves. Then ZiZi+1 is a KZi-flip or flop for all i and ZkY1 is a KZk-divisorial contraction. In particular, Y1, Z2, ..., Zk are all terminal.

    Proof. Assume first that k=1. If Z1Y1 is a KZ1-negative contraction, then we are done. Otherwise, Z1Y1 is a KZ1-trivial contraction. In this case, EZ1 is covered by KX1-trivial curves, which contradicts our assumption.

    Now assume that k>1. We know that Z1Z2 is a KZ1-flip or flop. Also, notice that a general curve on EZ1 is KZ1-negative. Hence a general curve on EZ2 is KZ2-negative by Corollary 2.22. Now the relative effective cone NE(Z2/X) is a two-dimensional cone. One of the boundaries of NE(Z2/X) corresponds to the flipped/flopped curve of Z1Z2 and is KZ2-non-negative. Since there is a KZ2-negative curve, we know that the other boundary of NE(Z2/X) is KZ2-negative. Therefore, if k=2, then Z2Y1 is a KZ2-divisorial contraction, and for k>2, Z2Z3 is a KZ2-flip. One can prove the statement by repeating this argument k2 more times.

    We are going to find the sufficient conditions for the assumptions of Lemma 3.1. Our final results are Lemmas 3.4 and 3.6.

    Let XAn(x1,...,xn)/G=W/G be the embedding such that YX is a weighted blow-up with respect to the weight w and this embedding. First, we show that after replacing W by a larger affine space, if necessary, we may assume that Z1YX can be viewed as a sequence of weighted blow-ups with respect to the embedding XW/G.

    Let V be a suitable open set which contains P=CenterYF such that Z1Y can be viewed as a weighted blow-up with respect to an embedding VAn1(y1,...,yn1)/G. For j=1, ..., n1, we define DjV to be the Weil divisor corresponding to yj=0. Then Dj,X=ϕDj is a Weil divisor on X. Since X is Q-factorial, Dj,X corresponds to a G-semi-invariant function sjOW. We can consider the embedding

    XW/G(xn+jsj=0)j=1,...,n1An(x1,...,xn+n1)/G=ˉW/G.

    Then, YX is also a weighted blow-up with respect the weight ˉw which is defined by ˉw(xj)=w(xj) if j<n, and ˉw(xn+j)=w(sj). The embedding XˉW is exactly what we need.

    Now, let WW be the first weighted blow-up. We may assume P=CenterYF is the origin of UiW. Let y1, ..., yn be the local coordinate system of Ui that is mentioned in Corollary 2.2. We know that E|Ui=(yi=0). Let f4, ..., fn be the defining equation of XW/G. Then f4, ..., fn define Y|Ui where fi=(ϕ|1Ui)(fi). Since Y has terminal singularities, the weighted embedding dimension of Y|Ui near P is less than 4. For 5jn, we may write fj=ξjyj+fj(y1,...y4) for some ξj which does not vanish on P. One can always assume that HY=(y3=0) and so i3.

    Let fj=fj|yi=y3=0. Then f4, ..., fn defines HE near P. If fj is irreducible as a G-semi-invariant function, then we let ηj=fj. Otherwise, let ηj be a G-semi-invariant irreducible factor of fj.

    Lemma 3.2. Assume that YX can be viewed as a four-dimensional weighted blow-up. Then η4=...=ηn=0 defines an irreducible component of HYE.

    Proof. Since YX can be viewed as a four-dimensional weighted blow-up, we know that i4 and fj=yj+fj|y3=yi=0 for all j>4. Hence, we have ηj=fj. One can see that the projection

    (η4=...=ηn=0)|HYEP(a1,...,an)P(a1,...,a4)(η4=0)|HYE

    is an isomorphism. Since η4 is an irreducible function, it defines an irreducible curve.

    Notice that ηj is a polynomial in y1, ..., yn. There exists ηjOW such that ηj=(ϕ|1Ui)(ηj). We assume that YX is a weighted blow-up with the weight 1r(a1,...,an) and Z1Y is a weighted blow-up with the weight 1r(a1,...,an).

    Lemma 3.3. Let Γ=(η4=...=ηn=0) and assume that Γ is an irreducible and reduced curve. Then

    KZ1.ΓZ1=a23vE(η4)...vE(ηn)rn3ma1...an+aivF(η4)...vF(ηn)rn4a1...an.

    Here, m is the integer in Lemma 2.1 corresponding to the weighted blow-up YX.

    Proof. Since ΓE and KZ1+HZ1 is numerically trivial over X, we only need to show that

    HZ1.ΓZ1=a23vE(η4)...vE(ηn)rn3ma1...anaivF(η4)...vF(ηn)rn4a1...an. (3.1)

    We know that HZ1.ΓZ1=H.ΓvF(HY)F.ΓZ1. We need to show that the first term of (3.1) equals H.Γ and the second term of (3.1) equals vF(HY)F.ΓZ1.

    We have an embedding YWPW(a1,...,an). Let Dj be the divisor on W which corresponds to ηj. Then Γ=D4..Dn.E.H is a weighted complete intersection, so ΓZ1=D4,Z1..Dn,Z1.EZ1.HZ1. To compute H.Γ, we view Γ as a curve inside P(a1,...,an) which is defined by H=D4=...=Dn=0. It follows that

    H.Γ=a23vE(η4)...vE(ηn)rn3ma1...an.

    To compute F.ΓZ1, one writes

    F.ΓZ1=F.(ψD4vF(η4)F)..(ψDnvF(ηn)F).(ψEvF(E)F).(ϕHYvF(HY)F)=(1)n1vF(η4)...vF(ηn)vF(E)vF(HY)Fn.

    Since Z1Y is a w-morphism, the integer λ in Section 2.2 is 1. Hence, we know that Fn=(1)n1rn1a1...an. Now, vF(E)=air and vF(HY)=a(Y,F)=1r, so

    vF(HY)F.ΓZ1=aivF(η4)...vF(ηn)rn4a1...an.

    Lemma 3.4. Notation and assumption as in Lemma 3.3. Assume that:

    (ⅰ) For all 4jn, there exists an integer δj so that xkjδj appears in ηj as a monomial for some positive integer kj. Moreover, the integers δ4, ..., δn are all distinct.

    (ⅱ) If ji, 3, δ4, ..., δn, then a3ajaj.

    Then KZ1.ΓZ10.

    Proof. Fix j4. From the construction and our assumption we know that that i, δ4, ..., δn are all distinct. One can see that rvE(ηj)=kjaδj and rvF(ηj)kjaδj. Thus, we have a relation

    rvE(ηj)aδjrvF(ηj)aδj.

    One can always assume that if j>4, ji, then δj=j. By interchanging the order of y1, ..., y4, we may assume that δ4=4. Now, if i<4, then we may assume that i=1. If i>4, then we may assume that δi=1. We can write

    a23vE(η4)...vE(ηn)rn3ma1...an=1maia3a2rvE(η4)aδ4...rvE(ηn)aδn

    and

    aivF(η4)...vF(ηn)rn4a1...an=1r1a2rvF(η4)aδ4...rvF(ηn)aδn.

    Since mai=r, a3a21a2 and rvE(ηj)aδjrvF(ηj)aδj, we know that

    a23vE(η4)...vE(ηn)rn3ma1...anaivF(η4)...vF(ηn)rn4a1...an,

    So KZ1.ΓZ10.

    Remark 3.5.

    (1) From the construction we know that if j5, ji, then one can choose δj=j.

    (2) If YX can be viewed as a four-dimensional weighted blow-up, then condition (ⅰ) of Lemma 3.4 always holds. Indeed, in this case one has i4, so η4 is a two-variable irreducible function, hence there exists δ44 such that yk4δ4η4 for some positive integer k4. One also has δj=j for all j>4. Thus, condition (ⅰ) of Lemma 3.4 holds.

    (3) If for ji, 3, δ4, ..., δn one has that ajai, then condition (ⅱ) of Lemma 3.4 holds. Indeed, by Lemma 2.1 we know that UiAn/τ,τ where τ corresponds to the vector v=1ai(a1,...,ai1,r,ai+1,an). Let ˉv be the vector corresponding to the cyclic action near PUi. Then vmˉv (mod Zn) and r=mai. Since Z1Y is a w-morphism, and since HY is defined by y3=0, we know that a3=1 and ˉva31r(a1,...,an) (mod Zn). One can see that a3ajaj (mod r). This implies that a3ajaj since

    ajaimai=r.

    Lemma 3.6. Assume that KZ1 is anti-nef over X and there exists uOX such that vE(u)<a(E,X)a(F,X)vF(u). Then ZiZi+1 is a KZi-flip or flop for all 1ik1 and ZkY1 is a terminal divisorial contraction.

    In particular, if there exists ji such that a3aj>aj, then the conclusion of this lemma holds.

    Proof. We only need to show that EZ1 is not covered by KZ1-trivial curves. Then the conclusion follows from Lemma 3.1.

    Assume that EZ1 is covered by KZ1-trivial curves. Since KZ1 is anti-nef, those KZ1-trivial curves are contained in the boundary of the relative effective cone NE(Z1/X). Hence, k=1 and Z1Y1 is a KZ1-trivial divisorial contraction. Notice that if CYE is a curve which does not contain P, then KZ1.CZ1=KY.CY<0, hence the curve CZ1 is not contracted by Z1Y1. Thus, Z1Y1 is a divisorial contraction to the curve CY1. Notice that, in this case, a(E,Y1)=0.

    By computing the discrepancy, one can see that the pull-back of FY1 on Z1 is FZ1+a(E,X)a(F,X)EZ1. It follows that for all uOX, one has that

    vE(u)a(E,X)a(F,X)vF(u).

    Hence, if there exists u such that vE(u)<a(E,X)a(F,X)vF(u), then EZ1 is not covered by KZ1-trivial curves, so ZkY1 is a terminal divisorial contraction.

    Now, by Lemma 3.7, we know that

    a(E,X)a(F,X)=ra3r+a3ai.

    Consider u=xj. For ji we know that vE(xj)=ajr and

    vF(xj)=vF(yjyajri)=ajr+ajairr=raj+ajairr.

    The inequality vE(u)a(E,X)a(F,X)vF(u) becomes

    ajrra3r+a3airaj+ajairr=1ra3(raj+ajai)r+a3ai,

    or, equivalently,

    aj(r+a3ai)a3(raj+ajai).

    This is equivalent to

    aja3aj.

    Hence, the condition a3aj>aj implies that vE(u)<a(E,X)a(F,X)vF(u).

    Lemma 3.7. One has that

    a(E,X)=a3r,a(F,X)=r+a3airr.

    Proof. Since a(E,X,H)=0, we know that a(E,X)=vE(H)=a3r. Then

    a(F,X)=1r+a3rair=r+a3airr.

    Lemma 3.8. Note that the assumption in Lemma 3.4 depends only on the first weighted blow-up YX. In other words, we can check whether the assumption holds or not by simply considering the embedding which defines the weighted blow-up YX instead of considering the (possibly) larger embedding which defines both YX and Z1Y. Likewise, to apply Lemma 3.6, we can simply look at the embedding that defines YX, if condition a3aj>aj already holds under this embedding.

    Notation 3.9.

    (1) We say that the condition (Ξ) holds if conditions (ⅰ) and (ⅱ) in Lemma 3.4 hold for all possible choices of Γ. We say that the condition (Ξ) holds if conditions (2) and (3) in Remark 3.5 hold for all possible choices of Γ. As explained in Remark 3.5, we know that the condition (Ξ) implies the condition (Ξ).

    (2) We say that the condition (Ξ) (resp. (Ξ)) holds if the condition (Ξ) (reps. (Ξ)) holds and the inequality in Lemma 3.4 is strict for all possible choices of Γ. Using the notation in Lemma 3.4, it is equivalent to say that either there exists ji, 3, δ4, ..., δn such that a3aj>aj, or there exists j4 such that

    rvE(ηj)aδj>rvF(ηj)aδj.

    (3) We say that the condition (Θu) holds for some function u if vE(u)<a(E,X)a(F,X)vF(u). We say that the condition (Θj) holds for some index j if a3aj>aj. In either case, Lemma 3.6 can be applied.

    Notation 3.10. We say that a divisorial contraction YX is linked to another divisorial contraction Y1X if the diagram

    exists, where Z1Y is a strict w-morphism over a non-Gorenstein point, ZkY1 is a divisorial contraction, and ZiZi+1 is a flip or a flop for all 1ik1. We use the notation YXY1 if YX is linked to Y1X.

    Furthermore, if all ZiZi+1 are all flips, or k=1, then we say that Y is negatively linked to Y1, and use the notation YXY1.

    Remark 3.11. At this point, it is not clear why Z1Y should be a divisorial contraction to a non-Gorenstein point. In fact, from the classification of divisorial contractions between terminal threefolds (cf. Tables in Section 4), one can see that if there are two different divisorial contractions YX and Y1X, then Y or Y1 always contain a non-Gorenstein point. It is natural to construct the diagram starting with the most singular point, which is always a non-Gorenstein point.

    Remark 3.12.

    (1) If (Ξ) or (Ξ) holds and (Θu) or (Θj) holds for some function u or index j, then by Lemmas 3.4 and 3.6 one has that YXY1.

    (2) Assume that (Ξ) or (Ξ) holds and (Θu) or (Θj) holds for some function u or index j. Then one has that YXY1.

    Lemma 3.13. Assume that

    X(x1(x1+p(x2,...,x4))+g(x2,...,x4)=0)A4/G,

    such that

    (1) vE(g)=a1r+vE(p)=2a1r1.

    (2) i=1, a2+a4=a1 and a3=1.

    Then YXY1.

    Proof. We know that a1>aj for j=2, ..., 4, so (Ξ) holds. Consider the embedding

    X(x1x5+g(x2,...,x4)=x5x1p(x2,...,x4)=0)A5(x1,...,x5)/G.

    Then YX can be viewed as a weighted blow-up with the weight 1r(a1,...,a5) with respect to this embedding, where a5=rvE(p). The origin of U1 is a cyclic quotient point of type 1a1(r,a2,...,a5). The only w-morphism is the weighted blow-up that corresponds to the weight w(y2,...y4)=1a1(a2,...,a4). One can see that a5=rvE(g)>rvE(p)=a5, hence (Θ5) holds. Moreover, one can see that η5=y5p(y2,0,y4), so rvF(η5)=rvE(η5)=rvE(p(x2,0,x4)), hence

    rvE(η5)a5>rvF(η5)a5.

    Thus, YXY1.

    Lemma 3.14. Assume that YX and Y1X are two divisorial contractions such that YXY1. Let E and F be the exceptional divisors of YX and Y1X, respectively. Assume that there exists uOX such that vF(u)=1r and vF(u)>0 where r is the Cartier index of CenterXE and u is the strict transform of u on Y. Then a(F,X)<a(E,X) if a(E,X)>1.

    Proof. Notice that we have

    vF(u)=vF(˜u)+vE(u)vF(E).

    Since vE(u)1r and vF(˜u)>0, we know that vF(E)<1. Thus

    a(F,X)=1r+a(E,X)vF(E)1r+a(E,X)r1r=a(E,X)+1a(E,X)r

    where r is the Cartier index of CenterYF. Hence, a(F,X)<a(E,X) if a(E,X)>1.

    The aim of this section is to prove the following proposition:

    Proposition 4.1. Let X be a terminal threefold and YX, YX be two different divisorial contractions to points over X. Then there exists Y1, ..., Yk, Y1, ..., Yk such that

    Y=Y1XY2X...XYk=YkX...XY1=Y.

    Proof. We need a case-by-case discussion according to the type of the singularity on X. Please see Propositions 4.2, 4.3, 4.5–4.10 and 4.15–4.18.

    We keep the notation in Section 3.

    In this subsection, we assume that X has cA/r singularities. Divisorial contractions over X are listed in Table 2.

    Proposition 4.2.

    (1) If YX is of type A1 with a>1, then YXY1 for some Y1X which is of type A1 with the discrepancy less than a.

    (2) If YX is of type A1 with a=1 and b>r, then YXY1 where Y1 is an A1 type weighted blow-up with the weight 1r(br,c+r,1,r). One also has Y1XY if we begin with Y1X and interchange the role of x and y. Moreover, YXY1 if and only if η4=y.

    (3) If YX is of type A2, then YXY1 where Y1X is a divisorial contraction of type A1.

    Proof. Assume first that YX is of type A1. We are going to prove (1) and (2). If both b and c are less than r, then a=k=1. In this case, there is exactly one divisorial contraction of type A1, so there is nothing to prove. Thus, we may assume that one of b or c, say b>r.

    The origin of the chart UxY is a cyclic quotient point. On this chart, one can choose (y1,...,y4)=(x,u,z,y) with i=1 and δ4=4. One can see that (Ξ) holds. Now the two action in Lemma 2.1 is given by

    τ=1b(r,c,a,r),τ=1b(β,β(b+c)r,baβr,bβ).

    Since Ux is terminal, there exists a vector τ"=1b(bδ,ϵ,1,δ) such that τaτ" (mod Z4) and τλτ" (mod Z4) for some integer λ. There is exactly one w-morphism over the origin of Ux which extracts the exceptional divisor F so that vF corresponds to the vector τ". One can also see that

    ϵb=vF(y)=vF(g)rkb

    where g is the strict transform of g on Ux, since if zrpuqg, then ap+qak and

    vF(zrpuq)=1b(rp+δq)=1ab(rap+δaq)rkaab=rkb

    for δar because δar(mod b) and b>r. Thus,

    a4=c<rkaaϵ=a3a4,

    hence (Θ4) holds, and there exists Y1X such that YXY1.

    We need to check whether (Ξ) holds or not. We have that f4=η4=y4+g. One always has that

    rvF(η4)a4=bϵbϵ=1.

    Now, g=0 if and only if

    rvE(η4)a4=rcrc=1,

    and δa=r if and only if

    a3a2=aδ=r=a2.

    Thus, (Ξ) holds if and only if g0 or a do not divide r.

    One can compute the discrepancy of Y1X using Lemma 3.7. We know that ai=bδ and a3=a, so

    a(F,X)=r+a3airr=rδa+barbar=a(E,X).

    If a=1, then r=δ, so a(F,X)=a(E,X)=1r. One can verify that Y1X is the weighted blow-up with the weight 1r(br,c+r,1,r). In this case, YXY1 if and only if g0. Hence, YXY1 if and only if η4=y. This proves (2).

    Now assume that a>1. We already know that δar. If δa>r, then a(F,X)<a(E,X) and YXY1, so (1) holds. Hence, one only needs to show that δar. If δa=r, then

    b=aβ+λr=a(β+λδ)

    where λ=baβr. One can see that bβ=(a1)β+λaδ. On the other hand, since τλτ" (mod Zn), we know that bβλδ (mod b). Hence, b divides

    bβλδ=(a1)(β+λδ).

    This is impossible since (a1)(β+λδ) is a positive integer and is less than b.

    Finally, assume that YX of type A2. In this case, one needs to look at the chart UxY, and we choose (y1,...,y4)=(x,z,y,u) with i=1 and δ4=4. One can see that (Ξ) holds. The origin of the chart Ux is a cAx/4 point of the form

    (x2y2+z3+u2+g(x,y,z,u)=0)A4(x,y,z,u)/14(1,1,2,3).

    From [9, Theorem 7.9], we know that there are exactly two w-morphisms over this point which are weighted blow-ups with the weights w±(x±y,xy,z,u)=14(5,1,2,3). For both these two w-morphisms, one has that a2=2, a3=3 and a2=2, so (Θ2) holds and (Ξ) holds since a3a2>a2. Thus, there exists Y1X so that YXY1. One can compute that the discrepancy of Y1X is one, so Y1X is of type A1. This proves (3).

    In this subsection, we assume that X has cAx/r singularities with r=2 or 4. Divisorial contractions over X are listed in Table 3.

    Proposition 4.3. (1) Assume that YX is of type Ax1 or Ax3. Then YX is the only divisorial contraction over X.

    (2) Assume that YX is of type Ax2 or Ax4. Then there are exactly two divisorial contractions over X. Let Y1X be another divisorial contraction. Then Y1X has the same type of YX, and one has that YXY1XY.

    Proof. The number of divisorial contractions follows from [9, Section 7, 8]. So we can assume that YX is of type Ax2 or Ax4, and Lemma 3.13 implies that YXY1.

    In this subsection, we assume that X has cD singularities. At first, we consider w-morphisms over X, which are listed in Table 4. Notice that for types D1, D2 or D5 in Table 4 there is at most one divisorial contraction over X which is of the given type. This is because the equations of type D1, D2 and D5 come from the normal form of cD-type singularities, which are unique, and the blowing-up weights are determined by the defining equations.

    Lemma 4.4. Assume that there exists two different divisorial contractions with discrepancy one over X. Then, one of the following holds:

    (1) One of the divisorial contractions is of type D1.

    (2) The two morphisms are of type D2 and D5, respectively.

    (3) Both of the divisorial contractions are of type D3.

    (4) Both of the divisorial contractions are of type D4.

    Proof. Assume that YX and Y1X are the two given divisorial contractions. It is enough to prove the following statements:

    (ⅰ) If YX is of type D4, then Y1X is of type D1 or D4.

    (ⅱ) If YX is of type D2, then Y1X is of type D1 or D5.

    (ⅲ) If YX is of type D5, then Y1X is not of type D3.

    Let E and F be the exceptional divisors of YX and Y1X, respectively. Then a(F,Y)<1, since otherwise a(F,X)>1. Thus, P=CenterFY is a non-Gorenstein point.

    First, assume that YX is of type D4. In this case, P may be the origin of Ux or the origin of Uy, and they are both cyclic quotient points. Exceptional divisors over P with discrepancy less than one are described in [4, Proposition 3.1]. The origin of Ux is a 1b+1(b,1,1) point. If P is this point, then, since z=0 defines a Du Val section, we have that vF(z)=a(F,X)=1. One can verify that vF(u)=vF(z)=1 and vF(x)=vF(y)=b. Now, vF(x)=b only when xpb(z,u)g(x,z,u) for some homogeneous polynomial p(z,u) of degree b. One can check that vF(x+p(z,u))=b+1. In this case, Y1X is also of type D4 after a change of coordinates xxp(z,u). If P is the origin of Uy, then it is a 1b(1,1,1) point. One can verify that vF(u)>1. This implies that Y1X is of type D1.

    Now, assume YX is of type D2. Then P is the origin of UyY, which is a cA/b point. Exceptional divisors of discrepancy less than one over P are described in [4, Proposition 3.4]. One can verify that if λ0 and k=b, then vF(u)=1. In this case, Y1X is of type D5. Otherwise, vF(u)=2, and so Y1X is of type D1.

    Finally, assume that YX is of type D5. One can see that Y1X cannot have type D3 since zbp(z,u). This finishes the proof.

    Proposition 4.5. Assume that there exist two different divisorial contractions with discrepancy one over X, say YX and Y1X.

    (1) If YX and Y1X are both of type D4, then YXY1XY.

    (2) If YX is of type D3 and Y1X is of type D1, then YXY1. If Y1X is of type D3, then there exists another divisorial contraction Y2X which is of type D1 so that YXY2XY1.

    (3) If YX is of type D2 and Y1X is of type D5, then YXY1.

    (4) If YX is of type D1 and Y1X is not of type D3, then YXY1.

    Proof. Assume first that YX and Y1X are both of type D4. Notice that, in this case, xpb(z,u)g(x,z,u). Thus, YXY1 by Lemma 3.13.

    Now assume that YX is of type D3. Consider the chart UtY which is defined by

    (x2+u+λyzktb+k2b2+g(z,u,t)=y2p(x,z,u)+t=0)
    A5(x,y,z,u,t)/12b+1(b+1,b,1,1,1).

    Notice that, using the notation in Section 3, we know that

    f4=η4=x2+u+g(0,u,0),f5=y2+p(x,0,u).

    η5 can be y±μub if p(x,0,u)=μ2u2b for some μC, and otherwise η5=f5. One can see that η4=η5=0 defines an irreducible and reduced curve. There is only one w-morphism over the origin of Ut which is defined by weighted blowing up the weight w(x,y,z,u,t)=12b+1(b+1,b,1,2b+2,2b). Now, in this case we choose (y1,...,y5)=(x,y,z,u,t) with i=5, δ4=4, δ5=2. One can see that (Ξ) holds and (Θ4) holds. Also, one has that rvE(η4)a4=2b+2 while rvF(η4)a4=1. Thus, (Ξ) holds and so there exists Y2X such that YXY2. One can compute that Y2X is of type D1. If Y1X is of type D1, then Y2=Y1 since there is at most one divisorial contraction with type D1. This proves statement (2).

    Now assume that YX is of type D2 and Y1X is of type D5. In this case, we consider the embedding corresponding to Y1X. Under this embedding, YX is given by the weighted blow-up with the weight (b,b,1,1,b) and the chart UyY is given by

    Uy=(x2t+g(y,z,u)=yu+zb+t=0)A5(x,y,z,u,t)/1b(0,1,1,1,0).

    We take (y1,...,y5)=(y,u,z,x,t) with δ4=4 and δ5=5. Then (Ξ) holds. The origin of Uy is a cA/b point and the weight w(y1,...,y5)=1b(b1,1,1,b,2b) defines a w-morphism over Uy. One can see that (Θ5) holds. Moreover, since a5=2b>b=a5, we know that (Ξ) holds. Thus, YXY1.

    Finally, assume that YX is of type D1 and Y1X is not of type D3. Let b and b1 be the integers in Table 4 corresponding to YX and Y1X, respectively. First, we claim that bb1. Indeed, if Y1X is of type D5, then zb1h(z,u), which implies that b1b+1. If Y1X is of type D2 or D4, then the inequality bb1 follows from Corollary 2.17. Now, the origin of the chart UuY is defined by

    (x2+y2+λyzkukb1+g(z,u)=0)A4(x,y,z,u)/12(b,b1,1,1),

    which is a cAx/2 point. We can take (y1,...,y4)=(u,y,z,x) with i=1 and δ4=4. w-morphisms over this point are fully described in [9, Section 8]. Since bb1, we know that 2kb1>b and the multiplicity of g(z,u) is greater than or equal to 2b. Hence, if F is the exceptional divisor of a w-morphism over Y, then

    vF(y)b>b1=vE(y).

    Thus, (Ξ) and (Θ2) holds and one has YXY1.

    Now we study divisorial contractions of discrepancy greater than one. All such divisorial contractions are listed in Table 5.

    Proposition 4.6. Assume that YX is a divisorial contraction with discrepancy a>1. Then there exists a divisorial contraction Y1X such that YXY1, and a(F,X)<a where F=exc(Y1X).

    Proof. First, notice that (Θu) holds in cases D6–D10 or D12, and (Θz) holds in case D11. This is because vE(u) or vE(z)=1 in those cases and a(E,X)a(F,X)=a>1.

    Now we list all cases in Table 5, write down the chart on Y we are looking at, and write down the variables y1, ..., yn. One can easily see that (Ξ) holds in all cases.

    (1) Assume that YX is of type D6. Consider the chart UxY and take (y1,...,y4)=(x,y,z,u) with δ4=4.

    (2) Assume that YX is of type D7. Consider the chart UtY and take (y1,...,y5)=(y,u,z,x,t), δ4=4 and δ5=1 or 2.

    (3) Assume that YX is of type D8 or D9. We consider the chart UtY and take (y1,...,y5)=(x,y,z,u,t) with δ4=4 and δ5=2.

    (4) Assume that YX is of type D10. We consider the chart UyY and take (y1,...,y4)=(y,u,z,x) with δ4=4.

    (5) Assume that YX is of type D11. We consider the chart UyY and (y1,...,y4)=(y,z,u+λy,x) for some λC with δ4=4.

    (6) Assume that YX is of type D12. We consider the chart UyY and (y1,...,y4)=(y,z,x+λy,u) for some λC with δ4=4.

    Now we know that there exists Y1X so that YXY1. Then Y1X is of one of types in Table 4 or Table 5. One can see that vF(z)=1 if Y1X is of types D1–D5, D7–D9 or D11 and vF(u)=1 if Y1X is of type D6, D10 or D12. Since CenterYF is the origin of the chart Ux, Uy or Ut, one can apply Lemma 3.14 to say that a(F,X)<a. This finishes the proof.

    In this subsection, we assume that X has cD/r singularities with r=2 or 3. We first study w-morphisms over X.

    Proposition 4.7. Assume that X has cD/3 singularities.

    (1) If YX is of type D14 or if YX is of type D13 and both zu2 and z2ug(y,z,u), then there is only one w-morphism over X.

    (2) If YX is of type D13 and zu2 or z2ug(y,z,u), then there are two or three w-morphisms over X. Say Y1X, ..., YkX are other w-morphisms with k=1 or 2. Then YXYiXY for all 1ik.

    Proof. The statement about the number of w-morphisms follows from [9, Section 9]. Now we may assume that YX is of type D13 and zu2 or z2ug(y,z,u). The chart UzY is defined by

    (x2+y3+g(y,z,u)=0)A4(x,y,z,u)/14(3,2,1,1)

    with u2 or zug(y,z,u). We can take (y1,...,y4)=(z,x,u+λz,y) for some λC with δ4=4. Now the w-morphism over Uz is a weighted blow-up with the weight w(y1,...y4)=14(3,5,1,2). One can see that (Θ2) and (Ξ) hold. Hence, we can get a divisorial contraction Y1X such that YXY1. One can compute that Y1X is also a w-morphism.

    If there are three w-morphisms over X, then the defining equation of X is of the form x2+y3+zu(z+u) as in [9, Section 9.A], so g(y,z,u)=u(z+u). One can make a change of coordinates uuz and again consider the weighted blow-up with the same weight 14(3,2,1,5). In this way, we can get a divisorial contraction Y2X which is different to Y1, and we also have that YXY2. This finishes the proof.

    Proposition 4.8. Assume that X has cD/2 singularities and YX is of type D15, D16 or D17.

    (1) If YX is of type D15, then there is only one w-morphism over X.

    (2) If YX is of type D17, then there exists exactly two w-morphisms over X. The other one, Y1X, is of type D16, and one has that YXY1.

    (3) If YX is of type D16 and there are no w-morphisms over X with type D17, then there are exactly three w-morphisms over X. They are all of type D16 and are negatively linked to each other.

    Proof. The statement about the number of w-morphisms follows from [10, Section 4]. Assume that YX is of type D17. Consider the chart UtY with (y1,...,y5)=(y,u,y+z,x,t) with δ4=4 and δ5=1. One can see that (Ξ) holds. Now the origin of Ut is a cyclic quotient point. Let F be the exceptional divisor of the w-morphism over Ut. Then one has that vF(y1,...y5)=15(6,2,1,3,3). One can see that (Θ1) holds.

    Assume that YX is of type D16 and there are no w-morphisms of type D17 over X. By [10, Section 4], we know that neither y4 nor z4g(y,z,u). Assume first that (b,c,d)=(1,1,4). Consider the chart UuY which has a cAx/4 singular point at the origin. We choose (y1,...,y4)=(y,u,y+z,x) with δ4=4. One can see that (Ξ) holds. Let w be the weight on Uu so that w(y1,...,y4)=14(5,2,1,3). Then the weighted blow-up with weight w gives a w-morphism. It follows that (Θ1) holds and also (Ξ) holds since a1=5>3=a1. Hence, there exists a w-morphism Y1X so that YXY1. If we interchange the roles of y and z, we can get another w-morphism Y2X with YXY2.

    Now assume that (b,c,d)=(3,1,2). Consider the chart UyY which is defined by

    (x2+zu+g(y,z,u)=0)A4(x,y,z,u)/13(0,1,1,2).

    Taking (y1,...,y4)=(y,u,y+z,x), then (Ξ) holds. Let w be the weight w(y1,...,y4)=13(1,5,1,3). Then the weighted blow-up with the weight w gives a w-morphism over Uy and (Θ2) and (Ξ) holds. If we take w to be another weight w(x,y,z,u)=13(3,1,4,2), then we get another w-morphism over Uy and (Θz) holds. Thus, we can get two different w-morphisms over X and Y is negatively linked to both of them.

    Proposition 4.9. Assume that X has cD/2 singularities, YX is of type D18–D22, and assume that there are two w-morphisms YX and Y1X.

    (1) Assume that YX is of type D18 and Y1X is not of type D20. Then YXY1.

    (2) Assume that both YX and Y1X are not of type D18. Then, one of the following holds:

    (2–1) YX is of type D19 and Y1X is of type D22. One has that YXY1.

    (2–2) Both YX and Y1X are of type D21 and YXY1XY.

    (2–3) Both YX and Y1X are of type D20 and there exists another w-morphism Y2X which is of type D18 so that YXY2XY1.

    Proof. The computation is similar to the proof of Proposition 4.5 after replacing the types D1–D5 by D18–D22, so we will omit the proof. Notice that an analog result of Lemma 4.4 can be proved by a similar computation, or can be directly followed by [10, Section 5].

    Now we consider non-w-morphisms over X. Notice that there is no divisorial contraction with discrepancy greater than 13 over cD/3 points. Divisorial contractions of discrepancy greater than 12 over cD/2 points are listed in Table 7.

    Proposition 4.10. Assume that r=2 and YX is a divisorial contraction with the discrepancy a2>1. Then there exists a divisorial contraction Y1X such that YXY1, and a(F,X)<a2 where F=exc(Y1X).

    Proof. First, assume that YX is of type D23. Consider the chart UxY which is defined by

    (x+y2u+zm+g(x,y,z,u)=0)A4(x,y,z,u)/1b+2(2,b,a,2).

    We take (y1,...,y4)=(x,y,z,u) with δ4=2 or 4. One can see that (Ξ) holds. Since a4=2 and a3, we know that (Θ4) holds. Thus, there exists Y1X such that YXY1. The origin of Ux is a cyclic quotient point. Let F be the exceptional divisor of the w-morphism over this point. Then vF(x)mb+2. It follows that

    a(F,X)=1b+2+a2vF(x)2+ma2b+4=1<a2.

    Assume that YX is of type D24. Consider the chart UtY which is defined by

    (x2+y+g(z,u,t)=yu+zm+p(z,u)+t=0)A4(x,y,z,u,t)/1b+4(c1,...,c5)

    where (c1,...,c5)=(b+2,b,a,2,2) if a, b are odd, and (c1,...,c5)=(b+3,b+2,a2,1,1) if a, b are even. We take (y1,...,y5)=(y,u,z,x,t) with δ4=4 and δ5=1 or 2. Then (Ξ) holds. Now the origin of Ut is a cyclic quotient point. Let F be the exceptional divisor over this point. Then vF(y1,...,y5)=1b+4(a1,...,a5) with a2+a4=b+4. It follows that a(a2+a4)>b+4=a2+a4, hence (Θj) holds for j=2 or 4. Thus, there exists Y1X so that YXY1. One has that

    a(F,X)=1b+4+a2vF(x)2+ma2b+812.

    Hence, Y1X is a w-morphism.

    Assume that YX is of type D25. The chart UyY is given by

    (x2+yu+z4b+g(y,z,u)=0)A4(x,y,z,u)/14b(0,2b1,1,2b+1).

    We take (y1,...,y4)=(y,u,z,x) with δ4=4. One can see that (Ξ) holds. The origin of Uy is a cA/4b point and there is only one w-morphism over this point. Let F be the exceptional divisor of the w-morphism. Then vF(y1,...,y4)=14b(2b1,2b+1,1,4b). Hence, (Θ2) holds. One can also compute that a(F,X)=12, hence there exists a w-morphism Y1X such that YXY1.

    Assume that YX is of type D26. The chart UzY is a cA/4 point given by

    (x2+yu+y4+z2b4+uczc4=0)A4(x,y,z,u)/14(0,1,1,3).

    We take (y1,...,y4)=(y,u,y+z,x) with δ4=4. One can see that (Ξ) holds. Now let w be the weight such that w(y1,...,y4)=14(1,3,1,4) if b=4 and w(y1,...,y4)=14(1,7,1,4) if b6. Hence, (Θ2) holds, and there exists Y1X such that YXY1. One can compute that a(F,X)=12.

    Assume that YX is of type D27. Consider the chart UtY which is defined by

    (x2+u+y4+z4=yz+u2+t=0)A5(x,y,z,u,t)/16(5,1,1,4,2).

    We take (y1,...,y5)=(u,y,y+z,x,t) with δ4=4 and δ5=1. In this case, (Ξ) holds. Now the origin of Ut is a cyclic quotient point. Let F be the exceptional divisor over this point which corresponds to a w-morphism. Then vF(y1,...,y5)=16(5,1,1,4,2). One can see that (Θ1) holds. Thus, there exists Y1X which extracts F so that YXY1. One can compute that a(F,X)=12.

    Finally, assume that YX is of type D28 or D29. The chart UtY is defined by

    (x2+u+g(y,z,u,t)=y2+p(x,z,u)+t=0)A4(x,y,z,u,t)/14b+2(1,1,a2b1,2,2),

    where a=2 in case D28 and a=4 in case D29. We take (y1,...,y5)=(x,y,z,u,t) with δ4=4 and δ5=2. Then (Ξ) holds. The origin of Ut is a cyclic quotient point. Let F be the exceptional divisor corresponding to the w-morphism over this point. Then vF(y1,...,y5)=14b+2(a1,...,a5) with a1+a2=4b+2, a2(2b+1a)1 (mod 4b+2) and a3=1. From the defining equation one can see that a4>1, hence (Θ4) holds. Thus, there exists a divisorial contraction Y1X which extracts F so that YXY1. We only need to show that a(F,X)<a2.

    Assume that YX is of type D28. In this case, a2 is the integer such that a2(2b1)1 (mod 4b+2). If b is odd, then a2=b since

    b(2b1)=2b2b=(4b+2)b12+1.

    One can see that a52b. If b is even, then a2=3b+1 since

    (3b+1)(2b1)=6b2b1=(4b+2)(32b1)+1.

    Hence, a1=b+1. Now, since xzb1 or z2bp(x,z,u), we also have that a52b. In either case we have

    a(F,X)=14b+2+a54b+22b+14b+2=12<1=a2.

    Finally, assume that YX is of type D29. We want to show that a5<4b+2. Then

    a(F,X)=14b+2+2a54b+214b+2+8b+24b+2<2=a2

    and we can finish the proof. If zbp(x,z,u), then a5b. If a2<2b+1, then a54b. Assume that zbp(x,z,u) and a22b+1. Then a12b+1 and xzb12p(x,z,u). Hence,

    a52b+1+b12<4b+2.

    In this subsection, we assume that X has cE singularities. First, we study w-morphisms over cE points. All w-morphisms over cE type points are listed in Tables 8 and 9.

    We assume that there exist two different w-morphisms over X, say YX and Y1X. Let F=exc(Y1X). Let P=CenterYF. One always has that a(F,Y)<1, so P is a non-Gorenstein point.

    Lemma 4.11. Assume that both YX and Y1X are of type E1–E13. Then YX is not of type E1 or E6.

    Proof. Assume that YX is of type E1. Then the only non-Gorenstein point on Y is the origin of

    Uy=(x2+y2+g(z,u)=0)A4(x,y,z,u)/12(0,1,1,1).

    This is a cAx/2 point. The exceptional divisor G of discrepancy less than one over this point is given by the weighted blow-up with the weight w(x,y,z,u)=12(2,3,1,1). One can compute that a(G,X)=2, hence there is only one w-morphism over X. Thus, YX is not of type E1.

    Assume that YX is of type E6. If X has cE8 singularities, then there is only one non-Gorenstein point on Y, namely the origin of Uy. If X has cE7 singularities, then the origin of Uz is also a non-Gorenstein point. Assume first that P is the origin of Uz. Then P is a cyclic quotient point of index two and there is only one exceptional divisor over P with discrepancy less than one. Hence, F should correspond to this exceptional divisor. One can compute that vF(x,y,z,u)=(3,3,1,1), so Y1X should be of type E17. Nevertheless, in this case one can see that vF(σ)vE(σ) for all σOX. This contradicts Corollary 2.17. Hence, P can not be the origin of Uz.

    We want to show that P is also not the origin of Uy. The chart Uy is defined by

    (x2+y(y+p(z,u))+g(y,z,u)=0)A4(x,y,z,u)/14(1,3,2,1).

    The origin of Uy is a cAx/4 point. Since (u=0) defines a Du Val section, we know that vF(u)=vF(y)+vF(u)=1. Hence, both vF(y) and vF(u)<1. This means that vF(y)3. Assume that YX and Y1X correspond to the same embedding XA4. Then, since vF(y)3, we know that Y1X is of type E1–E5. However, in those cases one always has that vF(σ)vE(σ) for all σOX. This contradicts Corollary 2.17. Thus, Y1X corresponds to a different embedding.

    Let ZY be a w-morphism over the origin of Uy. From the classification we know that ZY is a weighted blow-up with the weight w(x,y,z,u)=14(5,k,2,1) for k=3 or 7. One can compute that non-Gorenstein points on Z over Uy are cyclic quotient points. Let ˉZZ be an economic resolution over those cyclic quotient points. Then F appears on ˉZ since a(F,Y)<1. Moreover, ˉZX can be viewed as a sequence of weighted blow-ups with respect to the embedding XA4(x,y,z,u). We write (x1,...,x4)=(x,y,z,u) and let XA4(x1,...,x4) be the embedding corresponding to the weighted blow-up Y1X. One can always assume that x4=x4=u since vE(u)=1. We write xj=xj+qj. Since YX and Y1X correspond to different embeddings, there exists j<4 such that qj0 and vF(xj)>vF(xj)=vF(q). Since ˉZX can be viewed as a sequence of weighted blow-ups with respect to the embedding XA4(x,y,z,u), we know that the defining equation of ˉZ is of the form xj+qj+ˉh such that vF(xj)=vF(ˉh). Hence, there is exactly one j such that qj0, and the defining equation of X is of the form ξ(xj+qj)+h. One can see that either xj=z, or xj=y and qj=p.

    Now, if xj=z, then x1=x1=x and x2=x2=y. One can see that vF(x2)=vF(y)3. So, Y1X is of type E1–E5. In those cases, vF(xj)2, so vF(xj)=vF(qj)=1 and vF(xj)=2. Hence, Y1X is of type E3–E5 and vF(y)=vF(x2)2. Also, since vF(qj)=1, qj=λu for some λC. Therefore vF(z)=vF(xjqj)=1. But, then

    vF(z)vE(z)=12vF(y)vE(y).

    By Lemma 2.16, CenterYF can not be the origin of Uy. This leads to a contradiction.

    Finally, we assume that xj=y and qj=p. Notice that p=λ1zu+λ2u3, hence vF(p)2. If vF(z)=vF(x3)=1, then Y1X is of type E1 or E2, and so vF(xj)=2. However, we know that vF(p)2. This contradicts the assumption that vF(xj)>vF(qj)=vF(p). Hence, vF(z)2 and so vF(p)3. Since

    vF(y)=vF(xj)=vF(qj)=vF(p)3

    and vF(y)3 by the previous discussion, we know that vF(y)=3. Recall that we write

    Uy=(x2+y(y+p(z,u))+g(y,z,u)=0)A4(x,y,z,u)/14(1,3,2,1).

    Since vF(y)=3, vF(E)=vF(y)=34. This means that a(F,Y)=14, so F corresponds to a w-morphism over Uy. Nevertheless, as we mentioned before, w-morphisms over Uy can be obtained by a weighted blow-up with respect to the above embedding, hence Y1X and YX correspond to the same four-dimensional embedding, leading to a contradiction.

    Lemma 4.12. Assume that both YX and Y1X are of type E1–E13. If P is the origin of UxY, then YX is of type E2, E5 or E9, and Y1X has the same type. One has that YXY1XY.

    Proof. This assumption implies that the origin of Ux is contained in Y, so YX is of type E2, E5 or E9 and P is a cyclic quotient point. If YX is a weight blow-up with the weight (b,c,d,1), then b=c+d and

    Ux=(x+p(z,u)+g(x,y,z,u)=0)A4(x,y,z,u)/1b(1,c,d,1).

    Since a(F,Y)<1, F is the valuation described by [4, Proposition 3.1]. Hence, vF(y,z,u)=1b(c,d,a) with c+d=b and ab=a(F,Y). Since a(F,X)=a(F,Y)+vF(x)=1, we know that vF(x)=1ab. One can compute that

    vF(x,y,z,u)=(ba,c(acc)b,daddb,1).

    Since c<b and acc(mod b), we know that accb0, so vF(y)c=vE(y). Likewise, we know that vF(z)d=vE(z). One also has that vF(x)<vE(x) and vF(u)=vE(u).

    On the other hand, Corollary 2.17 says that there exists σOX such that vF(σ)>vE(σ). This can only happen when

    vF(x)=vF(p(z,u))<vF(x+p(z,u))=vF(g(x,y,z,u))

    and in this case one can choose σ=x+p(z,u)OX. Now, Y1X can be obtained by a weighted blow-up with respect to the embedding

    X(σ2σp(z,u)+y3+g(y,z,u)=0)A4(σ,y,z,u)

    and with the weight w1(σ,y,z,u)=(b1,c1,d1,1) where c1=c(acc)b and d1=daddb. Since c1c, d1d and b1=vF(v)>vE(v), by Lemma 2.16 we know that CenterY1E is the origin of U1,σ.

    Now, if we interchange Y and Y1, then the above argument yields that cc1 and dd1. Hence, c=c1 and d=d1 and so YX and Y1X are of the same type. One has that a=1 and c=c, d=d. Thus, F is the exceptional divisor of the w-morphism over P. Now we know that YXY1 by Lemma 3.13 and also Y1XY by the symmetry.

    Lemma 4.13. Assume that both YX and Y1X are of type E1–E13. Then P is not the origin of UyY.

    Proof. By Lemma 4.11, we know that YX is not of type E1 or E6, hence YX is of type E4, E8 or E11 and the origin of Uy is a cyclic quotient point. We assume that YX is a weighted blow-up with the weight (b,c,d,1). Following the same computation as in the proof of Lemma 4.12, we may write Y1X as a weighted blow-up with respect to the embedding

    X(x2+(σp(z,u))2σ+g(σ,z,u)=0)A4(x,σ,z,u)

    and with the weight (b1,c1,d1,1), such that CenterY1E is the origin of U1,σY1. Nevertheless, in this case one always has that b1<b since b>c. The symmetry between Y and Y1 yields that b>b1>b, which is impossible.

    Lemma 4.14. Assume that both YX and Y1X are of type E1–E13. Let XA4(x1,...,x4) be the embedding that corresponds to YX in Table 8. Then P is the origin of Ui for some i4.

    Proof. Assume that P is not the origin of Ui for all i4. Then Y has a non-Gorenstein point on UiUj for some ij. In this case, YX is of type E7 or E10–E13. For simplicity we assume that i=1 and j=2. If YX is a weighted blow-up with the weight (a1,...,a4), then we have the following observation:

    (1) a1=dk1 and a2=dk2 for some integers k1, k2 and d. We may assume that k2=2 and k1 is odd.

    (2) xk21 and xk12 appear in f where f is the defining equation of X. Moreover vE(xk21)=vE(xk12)=vE(f).

    (3) P is a cyclic quotient point of index d. On U1, the local coordinate system is given by (x1,x3,x4), where xl is the strict transform of xl on U1.

    Since F is a valuation of discrepancy less than 1 over P, we know that vF(x1,x3,x4)=1d(a1,a3,a4) with al<d for l=1, 3, and 4. One can compute that

    vF(x1,...,x4)=(k1a1,k2a1,1d(a3+a3a1),1d(a4+a4a1)),

    and Y1X can be obtained by the weighted blow-up with respect to the same embedding XA4(x1,...,x4) and with the weight vF. Nevertheless, one can easily see that vF(xl)vE(xl) for all 1l4. This contradicts Corollary 2.17.

    Proposition 4.15. Assume that both YX and Y1X are both of type E1–E13. Then YX is of type E2, E5 or E9, and Y1X has the same type. One has that YXY1XY.

    Proof. Let

    X(f(x,y,z,u)=0)A4(x,y,z,u)

    be the embedding corresponding to YX, and

    X(f1(x1,y1,z1,u1)=0)A4(x1,y1,z1,u1)

    be the embedding corresponding to Y1X. If CenterYF is the origin of UxY or CenterY1E is the origin of Ux1Y1, then the statement follows from Lemma 4.12. We do not consider these cases here. Then, Lemmas 16 and 17 imply that CenterYF=UzY and CenterY1E=Uz1Y1.

    We may assume that vE(f)vF(f1). Since vE(u)=vF(u1)=1, one can always assume that u=u1 and (u=0) defines a Du Val section. Lemma 2.16 implies that vE(z)>vE(z1) and vF(z)<vF(z1), hence zz1. We may write z1=z+h. If vE(h)vE(z), then we may replace z by z+h, which will lead to a contradiction. Hence, vE(h)<vE(z). Thus, h=λuk for some k<vE(z). Since (u=0) defines a Du Val section, we know that z4, yz3 or z5f. It follows that u4k, yu3k or u5k appear in either f or f1. This means that vE(f)4k, vE(y)+3k or 5k for some k<vE(z). One can easily check that for all the cases in Table 8 this inequality never holds. Thus, we get a contradiction.

    Proposition 4.16. Assume that YX is of type E14–E18. Then there exists Y1X which is of type E3 or E6 such that YXY1. Moreover, if YX is of type E15 or E17, then YXY1.

    Proof. Assume that YX is of type E14. Consider the chart

    Ut=(x2+y3+z+g(y,z,u,t)=p(x,y,z,u)+t=0)A5(x,y,z,u,t)/15(3,2,1,1,4).

    We choose (y1,...,y5)=(x,y,u,z,t) with δ4=1 and δ5=2 or 4, or δ4=2 and δ5=1 or 4. Then (Ξ) holds. Now, let F be the exceptional divisor that corresponds to the w-morphism over the origin of Ut. Then

    vF(y1,...,y5)=15(3,2,1,6,4).

    One can see that (Θ4) holds. Thus, there exists a divisorial contraction Y1X so that YXY1 which extracts F. One can compute that vF(x,y,z,u)=(3,2,2,1), so Y1X is of type E3.

    Assume that YX is of type E15. Consider the chart

    Ux=(x2+p(z,u)+y3+g(x,y,z,u)=0)A4(x,y,z,u)/14(3,2,1,1).

    We choose (y1,...,y4)=(x,z,u,y) with δ4=4. Then (Ξ) holds. Now the origin of Ux is a cAx/4 point. After a suitable change of coordinates, we may assume that u2p(z,u). Then the w-morphism over this point can be given by a weighted blow-up with the weight vF(y1,...,y4)=14(3,5,1,2). One can see that (Θ2) and (Ξ) hold. Hence, there exists a divisorial contraction Y1X such that YXY1. One also has vF(x,y,z,u)=(3,2,2,1), so Y1X is of type E3.

    Assume that YX is of type E16. Consider the chart

    Ut=(x2+y3+p(z,u)+g(y,z,u,t)=q(y,z,u)+t=0)A5(x,y,z,u,t)/14(3,2,1,1,3).

    We choose (y1,...,y5)=(y,z,u,x,t) with δ4=4 and δ5=1 or 2. Then (Ξ) holds. Now the origin of Ut is a cAx/4 point. After a suitable change of coordinates, we may assume that u2p(z,u). Then the weight vF(y1,...,y5)=14(2,5,1,3,3) defines a w-morphism over Ut. One can see that (Θ2) holds. Hence, there exists a divisorial contraction Y1X such that YXY1. One can compute that vF(x,y,z,u)=(3,2,2,1), so again Y1X is of type E3.

    Assume that YX is of type E17. Consider the chart

    Uy=(x2+y3+z3+g(y,z,u)=0)A4(x,y,z,u)/13(0,2,1,1).

    We can choose (y1,...,y4)=(y,z,u,x) with δ4=4. Then (Ξ) holds. The origin of Uy is a cD/3 point. Notice that y2u2g(y,z,u), so the w-morphism over Uy is given by the weighted blow-up with the weight vF(y1,...,y4)=13(2,4,1,3). One can see that (Θ2) and (Ξ) hold. One has that vF(x,y,z,u)=(3,2,1,1), so there exists Y1X which is of type E3 such that YXY1.

    Finally, assume that YX is of type E18. Consider the chart

    Ut=(x2+y+g(y,z,u,t)=y2+p(y,z,u)+t=0)A5(x,y,z,u,t)/17(5,3,2,1,6).

    We choose (y1,...,y5)=(y,z,u,x,t) with δ4=4 and δ5=1. Then (Ξ) holds. The origin of Ut is a cyclic quotient point. Let F be the exceptional divisor corresponding to the w-morphism over this point. Then

    vF(y1,...,y5)=17(10,2,1,5,6)

    (notice that the irreducibility of y2+p(y,z,u) implies that p(0,z,u)0, so vF(t)=67). One can see that (Θ1) holds. Thus, there exists a divisorial contraction Y1X which extracts F so that YXY1. One can compute that vF(x,y,z,u)=(5,4,2,1), and so Y1X is of type E6.

    Now we study divisorial contractions over cE points with discrepancy greater than one. Those divisorial contractions are given in Table 10.

    Proposition 4.17. Assume that YX is a divisorial contraction with discrepancy a>1. Then there exists a w-morphism Y1X such that YXY1.

    Proof. Assume first that YX is of type E19. The chart UyY is defined by

    (x2+(y+p(z,u))3+u3+g(y,z,u)=0)A4(x,y,z,u)/13(0,1,1,2).

    One can choose (y1,...,y4)=(x,y+p,z,u) with δ4=1. Then (Ξ) holds. The origin of Uy is a cD/3 point. The w-morphism over this point is given by the weighted blow-up with the weight

    w(y1,...,y4)=(3,4,1,2) or (6,4,1,5).

    One can see that (Θ4) holds. Thus, there exists Y1X such that YXY1. A direct computation shows that Y1X is a w-morphism.

    Assume that YX is of type E20. The chart UtY is defined by

    (x2+y+g(y,z,u,t)=y2+p(z,u)+t=0)A5(x,y,z,u,t)/17(5,3,2,2,6).

    We take (y1,...,y5)=(y,z,u,x,t) with δ4=4 and δ5=1. Then (Ξ) holds. The w-morphism over the origin of Ut is given by weighted blowing-up the weight w(y1,...,y5)=17(5,1,1,6,3). One can see that (Θ1) holds. Hence, there exists a divisorial contraction Y1X such that YXY1. One can compute that Y1X is a w-morphism.

    Finally assume that YX is of type E21. The chart UyY is defined by

    (x2+y+u7+g(y,z,u)=0)A4(x,y,z,u)/15(2,4,3,2).

    One can choose (y1,...,y4)=(y,z,u,x). Then (Ξ) holds. The w-morphism over Uy is given by the weighted blow-up with the weight w(y1,...,y4)=15(2,4,1,1). One can see that (Θ2) holds. Thus, there exists a divisorial contraction Y1X such that YXY1. One can compute that Y1X is a w-morphism.

    Finally, we need to study divisorial contractions over cE/2 points. All such divisorial contractions are listed in Table 11.

    Proposition 4.18. Let YX be a divisorial contraction.

    (1) Assume that there are two w-morphisms over X. Then:

    (1–1) If YX is of type E22 and there exists another w-morphism Y1X, then Y1X is of type E22 or E23 and YXY1.

    (1–2) If YX is of type E23, then there are exactly two w-morphisms. The other one, Y1X, is of type E22. Interchanging Y and Y1, we are back to Case (1–1).

    (1–3) If YX is of type E24, then there are exactly two w-morphisms. They are both of type E24 and are negatively linked to each other.

    (1–4) YX is not of type E25.

    (2) Assume that YX is of type E26. Then there is a w-morphism Y1X which is of type E22 such that YXY1.

    Proof. The statement about the number of w-morphisms follows from [9, Section 10]. First, assume that there exists two w-morphisms over X and YX is of type E22. Let F=exc(Y1X). The only non-Gorenstein point on Y is the origin of Uz, which is a cD/3 point defined by

    (x2+y3+g(y,z,u)=0)A4(x,y,z,u)/13(0,2,1,1).

    One can see that

    12=a(F,X)=a(F,Y)+12vF(z),

    hence a(F,Y)=vF(z)=13. Thus, F corresponds to a w-morphism over Uz. From Table 6 we know that

    vF(x,y,z,u+λz)=13(b,c,1,4)

    for some λC, where (b,c)=(3,2) or (6,5). Now one can choose (y1,...,y4)=(y,u+λz,u+ξz,x) with \delta_4 = 4 , where \xi\in {\mathbb C} is a number so that u+\xi z defines a Du Val section on X and \xi\neq \lambda . Thus, ( \Xi' ) and ( \Theta_2 ) hold and Y\underset{{X}}{\Rightarrow }Y_1 .

    Now assume that Y\rightarrow X is of type E24. The chart U_x\subset Y is defined by

    (x'+p(y', z', u')+{y'}^3+g'(x', y', z', u') = 0)\subset {\mathbb A}^4_{(x', y', z', u')}/\frac{1}{7}(5, 4, 3, 1).

    The origin is a cyclic quotient point and there is only one w -morphism over this point. Let F be the exceptional divisor corresponding to this w -morphism. By Lemma 3.13 we know that there exists a divisorial contraction Y_1\rightarrow X which extracts F such that Y\stackrel{-}{\underset{{X}}{\Rightarrow }}Y_1 . One can compute that a(F, X) = \frac{1}{2} . Hence, Y_1\rightarrow X is also a w -morphism.

    Finally, assume that Y\rightarrow X is of type E26. Consider the chart U_y\subset Y which is defined by

    ({x'}^2+y'+{z'}^4+{u'}^8+g'(y', z', u') = 0)\subset {\mathbb A}^4_{(x', y', z', u')}/\frac{1}{6}(1, 2, 5, 1).

    One can take (y_1, ..., y_4) = (y', z', u', x') with \delta_4 = 4 . One can see that ( \Xi' ) holds. Now, let F be the exceptional divisor corresponding to the w -morphism over U_t . Then v_F(y_1, ..., y_4) = \frac{1}{6}(2, 5, 1, 1) . Hence, ( \Theta_2 ) and ( \Xi'_- ) hold. Thus, there exists a divisorial contraction Y_1\rightarrow X which extracts F so that Y\stackrel{-}{\underset{{X}}{\Rightarrow }}Y_1 . One can compute that v_F(x, y, z, u) = \frac{1}{2}(3, 2, 3, 1) , so Y_1\rightarrow X is of type E22.

    We want to understand the change of singularities after running the minimal model program. The final result is the following proposition.

    Proposition 5.1.

    (1) Assume that Y\rightarrow X is a divisorial contraction between terminal and {\mathbb Q} -factorial threefolds.

    (1–1) If Y\rightarrow X is a divisorial contraction to a point, then

    gdep(X)\leq gdep(Y)+1\;\mbox{and}\;dep(X)\leq dep(Y)+1.

    If Y\rightarrow X is a divisorial contraction to a curve, then

    gdep(X)\leq gdep(Y)\;\mbox{and}\;dep(X)\leq dep(Y).

    (1-2) dep_{Gor}(X)\geq dep_{Gor}(Y) and the inequality is strict if the non-isomorphic locus on X contains a Gorenstein singular point.

    (2) Assume that X\dashrightarrow X' is a flip between terminal and {\mathbb Q} -factorial threefolds.

    (2–1)

    gdep(X) > gdep(X')\;\mbox{and}\;dep(X) > dep(X').

    (2–2) dep_{Gor}(X)\leq dep_{Gor}(X') and the inequality is strict if the non-isomorphic locus on X' contains a Gorenstein singular point.

    Corollary 5.2. Assume that

    X_0\dashrightarrow X_1\dashrightarrow...\dashrightarrow X_k

    is a process of the minimal model program. Then:

    (1) \rho(X_0/X_k)\geq gdep(X_k)-gdep(X_0) and the equality holds if and only if X_i\dashrightarrow X_{i+1} is a strict w -morphism for all i .

    (2) dep_{Gor}(X_k)\geq dep_{Gor}(X_0) .

    In particular, if X is a terminal {\mathbb Q} -factorial threefold and W\rightarrow X is a resolution of singularities, then \rho(W/X)\geq gdep(X) and the equality holds if and only if W is a feasible resolution of X .

    Proof. Statement (2) easily follows from the inequalities in Proposition 5.1. Assume that the sequence contains m flips, then \rho(X_0/X_k) = k-m . On the other hand, we know that

    gdep(X_{i+1})\leq\left\lbrace \begin{array}{ll} gdep(X_i)-1 & \mbox{if }X_i\dashrightarrow X_{i+1}\mbox{ is a flip} \\ gdep(X_i)+1 & \mbox{otherwise}\end{array}\right..

    It follows that gdep(X_k)\leq gdep(X_0)+k-2m , hence gdep(X_k)-gdep(X_0)\leq \rho(X_0/X_k) . Now gdep(X_k)-gdep(X_0) = \rho(X_0/X_k) if and only if m = 0 and gdep(X_{i+1}) = gdep(X_i)+1 for all i , which is equivalent to X_i\dashrightarrow X_{i+1} being a strict w -morphism for all i .

    Now assume that X is a terminal {\mathbb Q} -factorial threefold and W\rightarrow X is a resolution of singularities. We can run K_W -MMP over X and the minimal model is X itself. Since gdep(W) = 0 , one has that \rho(W/X)\geq gdep(X) and the equality holds if and only if W is a feasible resolution of X .

    The inequalities for the depth part are exactly Lemma 2.20. We only need to prove the inequalities for the generalized depth and the Gorenstein depth.

    Convention 5.3. Let \mathcal{S} be a set consisting of birational maps between {\mathbb Q} -factorial terminal threefolds. We say that ({\Large{ \ast }})_{ \mathcal{S}} holds if, for all Z\dashrightarrow V inside \mathcal{S} , one has that:

    (1) If Z\rightarrow V is a divisorial contraction to a point, then

    dep_{Gor}(V)\geq dep_{Gor}(Z)\geq dep_{Gor}(V)-(dep(Z)-dep(V)+1).

    (2) If Z\rightarrow V is a divisorial contraction to a smooth curve, then

    dep_{Gor}(V)\geq dep_{Gor}(Z)\geq dep_{Gor}(V)-(dep(Z)-dep(V)).

    (3) If Z\dashrightarrow V is a flip, then

    dep_{Gor}(V)\geq dep_{Gor}(Z)\geq dep_{Gor}(V)-(dep(Z)-dep(V)-1).

    (4) If Z\dashrightarrow V is a flop, then dep_{Gor}(V) = dep_{Gor}(Z) .

    Moreover, if there exists a Gorenstein singular point P\in V such that P is not contained in the isomorphic locus of Z\dashrightarrow V , then dep_{Gor}(V) > dep_{Gor}(Z) unless V\dashrightarrow Z is a flop.

    We say that ({\Large{ \ast }})^{(1)}_{ \mathcal{S}} holds if statement (1) is true, but statements (2) and (3) are unknown.

    If V\dashrightarrow Z is a flip or a divisorial contraction, we denote the condition ({\Large{ \ast }})_{V\dashrightarrow Z} = ({\Large{ \ast }})_{ \mathcal{S}} where \mathcal{S} is the set containing only one element V\dashrightarrow Z .

    It is easy to see that if Y\rightarrow X is a divisorial contraction, then ({\Large{ \ast }})_{Y\rightarrow X} holds if and only if the inequalities in Proposition 5.1 (1) hold. Likewise, if X\dashrightarrow X' is a flip, then ({\Large{ \ast }})_{X\dashrightarrow X'} holds if and only if the inequalities in Proposition 5.1 (2) hold.

    Remark 5.4. If Z\dashrightarrow V is a flop, then the singularities on Z and V are the same by [17, Theorem 2.18]. Hence, statement (4) is always true.

    Convention 5.5. Given n\in {\mathbb Z}_{\geq0} , we denote

    \mathcal{S}_n=\left\{\phi: Z \rightarrow V \left\lvert\, \begin{array}{c}\phi \text { is a flip, a flop or a divisorial contraction between } \\ \text { between } \mathbb{Q} \text {-factorial terminal threefolds, }gdep (Z) \leq n\end{array}\right.\right\}.

    Lemma 5.6. Assume that Y\rightarrow X and Y_1\rightarrow X are two divisorial contractions between terminal threefolds, such that Y\underset{{X}}{\Rightarrow }Y_1 . If ({\Large{ \ast }})_{ \mathcal{S}_{gdep(Y)-1}} holds, then gdep(Y_1)\leq gdep(Y) . Moreover, gdep(Y_1) = gdep(Y) if and only if Y_1\underset{{X}}{\Rightarrow }Y .

    Proof. We have a diagram

    such that Z_1\rightarrow Y is a strict w -morphism and Z_i\dashrightarrow Z_{i+1} is a flip or a flop for all 1\leq i\leq k-1 . Since gdep(Z_1) = gdep(Y)-1 and ({\Large{ \ast }})_{ \mathcal{S}_{gdep(Y)-1}} holds, we know that gdep(Z_2)\leq gdep(Y)-1 . Repeating this argument k-2 times, one can say that gdep(Z_k)\leq gdep(Y)-1 . Again, since ({\Large{ \ast }})_{ \mathcal{S}_{gdep(Y)-1}} holds, we know that gdep(Y_1)\leq gdep(Z_k)+1 = gdep(Y) .

    Now, gdep(Y_1) = gdep(Y) if and only if all the inequalities above are equalities. This is equivalent to Z_k\rightarrow Y_1 being a strict w -morphism and Z_i\dashrightarrow Z_{i+1} being a flop for all i = 1 , ..., k-1 , or k = 1 . In other words, we also have Y_1\underset{{X}}{\Rightarrow }Y .

    Corollary 5.7. Assume that Y\rightarrow X is a strict w -morphism over P\in X and Y_1\rightarrow X is another divisorial contraction over P . If ({\Large{ \ast }})_{ \mathcal{S}_{gdep(Y)}} holds, then there exist divisorial contractions Y_1\rightarrow X , ..., Y_k\rightarrow X such that

    Y_1\underset{{X}}{\Rightarrow }...\underset{{X}}{\Rightarrow }Y_k\underset{{X}}{\Rightarrow }Y.

    Proof. By Proposition 4.1, we know that there exists Y_1 , ..., Y_l , Y'_1 , ..., Y'_{l'} such that

    Y_1\underset{{X}}{\Rightarrow }...\underset{{X}}{\Rightarrow }Y_l = Y'_{l'}\underset{{X}}{\Leftarrow}...\underset{{X}}{\Leftarrow}Y'_1\underset{{X}}{\Leftarrow}Y.

    One can apply Lemma 5.6 to the sequence Y\underset{{X}}{\Rightarrow }Y'_1\underset{{X}}{\Rightarrow }...\underset{{X}}{\Rightarrow }Y'_{l'} and conclude that gdep(Y'_i)\leq gdep(Y) for all i . Since Y'_i\rightarrow X\in \mathcal{S}_{gdep(Y)} for all i = 1 , ..., l' , one has gdep(Y'_i)\geq gdep(X)-1 = gdep(Y) . Thus, gdep(Y'_i) = gdep(Y) and Lemma 5.6 says that one has

    Y'_{l'}\underset{{X}}{\Rightarrow }...\underset{{X}}{\Rightarrow }Y'_1\underset{{X}}{\Rightarrow }Y.

    Now we can take k = l+l'-1 and let Y_i = Y'_{l'-i+l} for l < i\leq k .

    Corollary 5.8. Assume that ({\Large{ \ast }})_{ \mathcal{S}_{n-2}} holds. Assume that P\in X is a cA/r point or a cAx/r point such that gdep(X) = n . Then every w -morphism over P is a strict w -morphism. In particular, one can always assume that the morphism Y_1\rightarrow X in Theorem 2.18 is a strict w -morphism.

    Proof. Propositions 4.2 and 4.3 say that if Y\rightarrow X and Y_1\rightarrow X are two different w -morphisms over P , then there exists Y_2\rightarrow X , ..., Y_k\rightarrow X such that

    Y_1\underset{X}{\Leftrightarrow}Y_2\underset{X}{\Leftrightarrow}...\underset{X}{\Leftrightarrow}Y_k = Y.

    We can assume that Y\rightarrow X is a strict w -morphism, so gdep(Y) = n-1 . Lemma 5.6 implies that Y_1 is also a strict w -morphism. Hence, every w -morphism over P is a strict w -morphism.

    Now assume that X is in the diagram in Theorem 2.18 and P is a non-Gorenstein point in the exceptional set of X\rightarrow W . If P is a cA/r or a cAx/r point, then we already know that every w -morphism over P is a strict w -morphism. Otherwise, by Remark 2.19 (2) we know that any w -morphism Y_1\rightarrow X over P induces a diagram in Theorem 2.18. Hence, we can choose Y_1\rightarrow X to be a strict w -morphism.

    Convention 5.9. Let DV be the set of symbols

    DV = \{A_i, D_j, E_k\}_{i\in {\mathbb N}, j\in {\mathbb N}_{\geq 4}, k = 6, 7, 8}.

    One can define an ordering on DV by

    A_i < A_{i'} < D_j < D_{j'} < E_k < E_{k'}\;{\mbox{for all}}\;i < i', j < j', k < k'.

    Given \square\in DV , define

    \mathcal{T}_{\square}=\left\{\begin{array}{c|c}X \text { is a terminal } & G E(P \in X) \leq \square \text { for all } \\ \mathbb{Q} \text {-factorial threefold } & \text { non-Gorenstein point } P \in X\end{array}\right\},
    \mathcal{T}_{\square, n}=\left\{X \in \mathcal{T}_{\square} \mid gdep(X) \leq n\right\}

    and

    \mathcal{T}_n = \bigcup\limits_{\square\in DV} \mathcal{T}_{\square, n}.

    Here, GE(P\in X) denotes the type of the general elephant near P . That is, the type of a general Du Val section H\in|-K_X| near an analytic neighborhood of P\in X .

    Convention 5.10. Let \mathcal{T} be a set of terminal threefolds. We say that the condition (\Pi)_{ \mathcal{T}} holds if for all X\in \mathcal{T} and for all strict w -morphisms Y\rightarrow X over non-Gorenstein points of X , one has that dep(Y) = dep(X)-1 .

    Remark 5.11.

    (1) Assume that Y\rightarrow X is a w -morphism over a non-Gorenstein point P . Then the general elephant of Y over X is better than the general elephant of X near P . This is because if H\in|-K_X| near P , then H_Y\in|-K_Y| and H_Y\rightarrow H is a partial resolution by [3, Lemma 2.7].

    (2) One has that (\Pi)_{ \mathcal{T}_{A_1}} always holds since if GE(P\in X) = A_1 for some non-Gorenstein point P , then P is a cyclic quotient point of index two (cf. [7, (6.4)]). In this case, there is only one w -morphism Y\rightarrow X over P and Y is smooth over X . Hence,

    gdep(P\in X) = dep(P\in X) = 1

    and gdep(Y) = dep(Y) = 0 over X . Thus, Y\rightarrow X is a strict w -morphism and also dep(Y) = dep(X)-1 .

    Lemma 5.12. Assume that Y\rightarrow X is a strict w -morphism over a non-Gorenstein point P . If ({\Large{ \ast }})_{ \mathcal{S}_{gdep(Y)}} holds and (\Pi)_{ \mathcal{T}_{\square}} holds for all \square < GE(P\in X) , then dep(Y) = dep(X)-1 .

    Proof. By the definition we know that dep(Y)\geq dep(X)-1 . Assume that dep(Y) > dep(X)-1 . Then there exists Y_1\rightarrow X such that dep(Y_1) = dep(X)-1 < dep(Y) . Corollary 5.7 says that there exist divisorial contractions Y_2\rightarrow X , ..., Y_k\rightarrow X such that Y_1\underset{{X}}{\Rightarrow }Y_2\underset{{X}}{\Rightarrow }...\underset{{X}}{\Rightarrow }Y_k\underset{{X}}{\Rightarrow }Y . From Remark 5.11 (1) we know that Y_i\in \mathcal{T}_{\square} for some \square < GE(P\in X) for all i , hence, if

    is the induced diagram of Y_i\underset{{X}}{\Rightarrow }Y_{i+1} , then dep(Z_{i, 1}) = dep(Y_i)-1 . By Lemma 2.20 we know that

    dep(Y_{i+1})\leq dep(Z_{i, k_i})+1\leq dep(Z_{i, 1})+1 = dep(Y_i)

    for all i . Hence, dep(Y_1)\leq dep(Y) . This leads to a contradiction as we assume that dep(Y) > dep(Y_1) .

    Lemma 5.13. Fix an integer n and assume that ({\Large{ \ast }})_{ \mathcal{S}_{n-1}} holds. Then (\Pi)_{ \mathcal{T}_n} holds.

    Proof. We need to show that for all X\in \mathcal{T}_n and for all strict w -morphism Y\rightarrow X over a non-Gorenstein point, one has that dep(Y) = dep(X)-1 . By Remark 5.11 (2) we know that (\Pi)_{ \mathcal{T}_{A_1}} holds, hence (\Pi)_{ \mathcal{T}_{A_m, n}} and (\Pi)_{ \mathcal{T}_{D_4, n}} hold for all m\in {\mathbb N} by Lemma 5.12 and by induction on m . Then, one can prove that (\Pi)_{ \mathcal{T}_{D_m, n}} and (\Pi)_{ \mathcal{T}_{E_6, n}} hold by again applying Lemma 5.12 and by induction on m . Statements (\Pi)_{ \mathcal{T}_{E_7}, n} and (\Pi)_{ \mathcal{T}_{E_8}, n} can be proved in the same way.

    Lemma 5.14. Fix an integer n and assume that ({\Large{ \ast }})_{ \mathcal{S}_{n-1}} holds. Assume that we have a diagram

    such that gdep(X) = n , Y_1\rightarrow X is a strict w -morphism, Y_i\dashrightarrow Y_{i+1} is a flip or a flop for i = 1 , ..., k-1 , and Y_k\rightarrow X' is a divisorial contraction. Then,

    (1) dep_{Gor}(X) = dep_{Gor}(Y_1)\leq dep_{Gor}(X') and gdep(X')\leq gdep(X) .

    (2) ({\Large{ \ast }})_{Y_i\dashrightarrow Y_{i+1}} holds for i = 1 , ..., k-1 .

    (3) ({\Large{ \ast }})_{Y_k\rightarrow X'} holds.

    Moreover, if the non-isomorphic locus of X\dashrightarrow X' on X' contains a Gorenstein singular point, then dep_{Gor}(X) < dep_{Gor}(X') .

    Proof. Since ({\Large{ \ast }})_{ \mathcal{S}_{n-1}} holds, we know that (\Pi)_{ \mathcal{T}_n} holds by Lemma 5.13. Hence, dep_{Gor}(Y_1) = dep_{Gor}(X) . We know that gdep(Y) = n-1 . Since ({\Large{ \ast }})_{ \mathcal{S}_{n-1}} holds, one can prove that gdep(Y_i)\leq gdep(Y_1)\leq n-1 for all i and hence ({\Large{ \ast }})_{Y_i\dashrightarrow Y_{i+1}} and ({\Large{ \ast }})_{Y_k\rightarrow X'} hold. Thus,

    gdep(X')\leq gdep(Y_k)-1\leq gdep(Y_1)-1\leq gdep(X)

    and

    dep_{Gor}(X) = dep_{Gor}(Y_1)\leq...\leq dep_{Gor}(Y_k)\leq dep_{Gor}(X').

    Now, if the non-isomorphic locus of X\dashrightarrow X' on X' contains a Gorenstein singular point, then either the non-isomorphic locus of Y_i\dashrightarrow Y_{i+1} on Y_{i+1} contains a Gorenstein singular point or the non-isomorphic locus of Y_k\rightarrow X' on X' contains a Gorenstein singular point. Hence, at least one of the above inequalities is strict. Thus, one has dep_{Gor}(X) < dep_{Gor}(X') .

    Lemma 5.15. Assume that ({\Large{ \ast }})_{ \mathcal{S}_{n-1}} holds. Then ({\Large{ \ast }})_{ \mathcal{S}_n}^{(1)} holds.

    Proof. Let Y\rightarrow X be a divisorial contraction to a point which belongs to \mathcal{S}_n . We know that gdep(Y) = n . Assume first that Y\rightarrow X is a strict w -morphism over a point P\in X . Notice that dep(Y)\geq dep(X)-1 by Lemma 2.20. If P is a non-Gorenstein point, then dep_{Gor}(Y) = dep_{Gor}(X) by Lemma 5.13, hence ({\Large{ \ast }})_{Y\rightarrow X} holds. If P is a Gorenstein point, then

    dep_{Gor}(X) = gdep(X)-dep(X) = gdep(Y)+1-dep(X) = dep_{Gor}(Y)+dep(Y)-dep(X)+1.

    Moreover, since dep(P\in X) = 0 , dep(Y)-dep(X)\geq 0 , hence

    dep_{Gor}(X) > dep_{Gor}(Y) = dep_{Gor}(X)-(dep(Y)-dep(X)+1).

    Thus, ({\Large{ \ast }})_{Y\rightarrow X} holds.

    In general, by Corollary 5.7 there exist Y_1\rightarrow X , ..., Y_k\rightarrow X such that Y_k\rightarrow X is a strict w -morphism and one has Y\underset{{X}}{\Rightarrow }Y_1\underset{{X}}{\Rightarrow }...\underset{{X}}{\Rightarrow }Y_k . By induction on k we may assume that ({\Large{ \ast }})_{Y_i\rightarrow X} holds for all i (notice that gdep(Y_i)\leq gdep(Y) = n for all i by Lemma 5.6). Now we have a diagram

    By Lemma 5.14 we know that dep_{Gor}(Y)\leq dep_{Gor}(Y_1) and ({\Large{ \ast }})_{Z_i\dashrightarrow Z_{i+1}} , ({\Large{ \ast }})_{Z_k\rightarrow Y_1} hold. Since ({\Large{ \ast }})_{Y_1\rightarrow X} holds, we know that dep_{Gor}(X)\geq dep_{Gor}(Y_1)\geq dep_{Gor}(Y) and

    \begin{align*} dep_{Gor}(Y)+(dep(Y)-dep(X)+1)& = dep_{Gor}(Z_1)+(dep(Z_1)-dep(X)+2)\\ &\geq gdep(Z_1)-dep(X)+2\\ &\geq gdep(Z_k)-dep(X)+2\\ &\geq gdep(Y_1)-dep(X)+1\\ &\geq dep_{Gor}(Y_1)+(dep(Y_1)-dep(X)+1)\\ &\geq dep_{Gor}(X). \end{align*}

    Moreover, if Y\rightarrow X is a divisorial contraction to a Gorenstein point, then Y_1\rightarrow X is also a divisorial contraction to a Gorenstein point. Hence, dep_{Gor}(Y_1) < dep_{Gor}(X) and we also have dep_{Gor}(Y) < dep_{Gor}(X) .

    Proof of Proposition 5.1. We need to say that ({\Large{ \ast }})_{ \mathcal{S}_n} holds for all n and we will prove this by induction on n . If n = 0 , then \mathcal{S}_0 consists only smooth blow-downs and smooth flops. One can see that ({\Large{ \ast }})_{ \mathcal{S}_0} holds. In general, assume that ({\Large{ \ast }})_{ \mathcal{S}_{n-1}} holds. By Lemma 5.15 we know that ({\Large{ \ast }})_{ \mathcal{S}_n}^{(1)} holds. Hence, it is enough to show that, given a flip X\dashrightarrow X' or a divisorial contraction to a curve X\rightarrow V such that gdep(X) = n , ({\Large{ \ast }})_{X\dashrightarrow X'} or ({\Large{ \ast }})_{X\rightarrow V} holds.

    If X\rightarrow V is a smooth blow-down, then dep_{Gor}(X) = dep_{Gor}(V) , and so there is nothing to prove. In general, we have a diagram as in Theorem 2.18:

    By Lemma 5.14 we know that dep_{Gor}(X)\leq dep_{Gor}(X') and ({\Large{ \ast }})_{Y_i\dashrightarrow Y_{i+1}} , ({\Large{ \ast }})_{Y_k\rightarrow X'} hold. One has that

    \begin{align*} dep_{Gor}(X)+(dep(X)-dep(X'))& = gdep(X)-dep(X')\\ & = gdep(Y_1)-dep(X')+1\\ &\geq gdep(Y_k)-dep(X')+1\\ &\geq gdep(X')-dep(X') = dep_{Gor}(X'). \end{align*}

    Moreover, if X\dashrightarrow X' is a flip, then either one of Y_i\dashrightarrow Y_{i+1} is a flip or Y_k\rightarrow X' is a divisorial contraction to a curve by [3, Remark 3.4]. This implies that either gdep(Y_1) > gdep(Y_k) or gdep(Y_k)\geq gdep(X') . If X\rightarrow V is a divisorial contraction to a curve, then Y_k\rightarrow X' is a divisorial contraction to a curve, hence one always has that gdep(Y_k)\geq gdep(X') . In conclusion, we have

    dep_{Gor}(X)\geq dep_{Gor}(X')-(dep(X)-dep(X')-1).

    If X\dashrightarrow X' is a flip, then one can see that ({\Large{ \ast }})_{X\dashrightarrow X'} holds. Now assume that X\rightarrow V is a divisorial contraction to a curve. Then X'\rightarrow V is a divisorial contraction to a point. We know that ({\Large{ \ast }})_{X'\rightarrow V} holds since ({\Large{ \ast }})_{ \mathcal{S}_n}^{(1)} holds and gdep(X')\leq n by Lemma 5.14. One can see that dep_{Gor}(X)\leq dep_{Gor}(X')\leq dep_{Gor}(V) and

    \begin{align*} dep_{Gor}(X)&\geq dep_{Gor}(X')-(dep(X)-dep(X')-1)\\ &\geq dep_{Gor}(V)-(dep(X')-dep(V)+1)-(dep(X)-dep(X')-1)\\ & = dep_{Gor}(V)-(dep(X)-dep(V)). \end{align*}

    Thus, ({\Large{ \ast }})_{X\rightarrow V} holds.

    In this section, we describe the difference between two different feasible resolutions of a terminal {\mathbb Q} -factorial threefold. The final result is the diagrams in Theorem 1.1.

    Lemma 6.1. Assume that X is a terminal threefold and Y\rightarrow X , Y_1\rightarrow X are two different strict w -morphisms over P\in X such that Y\underset{{X}}{\Rightarrow }Y_1 . Let

    be the corresponding diagram. Then one of the following holds:

    (1) Y\stackrel{-}{\underset{{X}}{\Rightarrow }}Y_1\stackrel{-}{\underset{{X}}{\Rightarrow }}Y and k = 1 .

    (2) k = 2 and Z_1\dashrightarrow Z_2 is a smooth flop.

    (3) k = 2 , Z_1\dashrightarrow Z_2 is a singular flop, P\in X is of type cA/r and both Y\rightarrow X and Y_1\rightarrow X are of type A1 in Table 2.

    Proof. Since Y\rightarrow X and Y_1\rightarrow X are both strict w -morphisms, we know that gdep(Y) = gdep(Y_1) . Hence, Z_i\dashrightarrow Z_{i+1} can not be a flip. Thus, if Y\stackrel{-}{\underset{{X}}{\Rightarrow }}Y_1 , then k = 1 .

    Now assume that Y\not{\stackrel{-}{\underset{{X}}{\Rightarrow }}}Y_1 . According to the result in Section 4, one has that:

    (ⅰ) If P is of type cA/r , then by Proposition 4.2 we know that Y\rightarrow X and Y_1\rightarrow X are both of type A1. Since Y\not{\stackrel{-}{\underset{{X}}{\Rightarrow }}}Y_1 , we know that f^{\circ}_4 = \eta_4 = y is irreducible. Hence, there is only one K_{Z_1} -trivial curve on Z_1 and so k = 2 .

    (ⅱ) P is not of type cAx/r since one always has that Y\stackrel{-}{\underset{{X}}{\Rightarrow }}Y_1 if Y\rightarrow X and Y_1\rightarrow X are two different w -morphisms over P by Proposition 4.3.

    (ⅲ) P is not of type cD by Proposition 4.5.

    (ⅳ) P is not of type cD/3 by Proposition 4.7.

    (ⅴ) If P is of type cD/2 , then by Propositions 4.8 and 4.9 we know that Y\rightarrow X is of type D17 in Table 6.

    We know that Z_1\rightarrow Y is a w -morphism over the origin of

    U_t = ({x'}^2+y'+g'(z', u', t') = z'u'+{y'}^3+t' = 0)\subset {\mathbb A}^5_{(x', y', z', u', t')}/\frac{1}{5}(3, 1, 1, 2, 3)

    and we take (y_1, ..., y_5) = (y', u', y'+z', x', t') . One can see that there are two curves contained in y_3 = y_5 = 0 , namely \Gamma_1 = ({x'}^2+g'(0, u', 0) = y' = y'+z' = u' = 0) and \Gamma_2 = ({x'}^2+g'(-y', -{y'}^2, 0) = u'+{y'}^2 = y'+z' = t' = 0) . We know that ( \Theta_1 ) holds. When computing the intersection number for \Gamma_2 one can take \delta_4 = 4 and \delta_5 = 2 , hence ( \Xi_- ) holds in this case. This implies that the proper transform \Gamma_{2, Z_1} of \Gamma_2 on Z_1 is a K_{Z_1} -trivial curve and is not contained in exc(Z_1\rightarrow Y) . Thus, Z_i\dashrightarrow Z_{i+1} is a flip along \Gamma_{2, Z_i} for some i , and hence gdep(Y_1) < gdep(Y) . This is a contradiction. Thus, this case will not happen.

    (ⅵ) If P is of type cE , then by Proposition 4.15 and Proposition 4.16 we know that Y\rightarrow X is of type E14, E16, or E18 in Table 9. In those cases, Y\rightarrow X are five-dimensional weighted blow-ups. Let \Gamma\subset Y be a curve contained in exc(Y\rightarrow X) such that the proper transform \Gamma_{Z_1} of \Gamma on Z_1 is a possibly K_{Z_1} -trivial curve. From the proof of Proposition 4.16, one can see that:

    (ⅵ–ⅰ) Y\rightarrow X is of type E14. In this case, Z_1\rightarrow Y is a w -morphism over the origin of

    U_t = ({x'}^2+{y'}^3+z'+g'(y', z', u', t') = p(x', y', z', u')+t' = 0)
    \subset {\mathbb A}^5_{(x', y', z', u', t')}/\frac{1}{5}(3, 2, 1, 1, 4).

    We choose (y_1, ..., y_5) = (x', y', u', z', t') with \delta_4 = 1 and \delta_5 = 2 or 4 , or \delta_4 = 2 and \delta_5 = 1 or 4 . We also know that ( \Theta_4 ) holds. If both \delta_4 and \delta_5\neq 4 , then ( \Xi_- ) holds, which implies that Y\stackrel{-}{\underset{{X}}{\Rightarrow }}Y_1 . This contradicts our assumption. Hence, \delta_5 = 4 . Now, \Gamma = ({x'}^2+{y'}^3 = z' = u' = t' = 0) .

    (ⅵ–ⅱ) Y\rightarrow X is of type E16. In this case Z_1\rightarrow Y is a w -morphism over the origin of

    U_t = ({x'}^2+{y'}^3+p(z', u')+g'(y', z', u', t') = q(y', z', u')+t' = 0)
    \subset {\mathbb A}^5_{(x', y', z', u', t')}/\frac{1}{4}(3, 2, 1, 1, 3).

    As in the proof of Proposition 4.16, we choose (y_1, ..., y_5) = (y', z', u', x', t') with \delta_4 = 4 and \delta_5 = 1 or 2 . We know that ( \Theta_2 ) holds. Hence, if \delta_5 = 1 , then ( \Xi_- ) holds, and so Y\stackrel{-}{\underset{{X}}{\Rightarrow }}Y_1 . This leads to a contradiction. Thus, \delta_5 = 2 and one has \Gamma = ({x'}^2+{y'}^3 = z' = u' = t' = 0) .

    (ⅵ–ⅲ) Y\rightarrow X is of type E18. In this case, Z_1\rightarrow Y is a w -morphism over the origin of

    U_t = ({x'}^2+y'+g'(y', z', u', t') = {y'}^2+p(y', z', u')+t = 0)
    \subset {\mathbb A}^5_{(x', y', z', u', t')}/\frac{1}{7}(5, 3, 2, 1, 6).

    We choose (y_1, ..., y_5) = (y', z', u', x', t') with \delta_4 = 4 and \delta_5 = 1 . We know that ( \Theta_1 ) holds. If {z'}^3\in p , then we can choose \delta_5 = 2 . Then, since ( \Theta_1 ) holds, we know that ( \Xi_- ) holds, and so Y\stackrel{-}{\underset{{X}}{\Rightarrow }}Y_1 . This contradicts our assumption. Hence, {z'}^3\not\in p . In this case, u divides p and X has cE_8 singularities, so {z'}^5\in g' . One can see that \Gamma = ({x'}^2+{z'}^5 = y' = u' = t' = 0) .

    Now the origin of U_t is a cyclic quotient point and the w -morphism over this point is a weighted blow-up with weights v_F(x', y', z', u', t') = \frac{1}{5}(3, 2, 6, 1, 4) , \frac{1}{4}(3, 2, 5, 1, 3) and \frac{1}{7}(5, 10, 2, 1, 6) , respectively. An easy computation shows that \Gamma_{Z_1} does not pass through any singular point of Z_1 , hence Z_1\dashrightarrow Z_2 is a smooth flop. Since there is only one K_{Z_1} -trivial curve, we know that k = 2 .

    (ⅶ) If P is of type cE/2 , then by Proposition 4.18 we know that Y\rightarrow X is of type E22 or E23 in Table 11. Moreover, if Y\rightarrow X is of type E23, then Y_1\rightarrow X is of type E22. Thus, interchanging Y and Y_1 if necessary, we can always assume that Y\rightarrow X is of type E22. As in the proof of Proposition 4.18, we know that Z_1\rightarrow Y is a w -morphism over the origin of U_z\subset Y , which is a cD/3 point defined by

    ({x'}^2+{y'}^3+g'(y', z', u') = 0)\subset {\mathbb A}^4_{(x', y', z', u')}/\frac{1}{3}(0, 2, 1, 1).

    The only possible K_{Z_1} -trivial curve \Gamma_{Z_1} is the lifting of the curve \Gamma = ({x'}^2+{y'}^3 = z' = u' = 0) on Z_1 . Moreover, Z_1\rightarrow Y is defined by a weighted blow-up with the weight

    v_F(x', y', z', u'+\lambda z') = \frac{1}{3}(b, c, 1, 4)

    for some \lambda\in {\mathbb C} , where (b, c) = (3, 2) or (6, 5) . If (b, c) = (6, 5) , then one can see that (\Xi_-) holds by considering the function y , hence Y\stackrel{-}{\underset{{X}}{\Rightarrow }}Y_1 , which contradicts our assumption. Hence, (b, c) = (3, 2) . In this case, an easy computation shows that \Gamma_{Z_1} does not pass through any singular point of Z_1 , so Z_1\dashrightarrow Z_2 is a smooth flop. Since there is only one K_{Z_1} -trivial curve, we know that k = 2 .

    We need to construct a factorization of the flop in Lemma 6.1 (3). Assume that

    X = (xy+f(z, u) = 0)\subset {\mathbb A}^4_{(x, y, z, u)}/\frac{1}{r}(\beta, -\beta, 1, 0)

    is a cA/r singularity with f(z, u) = z^{rk}+g(z, u) and Y\rightarrow X , Y_1\rightarrow X are two different strict w -morphisms with the factorization

    such that Z_1\dashrightarrow Z_2 is a flop.

    Remark 6.2. One always has that k > 1 since, if k = 1 , then there is only one w -morphism over X by classification Table 2.

    Lemma 6.3. Z_1\dashrightarrow Z_2 is a flop over

    V = (\bar{x}\bar{y}+\bar{u}f_1(\bar{z}, \bar{u}) = 0)\subset {\mathbb A}^4_{(\bar{x}, \bar{y}, \bar{z}, \bar{u})}/\frac{1}{r}(\beta, -\beta, 1, 0),

    where f_1 = f(zu^{\frac{1}{r}}, u)/u^k . Moreover, Z_1 = Bl_{(\bar{x}, \bar{u})}V and Z_2 = Bl_{(\bar{y}, \bar{u})}V .

    Proof. We may assume that Y\rightarrow X is a weighted blow-up with the weight w(x, y, z, u) = \frac{1}{r}(b, c, 1, r) with b > r . The chart U_u\subset Y is defined by

    (x_1y_1+f_1(z_1, u_1) = 0)\subset {\mathbb A}^4_{(x_1, y_1, z_1, u_1)}/\frac{1}{r}(b, c, 1, r)

    and the chart U_x\subset Y is defined by

    (y_2+f_2(x_2, z_2, u_2) = 0)\subset {\mathbb A}^4_{(x_2, y_2, z_2, u_2)}/\frac{1}{b}(b-r, c, 1, r)

    for some f_2 . As in the proof of Proposition 4.2, we know that Z_1\rightarrow Y is a weighted blow-up over the origin of U_x with the weight w'(x_2, y_2, z_2, u_2) = \frac{1}{b}(b-r, rk, 1, r) . The flopping curve \Gamma of Z_1\rightarrow Z_2 is the strict transform of the curve \Gamma_Y\subset Y such that \Gamma_Y|_{U_u} = (y_1 = z_1 = u_1 = 0) and \Gamma_Y|_{U_x} = (x_2 = y_2 = z_2 = 0) . One can see that \Gamma intersects exc(Z_1\rightarrow Y) at the origin of U'_u\subset Z_1 , which is defined by

    (y'+f'(x', z', u') = 0)\subset {\mathbb A}^4_{(x', y', z', u')}/\frac{1}{r}(b, 0, 1, -b).

    It is easy to see that \Gamma is contained in U'_u\cup U_u , and on U'_u we know that \Gamma is defined by (x' = z' = 0) .

    We have the following change of coordinates formula:

    x = x_1u_1^{\frac{b}{r}}, \quad y = y_1u_1^{\frac{c}{r}}, \quad z = z_1u_1^{\frac{1}{r}}, \quad u = u_1, \mbox{ and }
    x = x_2^{\frac{b}{r}}, \quad y = y_2x_2^{\frac{c}{r}}, \quad z = z_2x_2^{\frac{1}{r}}, \quad u = u_2x_2.

    Also,

    x_2 = x'{u'}^{\frac{b-r}{b}}, y_2 = y'{u'}^{\frac{rk}{b}}, \quad z_2 = z'{u'}^{\frac{1}{b}}\quad, u_2 = {u'}^{\frac{r}{b}}.

    One can see that

    x = {x'}^{\frac{b}{r}}{u'}^{\frac{b-r}{r}}, y = y'{x'}^{\frac{c}{r}}{u'}^{\frac{rk}{b}+\frac{c}{r}-\frac{c}{b}} = y'{x'}^{\frac{c}{r}}{u'}^{\frac{c}{r}+1}, \quad z = z'{x'}^{\frac{1}{r}}{u'}^{\frac{1}{r}}\mbox{ and }u = x'u'.

    It follows that

    u_1 = x'u', \quad z_1 = z', \quad y_1 = y'u'\mbox{ and }x_1 = {u'}^{-1}.

    If we choose an isomorphism

    U'_u\cong (y_1+u'f'(x', z', u') = 0)\subset {\mathbb A}^4_{(x', y_1, z', u')}/\frac{1}{r}(b, -b, 1, -b),

    then U_u\cup U'_u = Bl_{(x', u_1)}V , where

    V = (x'y_1+u_1f_1(z_1, u_1) = 0)\subset {\mathbb A}^4_{(x', y_1, z_1, u_1)}/\frac{1}{r}(b, -b, 1, 0)

    by noticing that

    \begin{align*} f'(x', z', u')& = f_2(x'{u'}^{\frac{b-r}{b}}, z'{u'}^{\frac{1}{b}}, {u'}^{\frac{r}{b}})/{u'}^{\frac{rk}{b}}\\ & = f(z_2x_2^{\frac{1}{r}}, u_2x_2)/x_2^k{u'}^{\frac{rk}{b}}\\ & = f(z'{x'}^{\frac{1}{r}}{u'}^{\frac{1}{r}}, x'u')/{x'}^k{u'}^k\\ & = f(z_1u_1^{\frac{1}{r}}, u_1)/{u_1}^k\\ & = f_1(z_1, u_1). \end{align*}

    Now we can choose (\bar{x}, \bar{y}, \bar{z}, \bar{u}) = (x', y_1, z_1, u_1) .

    Finally, we know that Y_1\rightarrow X is a weighted blow-up with the weight \frac{1}{r}(b-r, c+r, 1, r) and Z_2\rightarrow Y_1 is a w -morphism over the origin of U_{1, y}\subset Y_1 . Hence, the local picture of Z_2\rightarrow V can be obtained by a similar computation, but interchanging the role of x and y . One then has that Z_2 = Bl_{(\bar{y}, \bar{u})}V .

    In this subsection, we assume that

    V = (xy+uf(z, u) = 0)\subset {\mathbb A}^4_{(x, y, z, u)}/\frac{1}{r}(\beta, -\beta, 1, 0)

    with f(z, u) = z^{rk}+g(z, u) for some k > 1 . Let Z_1 = Bl_{(x, u)}V and Z_2 = Bl_{(y, u)}V such that Z_1\dashrightarrow Z_2 is a {\mathbb Q} -factorial terminal flop. Let w be the weight w(z, u) = \frac{1}{r}(1, r) and m = w(f(z, u)) . Then m\leq k . Let U_{1, x}\subset Z_1 be the chart

    (y+u_1f_1(z, u_1) = 0)\subset {\mathbb A}^4_{(x, y, z, u_1)}/\frac{1}{r}(\beta, -\beta, 1, -\beta)

    and U_{1, u}\subset Z_1 be the chart

    (x_1y+f(z, u) = 0)\subset {\mathbb A}^4_{(x_1, y, z, u)}/\frac{1}{r}(\beta, -\beta, 1, 0)

    with the relations u = u_1x , x = x_1u and f_1 = f(z, xu_1) . Similarly, let U_{2, y}\subset Z_2 be the chart

    (x+u_2f_2(z, u_2) = 0)\subset {\mathbb A}^4_{(x, y, z, u_2)}/\frac{1}{r}(\beta, -\beta, 1, \beta)

    and U_{2, u}\subset Z_2 be the chart

    (xy_2+f(z, u) = 0)\subset {\mathbb A}^4_{(x, y_2, z, u)}/\frac{1}{r}(\beta, -\beta, 1, 0)

    with the relations u = u_2y , y = y_2u and f_2 = f(z, yu_2) .

    Lemma 6.4. Let \phi:V'\rightarrow V be a strict w -morphism. Then:

    (1) The chart U'_u\subset V' is {\mathbb Q} -factorial.

    (2) If m = k then U'_z\subset V' is smooth. Otherwise, U'_z contains exactly one non- {\mathbb Q} -factorial cA point which is defined by xy+uf"(z, u) = 0 where f" = f(z^{\frac{1}{r}}, zu)/z^m . One then has that w(f") < m .

    (3) All other singular points on V' are cyclic quotient points.

    Proof. From Table 2, we know that V'\rightarrow V is a weighted blow-up with the weight \frac{1}{r}(b, c, 1, r) with b+c = r(m+1) . Statement (3) follows from direct computations. One can compute that

    U'_u\cong (xy+f'(z, u) = 0)\subset {\mathbb A}^4_{(x, y, z, u)}/\frac{1}{r}(\beta, -\beta, 1, 0)

    with f' = f(zu^{\frac{1}{r}}, u)/u^m . By [18, 2.2.7] we know that U'_u is {\mathbb Q} -factorial if and only if f' is irreducible as a {\mathbb Z}/r {\mathbb Z} -invariant function. If U'_u is not {\mathbb Q} -factorial, then f' = f'_1f'_2 for some non-unit {\mathbb Z}/r {\mathbb Z} -invariant functions f'_1 and f'_2 . Write

    f'_1 = \sum\limits_{(i, j)\in {\mathbb Z}^2_{\geq0}}\xi_{i, j}z^{ri}u^j\mbox{ and }f'_2 = \sum\limits_{(i, j)\in {\mathbb Z}^2_{\geq0}}\zeta_{i, j}z^{ri}u^j.

    Let

    m_1 = -\min\limits_{\xi_{i, j}\neq 0}\{j-i\}, \quad m_2 = -\min\limits_{\zeta_{i, j}\neq 0}\{j-i\}.

    Notice that if f' = \sum_{(i, j)}\sigma_{i, j}z^{ri}u^j , then j+m-i\geq 0 if \sigma_{i, j}\neq 0 because f' = f(zu^{\frac{1}{r}}, u)/u^m . Hence, we have the relation m_1+m_2\leq m . Now let

    f_1 = \sum\limits_{(i, j)}\xi_{ij}z^{ir}u^{j-i+m_1}\mbox{ and }f_2 = \sum\limits_{(i, j)}\zeta_{ij}z^{ir}u^{j-i+m_2}.

    Then f_1 and f_2 can be viewed as {\mathbb Z}/r {\mathbb Z} -invariant functions on V such that \phi ^{\ast} f_i = u^{m_i}f'_i for i = 1 , 2 . This means that f = u^{m-m_1-m_2}f_1f_2 is not irreducible. Nevertheless, the chart U_{1, u}\subset Z_1 is defined by x_1y+f(z, u) = 0 and has {\mathbb Q} -factorial singularities. This leads to a contradiction. Thus, U'_u is {\mathbb Q} -factorial.

    The chart U'_z is defined by xy+uf"(z, u) = 0 . When m = k , we know that f" is a unit along u = 0 , hence U'_z is smooth. If m < k , then f"(z, u) is a non-unit. In this case, the origin of U'_z is a non- {\mathbb Q} -factorial cA singularity.

    From the construction in [19], for any strict w -morphism V'\rightarrow V we have the following diagram

    where V"_i\rightarrow V' is a {\mathbb Q} -factorization and V"_i\dashrightarrow Z'_i is a composition of flips for i = 1 , 2 . By Lemma 6.4, we know that if m = k , then V"_1 = V' = V"_2 . Otherwise, V"_1\dashrightarrow V"_2 is a flop over the singularity xy+uf"(z, u) = 0 .

    First we discuss the factorization of V"_1\dashrightarrow Z'_1 . If m = k , then V"_1 = V' is covered by four affine charts U'_x , U'_y , U'_z and U'_u . The origin of U'_x and U'_y are cyclic quotient points and all other singular points are contained in U'_u . When m < k , the chart U'_z has a non- {\mathbb Q} -factorial point and there are two charts U"_{1, x} and U"_{1, u} over this point. We fix the following notation for the latter discussion.

    \bullet \bar{U}"_1 = U'_x = (y"_1+u"_1f"_1(x"_1, z"_1, u"_1) = 0) \subset {\mathbb A}^4_{(x"_1, y"_1, z"_1, u"_1)}/\frac{1}{b}(b-r, c, 1, r)

    with x = {x"_1}^{\frac{b}{r}} , y = y"_1{x"_1}^{\frac{c}{r}} , z = z"_1{x"_1}^{\frac{1}{r}} , u = u"_1x"_1 and f"_1 = f(z"_1{x"_1}^{\frac{1}{r}}, u"_1x"_1)/{x"_1}^m .

    \bullet \bar{U}"_2 = U'_y = (x"_2+u"_2f"_2(y"_2, z"_2, u"_2) = 0) \subset {\mathbb A}^4_{(x"_2, y"_2, z"_2, u"_2)}/\frac{1}{c}(b, c-r, 1, r)

    with x = x"_2{y"_2}^{\frac{b}{r}} , y = {y"_2}^{\frac{c}{r}} , z = z"_2{y"_2}^{\frac{1}{r}} , u = u"_2y"_2 and f"_2 = f(z"_2{y"_2}^{\frac{1}{r}}, u"_2y"_2)/{y"_2}^m .

    \bullet \bar{U}"_3 = U'_u = (x"_3y"_3+f"_3(z"_3, u"_3) = 0) \subset {\mathbb A}^4_{(x"_3, y"_3, z"_3, u"_3)}/\frac{1}{r}(b, c, 1, r)

    with x = {x"_3}{u"_3}^{\frac{b}{r}} , y = y"_3{u"_3}^{\frac{c}{r}} , z = z"_3{u"_3}^{\frac{1}{r}} , u = u"_3 and f"_3 = f(z"_3{u"_3}^{\frac{1}{r}}, u"_3)/{u"_3}^m .

    If m = k , define \bar{U}"_4 = U'_z . In this case this chart is smooth. When m < k we define

    \bullet \bar{U}"_4 = U"_{1, x} = (y"_4+u"_4f"_4(x"_4, z"_4, u"_4) = 0) \subset {\mathbb A}^4_{(x"_4, y"_4, z"_4, u"_4)}

    with x = {x"_4}{z"_4}^{\frac{b}{r}} , y = y"_4{z"_4}^{\frac{c}{r}} , z = {z"_4}^{\frac{1}{r}} , u = x"_4u"_4z"_4 and f"_4 = f({z"_4}^{\frac{1}{r}}, x"_4u"_4z"_4)/{z"_4}^m .

    \bullet \bar{U}"_5 = U"_{1, u} = (x"_5y"_5+f"_5(z"_5, u"_5) = 0) \subset {\mathbb A}^4_{(x"_5, y"_5, z"_5, u"_5)}

    with x = {x"_5}u"_5{z"_5}^{\frac{b}{r}} , y = y"_5{z"_5}^{\frac{c}{r}} , z = {z"_5}^{\frac{1}{r}} , u = u"_5z"_5 and f"_5 = f({z"_5}^{\frac{1}{r}}, u"_5z"_5)/{z"_5}^m .

    On the other hand, we will see later that it is enough to assume that Z'_1\rightarrow Z_1 is a divisorial contraction over the origin of U_{1, u}\subset Z_1 . This means that Z'_1 is covered by five affine charts U_{1, x} , U'_{1, x} , U'_{1, y} , U'_{1, z} and U'_{1, u} where the latter four charts correspond to the weighted blow-up Z'_1\rightarrow Z_1 . Notice that exc(Z'_1\rightarrow Z_1) and exc(V'\rightarrow V) are the same divisor since V"_1 and Z'_1 are isomorphic in codimension one. One can see that Z'_1\rightarrow Z_1 is a weighted blow-up with the weight w'(x_1, y, z, u) = \frac{1}{r}(b-r, c, 1, r) . Again, we use the following notation:

    \bullet \bar{U}'_1 = U'_{1, x} = (y'_1+f'_1(x'_1, z'_1, u'_1) = 0) \subset {\mathbb A}^4_{(x'_1, y'_1, z'_1, u'_1)}/\frac{1}{b-r}(b-2r, c, 1, r)

    with x = {x'_1}^{\frac{b}{r}}u'_1 , y = y'_1{x'_1}^{\frac{c}{r}} , z = z'_1{x'_1}^{\frac{1}{r}} , u = u'_1x'_1 and f'_1 = f(z'_1{x'_1}^{\frac{1}{r}}, u'_1x'_1)/{x'_1}^m .

    \bullet \bar{U}'_2 = U'_{1, y} = (x'_2+f'_2(y'_2, z'_2, u'_2) = 0) \subset {\mathbb A}^4_{(x'_2, y'_2, z'_2, u'_2)}/\frac{1}{c}(b-r, c-r, 1, r)

    with x = x'_2{y'_2}^{\frac{b}{r}}u'_2 , y = {y'_2}^{\frac{c}{r}} , z = z'_2{y'_2}^{\frac{1}{r}} , u = u'_2y'_2 and f'_2 = f(z'_2{y'_2}^{\frac{1}{r}}, u'_2y'_2)/{y'_2}^m .

    \bullet \bar{U}'_3 = U'_{1, u} = (x'_3y'_3+f'_3(z'_3, u'_3) = 0) \subset {\mathbb A}^4_{(x'_3, y'_3, z'_3, u'_3)}/\frac{1}{r}(b-r, c, 1, r)

    with x = {x'_3}{u'_3}^{\frac{b}{r}} , y = y'_3{u'_3}^{\frac{c}{r}} , z = z'_3{u'_3}^{\frac{1}{r}} , u = u'_3 and f'_3 = f(z'_3{u'_3}^{\frac{1}{r}}, u'_3)/{u'_3}^m .

    \bullet \bar{U}'_4 = U_{1, x} = (y'_4+u'_4f'_4(z'_4, u'_4) = 0) \subset {\mathbb A}^4_{(x'_4, y'_4, z'_4, u'_4)}/\frac{1}{r}(\beta, -\beta, 1, -\beta)

    with x = x'_4 , y = y'_4 , z = z'_4 , u = u'_4x'_4 and f'_4 = f(z'_4, u'_4x'_4) .

    \bullet \bar{U}'_5 = U'_{1, z} = (x'_5y'_5+f'_5(z'_5, u'_5) = 0) \subset {\mathbb A}^4_{(x'_5, y'_5, z'_5, u'_5)}

    with x = {x'_5}{z'_5}^{\frac{b}{r}}u'_5 , y = y'_5{z'_5}^{\frac{c}{r}} , z = {z'_5}^{\frac{1}{r}} , u = u'_5z'_5 and f'_5 = f({z'_5}^{\frac{1}{r}}, u'_5z'_5)/{z'_5}^m .

    Lemma 6.5. Assume that Z'_1\rightarrow Z_1 is a divisorial contraction over the origin of U_{1, u} . Then:

    (1) gdep(Z'_1) = gdep(V"_1)-1 .

    (2) All the singular points on the non-isomorphic loci of V"_1\dashrightarrow Z'_1 on both V"_1 and Z'_1 are cyclic quotient points.

    (3) The flip V"_1\dashrightarrow Z'_1 is of type IA in the convention of [20, Theorem 2.2].

    Proof. It is easy to see that \bar{U}'_i\cong\bar{U}"_i for i = 2 , 3 and i = 5 if m < k . Since \bar{U}'_4 is smooth, the only singular point contained in the non-isomorphic locus of V"_1\dashrightarrow Z'_1 on V"_1 is the origin of \bar{U}"_1 . This point is a cyclic quotient point of index b , so it has generalized depth b-1 .

    On the other hand, singular points on the non-isomorphic locus of V"_1\rightarrow Z'_1 on Z'_1 are origins of \bar{U}'_1 and \bar{U}'_4 . They are cyclic quotient points of indices b-r and r , respectively. One can then see that

    gdep(V"_1)-gdep(Z'_1) = b-1-(b-r-1+r-1) = 1.

    Now we know that that the flipping curve contains only one singular point which is a cyclic quotient point. Also, the general elephant of the flip is of A -type since it comes from the factorization of a flop over a cA/r point. Thus, the flip is of type IA by the classification from [20], Theorem 2.2].

    Lemma 6.6. Assume that is a flip of type IA. Assume that

    is the factorization in Theorem 2.18. Then:

    (1) If S_1\dashrightarrow S_2 is a flop, then it is a Gorenstein flop.

    (2) If S_1\dashrightarrow S_2 is a flip, or S_1\dashrightarrow S_2 is a flop and S_2\dashrightarrow S_3 is a flip, then the flip is of type IA.

    Proof. Let C\subset T be the flipping curve. Since T\dashrightarrow T' is a type IA flip, there is exactly one non-Gorenstein point which is contained in C , and this point is a cA/r point. From the construction, we know that S_1\rightarrow T is a w -morphism over this cA/r point. Also, there exists a Du Val section H\in|-K_T| such that C\not\subset H . We know that H_{S_1}\in|-K_{S_1}| by [3, Lemma 2.7 (2)]. Hence, all non-Gorenstein point of S_1 is contained in H_{S_1} . Now assume that C_{S_1} contains a non-Gorenstein point. Then H_{S_1} intersects C_{S_1} non-trivially. Since C_{S_1}\not\subset H_{S_1} , we know that H_{S_1}.C_{S_1} > 0 , hence C_{S_1} is a K_{S_1} -negative curve and so S_1\dashrightarrow S_2 is a flip. Thus, if S_1\dashrightarrow S_2 is a flop, then it is a Gorenstein flop.

    If S_1\dashrightarrow S_2 is a flip, then C_{S_1} passes through a non-Gorenstein point since 0 > K_{S_1}.C_{S_1} > -1 by [21, Theorem 0]. Since C\subset T passes through exactly one non-Gorenstein point, C_{S_1} passes through exactly one non-Gorenstein point and this point is contained in E = exc(S_1\rightarrow T) . Since S_1\rightarrow T is a w -morphism over a cA/r point, an easy computation shows that E contains only cA/r singularities. Also, we know that H_{S_1} is a Du Val section which does not contain C_{S_1} . Thus, S_1\dashrightarrow S_2 is also a flip of type IA by the classification [20, Theorem 2.2].

    Now assume that S_1\dashrightarrow S_2 is a flop and S_2\dashrightarrow S_3 is a flip. Then the flipping curve \Gamma of S_2\dashrightarrow S_3 is contained in E_{S_2} since all K_{S_2} -negative curves over V are contained in E_{S_2} . Since S_1\dashrightarrow S_2 is a flop, we know that E_{S_2} contains only cA/r singularities [17, Theorem 2.18] and H_{S_2}\in|-K_{S_2}| is also a Du Val section. If \Gamma\not\subset H_{S_2} , then S_2\dashrightarrow S_3 is a flip of type IA by the classification from [20, Theorem 2.2].

    Thus, we only need to prove that \Gamma\not\subset H_{S_2} . Assume that \Gamma\subset H_{S_2}\cap E_{S_2} . Then, since H_{S_1} does not intersect C_{S_1} (otherwise C_{S_1} is a K_{S_1} -negative curve and then S_1\dashrightarrow S_2 is not a flop), we know that \Gamma does not intersect the flopping curve C'_{S_2}\subset S_2 . Let B\subset E_{S_2} be a curve which intersects C'_{S_2} non-trivially. Then \Gamma_{S_1}\equiv \lambda B_{S_1} for some \lambda\in {\mathbb Q} since the both curves are contracted by S_1\rightarrow T . Hence, for all divisors D\subset S_2 such that D.C'_{S_2} = 0 , we know that D.\Gamma = \lambda D.B . On the other hand, S_1\dashrightarrow S_2 is a K_{S_1}+E -anti-flip since C_{S_1} is not contained in E . By Corollary 2.22 we know that (K_{S_2}+E_{S_2}).B_{S_2} > (K_{S_1}+E).B_{S_1} , hence E_{S_2}.B_{S_2} > E.B_{S_1} . Now we know that \rho(S_2/V) = 2 and B is not numerically equivalent to a multiple of C'_{S_2} , hence we may write \Gamma\equiv \lambda B+\mu C'_{S_2} for some \mu\in {\mathbb Q} . Since

    E_{S_2}.\Gamma = E_{S_1}.\Gamma_{S_1} = \lambda E.B_{S_1} < \lambda E_{S_2}.B_{S_2}

    and E_{S_2}.C'_{S_2} < 0 , we know that \mu > 0 . Hence, \Gamma is not contained in the boundary of the relative effective cone NE(S_2/V) . Thus, \Gamma cannot be the flipping curve of S_2\dashrightarrow S_3 . This leads to a contradiction.

    Lemma 6.7. Assume that T\dashrightarrow T' is a three-dimensional terminal {\mathbb Q} -factorial flip which satisfies conditions (1)–(3) of Lemma 6.5. Then, the factorization in Theorem 2.18 for T\dashrightarrow T' is one of the following diagrams:

    (1)

    where S_1\dashrightarrow S_2 is a flip which also satisfies conditions (1)–(3) of Lemma 6.5 and S_2\rightarrow T' is a strict w -morphism.

    (2)

    where S_1\dashrightarrow S_2 is a smooth flop and S_2\dashrightarrow S_3 is a flip which also satisfies conditions (1)–(3) of Lemma 6.5 and S_3\rightarrow T' is a strict w -morphism.

    (3)

    where S_1\dashrightarrow S_2 is a smooth flop and S_2 = Bl_{C'}T' where C' is a smooth curve contained in the smooth locus of T' .

    Proof. We have the factorization

    such that S_1\dashrightarrow S_2 is a flip or a flop and S_i\dashrightarrow S_{i+1} is a flip for all 2\leq i\leq k-1 . One has that

    gdep(S_k)\leq gdep(S_1) = gdep(T)-1 = gdep(T')\leq gdep(S_k)+1.

    If gdep(S_k) = gdep(S_1) , then k = 2 and S_1\dashrightarrow S_2 is a flop. By [3, Remark 3.4] we know that S_2\rightarrow T' is a divisorial contraction to a curve. Since singular points on the non-isomorphic locus of T\dashrightarrow T' are all cyclic quotient points and there is no divisorial contraction to a curve which passes through a cyclic quotient point [8, Theorem 5], we know that S_2\rightarrow T' is a divisorial contraction to a curve C' contained in the smooth locus. Now, we also have that gdep(S_2) = gdep(T') , hence S_2 is smooth over T' and so C' is also a smooth curve.

    Now assume that gdep(S_k) < gdep(S_1) . Then gdep(S_k) = gdep(S_1)-1 = gdep(T')-1 , hence either k = 2 or k = 3 and S_1\dashrightarrow S_2 is a flop. Also, S_k\rightarrow T' is a w -morphism. Since singular points on the non-isomorphic locus of T\dashrightarrow T' are all cyclic quotient points, singular points on the exceptional divisor of S_k\rightarrow T' are all cyclic quotient points. Since flops do not change singularities [17, Theorem 2.4], we know that the singular points on the non-isomorphic locus of S_i\dashrightarrow S_{i+1} are all cyclic quotient points for i = 1 , ..., k-1 . If S_1\dashrightarrow S_2 is a flop, then by Lemma 6.6 we know that it is a Gorenstein flop. Since cyclic quotient points are not Gorenstein, we know that the flop is in fact a smooth flop. Now assume that S_i\dashrightarrow S_{i+1} is a flip for i = 1 or 2 . Then again, by Lemma 6.6 we know that it is a flip of type IA. Hence, conditions (1)–(3) of Lemma 6.5 are satisfied for this flip.

    Corollary 6.8. Assume that T\dashrightarrow T' is a three-dimensional terminal {\mathbb Q} -factorial flip which satisfies conditions (1)–(3) of Lemma 6.5. Then we have a factorization

    where \tilde{{T}}\rightarrow T and \tilde{{T'}}\rightarrow T are feasible resolutions, \bar{T'} = Bl_{C'}\tilde{{T'}} where C'\subset \tilde{{T'}} is a smooth curve, and \tilde{{T}}\dashrightarrow \bar{T'} is a sequence of smooth flops.

    Proof. For convenience we denote the diagram by (A)_{T\dashrightarrow T'} .

    We know that the factorization of T\dashrightarrow T' is of the form (1)–(3) in Lemma 6.7. Notice that, since the only singular points on the non-isomorphic locus of T\dashrightarrow T' are cyclic quotient points, the feasible resolutions \tilde{{T}} and \tilde{{T'}} are uniquely determined. Hence, \tilde{{T}} is also a feasible resolution of S_1 . If the factorization of T\dashrightarrow T' is of type (3) in Lemma 6.7, then the non-isomorphic locus of S_1\dashrightarrow T' contains no singular points. This means that S_2\rightarrow T' induces a smooth blow-up \bar{T'}\rightarrow \tilde{{T'}} on \tilde{{T'}} , and S_1\dashrightarrow S_2 induces a smooth flop \tilde{{T}}\dashrightarrow\bar{T'} . Thus, (A)_{T\dashrightarrow T'} exists.

    Now, if the factorization of T\dashrightarrow T' is of type (1), then \tilde{{T'}} is a feasible resolution of S_2 . Since gdep(S_1) = gdep(T)-1 , we may utilize induction on gdep(T) and assume that (A)_{S_1\dashrightarrow S_2} exists, and then (A)_{T\dashrightarrow T'} can be induced by (A)_{S_1\dashrightarrow S_2} . If the factorization of T\dashrightarrow T' is of type (2), then again by induction we may assume that

    exists. Since S_3\rightarrow T' is a strict w -morphism, we know that \tilde{{S_3}} = \tilde{{T'}} . Also, since S_1\dashrightarrow S_2 is a smooth flop, it induces a smooth flop \tilde{{T}} = \tilde{{S_1}}\dashrightarrow \tilde{{S_2}} . If we let \bar{T'} = \bar{S_3} and let \tilde{{T}}\dashrightarrow \bar{T'} be the composition \tilde{{T}}\dashrightarrow \tilde{{S_2}}\dashrightarrow \bar{T'} , then we get the diagram (A)_{T\dashrightarrow T'} .

    Definition 6.9. Let W_1\dashrightarrow W_2 be a birational map between smooth threefolds. We say that W_1\dashrightarrow W_2 is of type \Omega_0 if it is a composition of smooth flops. We say that W_1\dashrightarrow W_2 is of type \Omega_n if there exists the diagram

    such that:

    (1) \bar{W}_i = Bl_{C_i}W_i for some smooth curve C_i\subset W_i for i = 1 , 2 .

    (2) \bar{W'}_i\dashrightarrow \bar{W}_i is a composition of smooth flops for i = 1 , 2 .

    (3) \bar{W'_1}\dashrightarrow \bar{W'_2} has the factorization

    \bar{W'_1} = \bar{W}'_{1, 1}\dashrightarrow \bar{W}'_{1, 2}\dashrightarrow...\dashrightarrow\bar{W}'_{1, m} = \bar{W'_2}

    such that \bar{W}'_{1, j}\dashrightarrow\bar{W}'_{1, j+1} is a birational map of type \Omega_{m_j} for some m_j < n .

    Example 6.10. In the following diagrams, dashmaps stand for smooth flops and all other maps are blowing-down smooth curves.

    (1)

    (2)

    (3)

    In diagram (1), W_1\dashrightarrow W_2 is of type \Omega_1 . In both diagram (2) and (3), W_1\dashrightarrow W_2 is of type \Omega_2 .

    Definition 6.11. Let W\dashrightarrow W' be a birational map between smooth threefolds. We say that W\dashrightarrow W' has an \Omega -type factorization if there exists birational maps between smooth threefolds

    W = W_1\dashrightarrow W_2\dashrightarrow...\dashrightarrow W_k = W'

    such that W_i\dashrightarrow{W_{i+1}} is of type \Omega_{n_i} for some n_i\in {\mathbb Z}_{\geq0} .

    Proposition 6.12. Assume that X is a {\mathbb Q} -factorial terminal threefold and W\rightarrow X , W'\rightarrow X are two different feasible resolutions. Then the birational map W\dashrightarrow W' has an \Omega -type factorization.

    Proof. First, notice that if gdep(X) = 1 , then X has either a cyclic quotient point of index 2, or a cA_1 point defined by xy+z^2+u^n for n = 2 or 3 by [22, Corollary 3.4]. In those cases, there is exactly one feasible resolution (which is obtained by blowing-up the singular point). Hence, one may assume that gdep(X) > 1 . Let Y\rightarrow X (resp. Y'\rightarrow X ) be the strict w -morphism which is the first factor of W\rightarrow X (resp. W'\rightarrow X ). If Y = Y' , then W and W' are two different feasible resolutions of Y . In this case, the statement can be proved by induction on gdep(X) . Thus, we may assume that Y\neq Y' .

    Since both Y and Y' are strict w -morphisms, by Corollary 5.7 there exists a sequence of strict w -morphisms Y_1 = Y\rightarrow X , Y_2\rightarrow X , ..., Y_k = Y'\rightarrow X such that Y_i\underset{{X}}{\Rightarrow }Y_{i+1} for i = 1 , ..., k-1 . For each 2\leq i\leq k-1 , let W_i\rightarrow Y_i be a feasible resolution. Then W_i is also a feasible resolution of X and it is enough to prove that our statement holds for W_i and W_{i+1} , for all i = 1 , ..., k-1 . Thus, we may assume that Y\underset{{X}}{\Rightarrow }Y' .

    We have the diagram

    Lemma 6.1 says that there are three possibilities. If k = 1 , then W and W' are two different feasible resolutions of Z_1 . Since gdep(Z_1) = gdep(X)-2 , again by induction on gdep(X) we know that W\dashrightarrow W' has an \Omega -type factorization. Assume that k = 2 and Z_1\dashrightarrow Z_2 is a smooth flop. Then it induces a smooth flop \tilde{{Z}}_1\dashrightarrow \tilde{{Z}}_2 where \tilde{{Z}}_i\rightarrow Z_i is a feasible resolution of Z_i for i = 1 , 2 . Also, we know that W and \tilde{{Z}}_1 are two feasible resolutions of Y , and W' and \tilde{{Z}}_2 are two feasible resolutions of Y' . Again, by induction on gdep(X) , we know that both W\dashrightarrow \tilde{{Z}}_1 and \tilde{{Z}}_2\dashrightarrow W' have \Omega -type factorizations, hence W\dashrightarrow W' does as well.

    Finally, assume that we are in the case of Lemma 6.1 (3), namely that X has a cA/r singularity and Z_1\dashrightarrow Z_2 is a singular flop. By Lemma 6.3 we know that Z_1\dashrightarrow Z_2 is a flop over

    V = (xy+uf(z, u) = 0)\subset {\mathbb A}^4_{(x, y, z, u)}/\frac{1}{r}(\beta, -\beta, 1, 0).

    We have the factorization of the flop Z_1\dashrightarrow Z_2

    We use the notation at the beginning of this subsection. Assume that m > 1 . Then we can choose V'\rightarrow V to be the weighted blow-up with the weight w'(x, y, z, u) = \frac{1}{r}(\beta+r, mr-\beta, 1, r) . In this case, Z'_i\rightarrow Z_i is a divisorial contraction over the origin of U_{i, u} for i = 1 , 2 since both w'(x_1) and w'(y_2) > 0 . When m = 1 , let V'\rightarrow V be the weighted blow-up with the weight w'(x, y, z, u) = \frac{1}{r}(r+\beta, r-\beta, 1, r) . Then, Z'_1\rightarrow Z_1 is a divisorial contraction over the origin of U_{1, u} , but Z'_2\rightarrow Z_2 is a divisorial contraction over the origin of U_{2, y} . Since m = 1 and k > 1 by Remark 6.2, we know that f(z, u) = \lambda u+z^{rk} for some unit \lambda . If we let \bar{u} = \lambda u+z^{rk} , then we can write the defining equation of V as xy+\bar{u}\bar{f}(z, \bar{u}) where \bar{f} = \frac{1}{\lambda}(\bar{u}-z^{rk}) . One has that Z_2\cong Bl_{(x, \bar{u})}V , and under this notation one also has that Z'_2\rightarrow Z_2 is a divisorial contraction over the origin of U_{2, \bar{u}} . In conclusion, Corollary 6.8 holds for both V"_1\dashrightarrow Z'_1 and V"_2\dashrightarrow Z'_2 .

    Let \tilde{{Z'_i}}\rightarrow Z'_i be a feasible resolution. If V"_1 = V"_2 , then one can see that \tilde{{Z'_1}}\dashrightarrow \tilde{{Z'_2}} is of type \Omega_1 . Assume that V"_1\dashrightarrow V"_2 is a flop. Notice that Z'_i\rightarrow Z_i is a w -morphism since a(E, Z_i) = a(E, V) where E = exc(Z'_i\rightarrow Z_i) = exc(V'\rightarrow V) . One has that

    gdep(V"_i) = gdep(Z'_i)+1 = gdep(Z_i)

    where the second equality follows from Corollary 5.8. Now we know that V"_1\dashrightarrow V"_2 is a flop over V' with gdep(V') < gdep(V) . By induction on gdep(V) , we may assume that \tilde{{V"_1}}\dashrightarrow \tilde{{V"_2}} has an \Omega -type factorization where \tilde{{V"_i}}\rightarrow V"_i is a feasible resolution corresponding to the diagram in Corollary 6.8. One can see that \tilde{{Z'_2}}\dashrightarrow \tilde{{Z'_2}} can be connected by a diagram of the form \Omega_n for some n\in {\mathbb N} . Finally, we know that W and \tilde{{Z'_1}} (resp. W' and \tilde{{Z'_2}} ) are feasible resolutions of Y (resp. Y' ). Again, by induction on gdep(X) , we may assume that W\dashrightarrow \tilde{{Z'_1}} and W'\dashrightarrow \tilde{{Z'_2}} have \Omega -type factorizations. Hence, W\dashrightarrow W' has an \Omega -type factorization.

    Remark 6.13. Assume that X = (xy+z^m+u^k = 0)\subset {\mathbb A}^4 is a cA singularity. Then there exists feasible resolutions W and W' such that the birational map W\dashrightarrow W' is connected by \Omega_n for n = \lceil {\frac{m}{k-m}}\rceil .

    In this section, we prove our main theorems. First, we recall some definitions which are defined in the introduction section.

    Definition 7.1. Let X be a projective variety. We say that a resolution of singularities W\rightarrow X is a P-minimal resolution if for any smooth model W'\rightarrow X one has that \rho(W)\leq \rho(W') .

    Definition 7.2. Let W\dashrightarrow W' be a birational map between smooth varieties. We say that this birational map is a P-desingularization of a flop if there exists a flop X\dashrightarrow X' such that W\rightarrow X and W'\rightarrow X' are P-minimal resolutions.

    Proposition 7.3. Assume that X is a threefold. Then, W\rightarrow X is a P-minimal resolution if and only if W is a feasible resolution of a terminalization of X . In particular, if X is a terminal and {\mathbb Q} -factorial threefold, then P-minimal resolutions of X coincide with feasible resolutions.

    Proof. Let W\rightarrow X be a resolution of singularities and let W\dashrightarrow X_W be the K_W -MMP over X . Then X_W is a terminalization of X . We know that \rho(W/X_W)\geq gdep(X_W) by Corollary 5.2. Assume first that W\rightarrow X is P -minimal. Let W_1\rightarrow X_W be a feasible resolution of X_W , then \rho(W_1/X) = gdep(X_W)\leq \rho(W/X_W) . Since W_1 is also a smooth resolution of X , the inequality is an equality. Therefore, \rho(W/X_W) = gdep(X_W) , which implies that W\dashrightarrow X_W is a sequence of strict w -morphisms by Corollary 5.2, or, equivalently, W\rightarrow X_W is a feasible resolution.

    Conversely, assume that W\rightarrow X is not P -minimal, but it is a feasible resolution of some X_W which is a terminalization of X . There exists a P -minimal resolution W'\rightarrow X such that \rho(W/X) > \rho(W'/X) . From the above argument, there exists a terminalization X_{W'} of X such that W'\rightarrow X_{W'} is a feasible resolution. Hence, \rho(W'/X) = gdep(X') . However, since terminalizations are connected by flops [23, Theorem 1] and flops do not change singularities by [17, Theorem 2.4], we know that gdep(X_W) = gdep(X_{W'}) . This means that \rho(W/X_W) > gdep(X) , so W can not be a feasible resolution of X_W . This is a contradiction.

    Proof of Theorem 1.1. Let X be a threefold and W\rightarrow X , W'\rightarrow X be two P-minimal resolutions. By Proposition 7.3, we know that W (resp. W' ) is a feasible resolution of a terminalization X_W\rightarrow X (resp. X_{W'}\rightarrow X ). If X_W\not\cong X_{W'} , then X_W and X_{W'} are connected by flops [23, Theorem 1], hence W\dashrightarrow W' is connected by P-desingularizations of terminal {\mathbb Q} -factorial flops.

    Now assume that X_W = X_{W'} . Then W and W' are two different feasible resolutions of X_W . The first two paragraphs in the proof of Proposition 6.12 and Lemma 6.7 imply that W\dashrightarrow W' can be also connected by P-desingularizations of terminal {\mathbb Q} -factorial flops. Moreover, Proposition 6.12 says that those P-desingularizations of flops can be factorized into compositions of diagrams of the form \Omega_i . This finishes the proof.

    Remark 7.4. Assume that X is a terminal {\mathbb Q} -factorial threefold and W\rightarrow X , W'\rightarrow X are two different P-minimal resolutions. We know that W and W' can be connected by P-desingularizations of flops. Let W_i\dashrightarrow W_{i+1} be a P-desingularization of a flop X_i\dashrightarrow X_{i+1} which appears in the factorization of W\dashrightarrow W' . Then, from the construction we know that gdep(X_i) < gdep(X) .

    Now we compare an arbitrary resolution of singularities to a P-minimal resolution.

    Definition 7.5. Let W\dashrightarrow X be a birational map where W is a smooth threefold and X is a terminal threefold. We say that the birational map has a bfw-factorization if W\dashrightarrow X can be factorized into a composition of smooth blow-downs, P-desingularizations of flops, and strict w -morphisms.

    Remark 7.6. If X_2\rightarrow X_1 is a strict w -morphism and X_1\dashrightarrow X is a smooth blow-down or a P-desingularization of a flop, then on X_1 the indeterminacy locus of X_1\dashrightarrow X is disjoint to the indeterminacy locus of X_1\dashrightarrow X_2 since the former one lies on the smooth locus of X_1 and the latter one is a singular point. Hence, there exists X_2\dashrightarrow X'_1\rightarrow X where X_2\dashrightarrow X'_1 is a smooth blow-down or a P-desingularization of a flop, and X'_1\rightarrow X is a strict w -morphism. In other words, W\dashrightarrow X has a bfw-factorization if and only if there exists a birational map W\dashrightarrow \bar{X} which is a composition of smooth blow-downs and P-desingularization of flops, where \bar{X} is a feasible resolution of X .

    Proposition 7.7. Assume that a birational map W\dashrightarrow X has a bfw-factorization where W is a smooth threefold and X is a terminal threefold.

    (1) If X\dashrightarrow X' is a flop, then there is a birational map W\dashrightarrow X' which has a bfw-factorization.

    (2) If Y\rightarrow X is a strict w -morphism, then there exists a birational map W\dashrightarrow Y which also has a bfw-factorization.

    (3) If X\dashrightarrow X' is a flip or a divisorial contraction, then the induced birational map W\dashrightarrow X' has a bfw-factorization.

    Proof. Assume first that X\dashrightarrow X' is a flop. By Remark 7.6 we know that there exists a bfw-map W\dashrightarrow \bar{X} , where \bar{X} is a feasible resolution of X . Let \bar{X}'\rightarrow X' be a feasible resolution of X' . Then \bar{X}'\rightarrow X' is a composition of strict w -morphisms and the induced birational map \bar{X}\dashrightarrow \bar{X}' is a P-desingularization of the flop X\dashrightarrow X' . It follows that the composition

    W\dashrightarrow \bar{X}\dashrightarrow \bar{X}'\rightarrow X'

    is a bfw-map. This proves (1).

    We will prove (2) and (3) by induction on gdep(X) . If gdep(X) = 0 , then X is smooth. In this case, there is no strict w -morphism Y\rightarrow X or flip X\dashrightarrow X' . Assume that X\rightarrow X' is a divisorial contraction. If X' is smooth, then it is a smooth blow-down by [22, Theorem 3.3, Corollary 3.4], and if X' is singular, then X\rightarrow X' should be a strict w -morphism since in this case X' is terminal {\mathbb Q} -factorial and X is a P-minimal resolution of X' . Now we may assume that gdep(X) > 0 and statements (2) and (3) hold for threefolds with generalized depth less then gdep(X) .

    Let

    W = X_k\dashrightarrow X_{k-1}\dashrightarrow ...\dashrightarrow X_1\dashrightarrow X_0 = X

    be a sequence of birational maps so that X_{i+1}\dashrightarrow X_i is a smooth blow-down, a P-desingularization of a flop, or a strict w -morphism for all 1\leq i\leq k-1 . By Remark 7.6 we can assume that X_1\rightarrow X is a strict w -morphism. Now, given a strict w -morphism Y\rightarrow X , if Y\cong X_1 , then there is nothing to prove. Otherwise, by Corollary 5.7 there exists a sequence of strict w -morphisms Y_2\rightarrow X , ..., Y_{m-1}\rightarrow X such that

    X_1 = Y_1\underset{{X}}{\Rightarrow }Y_2\underset{{X}}{\Rightarrow }...\underset{{X}}{\Rightarrow }Y_{m-1}\underset{{X}}{\Rightarrow }Y_m = Y.

    For each 1\leq i\leq m-1 , one has the factorization

    such that Z_{i, 1}\dashrightarrow Z_{i, k_i} is a composition of flops, Z_{i, 1}\rightarrow Y_i is a strict w -morphism, and

    gdep(Z_{i, k_i}) = gdep(Z_{i, 1}) < gdep(Y_i) < gdep(X).

    By the induction hypothesis, we know that if there exists a bfw-map W\dashrightarrow Y_i , then there exists a bfw-map W\dashrightarrow Y_{i+1} . Now one can prove statement (2) by induction on m .

    Assume that X\dashrightarrow X' is a flip. Then we have a factorization

    as in Theorem 2.18. Since Y_1\rightarrow X is a strict w -morphism by Corollary 5.8, there exists a bfw-map W\dashrightarrow Y_1 . Since

    gdep(Y_k)\leq ...\leq gdep(Y_1) < gdep(X),

    the induction hypothesis implies that there exists a bfw-map W\dashrightarrow X' .

    Finally, assume that X\rightarrow X' is a divisorial contraction. If it is a smooth blow-down or a strict w -morphism, then there is nothing to prove. Otherwise, there exists a diagram

    such that Y_1\rightarrow X is a strict w -morphism and Y_i\dashrightarrow Y_{i+1} is a flip or a flop for all 1\leq i\leq k-1 . One has that

    gdep(Y_k)\leq ...\leq gdep(Y_1) < gdep(X),

    hence there exists a bfw-map W\dashrightarrow Z . If X\rightarrow X' is a divisorial contraction to a curve, then Y_k\rightarrow Z is a divisorial contraction to a curve as in Theorem 2.18. In this case, we also have gdep(Z)\leq gdep(Y_k) < gdep(X) , so there exists a bfw-map W\dashrightarrow X' . If X\rightarrow X' is a divisorial contraction to a point, then the discrepancy of Z\rightarrow X' is less than the discrepancy of X\rightarrow X' unless X\rightarrow X' is a w -morphism. Also, when X\rightarrow X' is a w -morphism, we know that gdep(Z) < gdep(X) by Lemma 5.6. Thus, we can prove statement (3) by induction on the generalized depth and the discrepancy of X over X' .

    One can easily see the following corollary:

    Corollary 7.8. Assume that W is a smooth threefold and W\dashrightarrow X is a birational map which is a composition of steps of MMP. Then, this birational map can be factorized into a composition of smooth blow-downs, P-desingularizations of flops, and strict w -morphisms.

    Proof of Theorem 1.2. By Corollary 7.8 and Remark 7.6 we know that there exists a feasible resolution \tilde{{X}}_W\rightarrow X_W such that W\dashrightarrow \tilde{{X}}_W is a composition of smooth blow-downs and P-desingularizations of flops, where X_W is a minimal model of W over X . By Proposition 7.3 we know that \tilde{{X}}_W is also a P-minimal resolution of X , hence the birational map \tilde{{X}}_W\dashrightarrow \tilde{{X}} is connected by P-desingularizations of flops. Thus, the composition W\dashrightarrow \tilde{{X}}_W\dashrightarrow \tilde{{X}} is connected by smooth blow-downs and P-desingularizations of flops.

    Proof of Corollary 1.3. Let

    W = \tilde{{X}}_k\dashrightarrow ... \dashrightarrow \tilde{{X}}_1\dashrightarrow \tilde{{X}}_0 = \tilde{{X}}

    be a sequence of smooth blow-downs and P-desingularization of flops as in Theorem 1.2. We only need to show that if \tilde{{X}}_{i+1}\dashrightarrow \tilde{{X}}_i is a P-desingularization of a flop X_{i+1}\dashrightarrow X_i , then b_j(\tilde{{X}}_{i+1}) = b_j(\tilde{{X}}_i) for all j = 0 , ..., 6 .

    By [24, Lemma 2.12] we know that b_j(X_{i+1}) = b_j(X_j) for all j . Since X_i and X_{i+1} have the same analytic singularities [17, Theorem 2.4], there exists a feasible resolution \tilde{{X}}'_{i+1}\rightarrow X_{i+1} such that b_j(\tilde{{X}}_i) = b_j(\tilde{{X}}'_{i+1}) for all j . Now, \tilde{{X}}'_{i+1} and \tilde{{X}}_{i+1} are two different P-minimal resolutions of X_{i+1} , so they can be connected by P-desingularizations of flops with smaller generalized depth by Remark 7.4. By induction on the generalized depth, one can see that b_j(\tilde{{X}}'_{i+1}) = b_j(\tilde{{X}}_{i+1}) . Hence, b_j(\tilde{{X}}_{i+1}) = b_j(\tilde{{X}}_i) for all j = 0 , ..., 6 .

    This section is dedicated to exploring minimal resolutions for singularities in higher dimensions and the potential applications of our main theorems.

    In three dimensions, P -minimal resolutions appear to be a viable generalization of minimal resolutions for surfaces. However, in higher dimensions, P -minimal resolutions are not good enough. For example, let X\dashrightarrow X' be a smooth flip (eg. a standard flip [1, Section 11.3]). Then, X and X' are both P -minimal resolutions of the underlying space, but X' is better than X . It is reasonable to assume that X' is a minimal resolution, while X is not. Inspired by Corollary 1.3, we define a new kind of minimal resolution:

    Definition 8.1. Let X be a projective variety over complex numbers. We say that a resolution of singularities W\rightarrow X is a B -minimal resolution if for any smooth model W'\rightarrow X one has that b_i(W)\leq b_i(W') for all 0\leq i\leq 2\dim X .

    As stated in Corollary 1.3, B-minimal resolutions coincide with P-minimal resolutions in dimension three. Our main theorems say that B-minimal resolutions of threefolds satisfy certain nice properties. It is logical to anticipate that B-minimal resolutions of higher-dimensional varieties share similar properties.

    Conjecture. For any projective variety X over the complex numbers, one has that:

    (1) B-minimal resolutions of X exist.

    (2) Two different B-minimal resolutions are connected by desingularizations of {\mathbb Q} -factorial terminal flops.

    (3) If \tilde{{X}}\rightarrow X is a B-minimal resolution and W\rightarrow X is an arbitrary resolution of singularities, then W\dashrightarrow \tilde{{X}} can be connected by smooth blow-downs, smooth flips, and desingularizations of {\mathbb Q} -factorial terminal flops.

    Let

    \mathcal{X}_3 = \{\mbox{ smooth threefolds }\}/\sim,

    where W_1\sim W_2 if W_1\dashrightarrow W_2 is connected by P-desingularizations of Q -factorial terminal flops. For \eta_1 , \eta_2\in \mathcal{X}_3 we say that \eta_1 > \eta_2 if there exist W_1 and W_2 so that \eta_i = [W_i] and W_1\rightarrow W_2 is a smooth blow-down. Then, Theorems 1.1 and 1.2 imply the following.

    Corollary 8.2. Given a threefold X , let

    \mathcal{X}_{3, X} = \left\lbrace {[W]\in \mathcal{X}_3} | {{\mbox{There exists a birational morphism }}W\rightarrow X} \right\rbrace.

    Then \mathcal{X}_{3, X} has a unique minimal element.

    In other words, if we consider the resolution of singularities inside \mathcal{X}_3 , then there is a unique minimal resolution, which behaves similarly to the minimal resolution of a surface.

    As a consequence, inside the space \mathcal{X}_3 the following strong factorization theorem holds.

    Theorem 8.3 (Strong factorization theorem for \mathcal{X}_3 ). Assume that W_1 and W_2 are smooth threefolds which are birational to each other. Then there exists a smooth threefold \bar{W} such that inside \mathcal{X}_3 one has [\bar{W]}).\geq[W_i] for i = 1 , 2 .

    Proof. Let W_1\leftarrow \bar{W}\rightarrow W_2 be a common resolution. Then [\bar{W}]).\in \mathcal{X}_{3, W_i} for i = 1 , 2 . Since the minimal element of \mathcal{X}_{3, W_i} is [W_i] itself, one has that [\bar{W]}).\geq[W_i] for i = 1 , 2 .

    One can characterize a surface singularity by the information of exceptional curves on the minimal resolution. One may ask, does a similar phenomenon happen for higher-dimensional singularities? Since for higher-dimensional singularities there is no unique minimal resolution, what we really want to study is the following object.

    Definition 8.4. Let X be a projective threefold over the complex numbers. We say that a divisorial valuation v_E over X is an almost essential valuation if for any P-minimal resolution \tilde{{X}}\rightarrow X one has that \mbox{Center}_{\tilde{{X}}}E is an irreducible component of the exceptional locus of \tilde{{X}}\rightarrow X .

    This name comes from the "essential valuation" in the theories of arc spaces.

    Definition 8.5. Let X be a variety. We say that a divisorial valuation v_E over X is an essential valuation if for any resolution of singularities W\rightarrow X one has that \mbox{Center}_{W}E is an irreducible component of the exceptional locus of W\rightarrow X .

    From the definition, one can see that essential valuations are almost essential, but an almost essential valuation may not be essential.

    Example 8.6. Let X = (xy+z^2+u^{2n+1})\subset {\mathbb A}^4 for some n > 2 . There is exactly one w -morphism X_1\rightarrow X over the singular point, which is obtained by blowing-up the origin. There is only one singular point on X_1 , which is defined by xy+z^2+u^{2(n-1)+1} . Blowing-up the singular point n-1 more times, we get a resolution of singularities \tilde{{X}}\rightarrow X . From the construction we know that \tilde{{X}} is a unique feasible resolution of X . Since X is terminal and {\mathbb Q} -factorial, \tilde{{X}} is the unique P-minimal resolution of X . Hence, almost essential valuations of X are those divisorial valuations which appear on \tilde{{X}} . One can compute that exc(\tilde{{X}}\rightarrow X) = E_1\cup...\cup E_n such that v_{E_i}(x, y, z, u) = (i, i, i, 1) . On the other hand, by [25, Lemma 15] we know that essential valuations of X are v_{E_1} and v_{E_2} . Hence, v_{E_3} , ..., v_{E_n} are almost essential valuations which are not essential.

    Notice that the set of essential valuations does not really characterize the singularity since it is independent of n . The set of almost essential valuations carries more information of the singularity.

    Let X be a smooth variety. The bounded derived category of coherent sheaves of X , denoted by D^b(X) , is an interesting subject of investigation. One possible method to study D^b(X) is to construct a semi-orthogonal decomposition of D^b(X) (refer to [26] for more information). Orlov [27] proved that a smooth blow-down yields a semi-orthogonal decomposition. In particular, if X is a smooth surface, then the K_X -MMP is a series of smooth blow-downs, thereby resulting in a semi-orthogonal decomposition of D^b(X) .

    Now assume that X is a smooth threefold and let

    X = X_0\dashrightarrow X_1\dashrightarrow ...\dashrightarrow X_k

    be the process of K_X -minimal model program. According to Corollary 7.8 and Remark 7.6, \tilde{{X}}_i\dashrightarrow \tilde{{X}}_{i+1} can be factored into a composition of smooth blow-downs and P-desingularizations of flops, where \tilde{{X}}_i\rightarrow X_i is a P-minimal resolution of X_i . If every P-desingularizations of flops that appears in the factorization is a smooth flop, then the sequence induces a semi-orthogonal decomposition of D^b(X) since smooth flops are derived equivalent [28].

    Example 8.7. Let X_1\dashrightarrow X_2 be the flip which is a quotient of an Atiyah flop by an {\mathbb Z}/2 {\mathbb Z} -action [16, Example 2.7]. Then X_2 is smooth and X_1 has a \frac{1}{2}(1, 1, 1) singular point. Let X\rightarrow X_1 be the smooth resolution obtained by blowing-up the singular point. Then X\rightarrow X_1\dashrightarrow X_2 is a sequence of MMP.

    The factorization of the flip is exactly diagram (3) in Lemma 6.7, namely the diagram

    where X\dashrightarrow X' is a smooth flop and X'\rightarrow X_2 is a blow-down of a smooth curve. We know that there exists an equivalence of category \Phi: D^b(X')\rightarrow D^b(X) and a semi-orthogonal decomposition D^b(X') = \langle {D_{-1}, D^b(X_2)}\rangle . Hence, D^b(X) = \langle {\Phi(D_{-1}), \Phi(D^b(X_2))}\rangle is a semi-orthogonal decomposition.

    In general, a P-desingularization of a flop \tilde{{X}}_i\dashrightarrow \tilde{{X}}_{i+1} may not be derived equivalent since \tilde{{X}}_i and \tilde{{X}}_{i+1} may not be isomorphic in codimension one. Nevertheless, due to the symmetry between \tilde{{X}}_i and \tilde{{X}}_{i+1} , one might expect that a semi-orthogonal decomposition on D^b(\tilde{{X}}_i) will result in a semi-orthogonal decomposition on D^b(\tilde{{X}}_{i+1}) . It still hopeful that our approach will be effective for all smooth threefolds.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    The author thanks Jungkai Alfred Chen for his helpful comments. The author thanks the referees for carefully reading the paper and for their useful comments. The author is supported by KIAS individual Grant MG088901.

    The authors declare there is no conflicts of interest.



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