1.
Introduction
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 ˉS→S such that ρ(ˉS/S) is minimal. The minimal resolution ˉS is unique, and any birational morphism S′→S from a smooth surface S′ to S factors through ˉS→S.
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 X⇢X′ 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 X⇢X′ as a possibly singular flop over W. Let ˜X→X and ~X′→X′ 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 X⇢X′ (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 W→X is a birational morphism from a smooth threefold W to X. Then, for any P-minimal resolution ˜X of X, one has a factorization
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 W→X 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 X⇢X′ 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 X⇢X′ 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 W→X be a resolution of singularities. One can run KW-MMP over X as
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 W→X 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
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.
2.
Preliminaries
2.1. Notation and conventions
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 U⊂X and V⊂Y such that U and V are isomorphic. If X and Y are birational, we say that ϕ:X⇢Y is a birational map, and we will denote ϕ|U to be the isomorphism U→V. If ϕ:X→Y is a morphism between X and Y and there exists a Zariski open set U⊂X 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 Y→X 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 OW→1rZ≥0 such that
Assume that ϕ:X⇢Y is a birational map. Let U⊂X be the largest open set such that ϕ|U is an isomorphism and Z⊂X be an irreducible subset such that Z intersects U non-trivially. We will denote by ZY the closure of ϕ|U(Z|U).
2.2. Weighted blow-ups
Let W=An and G be a finite cyclic group, such that ˉW=W/G≅An(x1,...,xn)/1r(a1,...,an). There is an elementary way to construct a birational morphism W′→ˉW, 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,v⟩Z, where e1, ..., en is the standard basis of Rn and v=1r(a1,...,an). Let σ=⟨e1,...,en⟩R≥0. We have ˉW≅SpecC[N∨∩σ∨].
Let w=1r(b1,...,bn) be a vector such that bi=λai+kir for λ∈N and ki∈Z with bi≠0. We define a weighted blow-up of ˉW with weight w to be the toric variety defined by the fan consisting of the cones
Let Ui be the toric variety defined by the cone σi and lattice N, namely
Lemma 2.1. One has that
where τ is the action given by
and τ′ is the action given by
Here, ξk denotes a k-th roots of unity for any positive integer k.
In particular, the exceptional divisor of W′→ˉW 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 j≠i and Tiw=ei. One can see that
and
Under this linear transformation, σi becomes the standard cone ⟨e1,...,en⟩R≥0. Note that
Hence, ei∈TiN and TiN=⟨e1,...,en,Tiei,Tiv⟩Z. Now Tiei corresponds to the action τ and Tiv corresponds to the action τ′. This means that Ui≅An/⟨τ,τ′⟩.
The computation above shows that τki=τ′λ. If we glue (xi=0)⊂An/⟨τ⟩ together, then we get a weight projective space P(b1,...,bn). The relation τki=τ′λ 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 Ui→ˉW 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
is a complete intersection and S′ is the proper transform of S on W′. Assume that the exceptional locus E of S′→S is irreducible and reduced. Then
Proof. Assume first that k=0. Denote ϕ:W′→ˉW. Then, on Ui, we have
hence KW′=ϕ∗KˉW+(b1+...+bnr−1)F where F=exc(W′→ˉW).
Now the statement follows from the adjunction formula. □
Corollary 2.4. Let F=exc(W′→ˉW). Then
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
□
Definition 2.5. Let ϕi:Ui→ˉW be the morphism in Corollary 2.2. For any G-semi-invariant function u∈OW, we can define the strict transform of u on Ui by (ϕ−1i)∗(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
We say that Y→X 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 Y→X 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
is a regular function on An. We denote
2.3. Terminal threefolds
2.3.1. Local classification
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 P∈X 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 P∈X be a point of threefold. Then P∈X is an isolated compound Du Val point if and only if P∈X is terminal and KX is Cartier near P.
Theorem 2.10 ([6, cf. [7,Theorem 6.1]). Let P∈X be a germ of three-dimensional terminal singularity such that KX has Cartier index r>1. Then
such that f, r and ai are given by Table 1.
Assume that P∈X 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.
2.3.2. Classification of divisorial contractions to points
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 Y→X is a divisorial contraction to a point between terminal threefolds. Then there exists an embedding X↪W 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 Y→X is a weighted blow-up with respect to w.
The defining equation of X⊂W and the weight are given in Table 2 to Table 11.
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 g≥m represents a function g∈OW such that w(g)=m. The notation pm represents a function p∈OW 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 (o∈X) be a germ of three-dimensional Gorenstein terminal singular point with type cD or cE.
Step1: Construct a divisorial contraction X1→X which contracts an exceptional divisor of discrepancy one to o. We refer to [2, Section 4, Section 6] for the explicit construction. X1→X 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(X1→X) is an exceptional divisor of discrepancy one. Assume that E≠exc(X1→X). Then an easy computation on discrepancies shows that a(E,X1)<1. In particular, CenterX1E is a non-Gorenstein point. Since X1→X 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 E∈S. 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 E∈S, the valuation of E on X can be calculated. One can construct a weighted blow-up YE→X with respect to this valuation. If YE do not have terminal singularities, then we remove E form S. Now, YE→X for all E∈S, together with X1→X, are all divisorial contractions to o with discrepancy one.
2.3.3. The depth
Definition 2.12. Let Y→X be a divisorial contraction which contracts a divisor E to a point P. We say that Y→X 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 P∈X, dep(P∈X), is the minimal length of the sequence
such that Xm is Gorenstein and Xi→Xi−1 is a w-morphism for all 1≤i≤m.
The generalized depth of a terminal singularity P∈X, gdep(P∈X), is the minimal length of the sequence
such that Xn is smooth and Xi→Xi−1 is a w-morphism for all 1≤i≤n. The variety Xn is called a feasible resolution of P∈X.
The Gorenstein depth of a terminal singularity P∈X, depGor(P∈X), is defined by gdep(P∈X)−dep(P∈X).
For a terminal threefold we can define
and
Remark 2.14. In the above definition, the existence of a sequence
such that Xm is Gorenstein follows from [10, Theorem 1.2]. The existence of a sequence
such that Xn is smooth follows from [2, Theorem 2].
Definition 2.15. Assume that Y→X is a w-morphism such that gdep(Y)=gdep(X)−1. Then we say that Y→X is a strict w-morphism.
Lemma 2.16. Assume that Y→X is a divisorial contraction which is a weighted blow-up with the weight w(x1,...,xn)=1r(a1,...,an) with respect to an embedding X↪An(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 CenterYE∩Ui non-trivially if and only if biai≤bjaj for all 1≤j≤n. 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:
One can see that
We know that CenterEY intersects Ui non-trivially if and only if bir−ajbirai≥0 for all j≠i, or, equivalently, bjaj≥biai for all j≠i. □
Corollary 2.17. Assume that Y→X and Y1→X are two different w-morphisms over the same point. Let E and F be the exceptional divisors of Y→X and Y1→X, respectively. Then there exists u∈OX such that vE(u)<vF(u).
Proof. Let X→An(x1,...,xn)/G be the embedding so that Y1→X 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
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 CenterY1E∩Un 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. □
2.4. Chen-Hacon factorizations
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 X⇢X′ is a flip over V, or X→V 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 Y1→X is a w-morphism, Yk→X′ is a divisorial contraction, Y1⇢Y2 is a flip or a flop, and Yi⇢Yi+1 is a flip for i>1. If X→V is divisorial, then Yk→X′ is a divisorial contraction to a curve and X′→V 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 Y1⇢Y2. Then CX is a flipping curve of X⇢X′. 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 X→V contains a non-Gorenstein point P which is not a cA/r or a cAx/r point. Then Y1→X 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 Y→X is a divisorial contraction to a point, then dep(Y)≥dep(X)−1.
2. If Y→X is a divisorial contraction to a curve, then dep(Y)≥dep(X).
3. If X⇢X′ is a flip, then dep(X)>dep(X′).
2.5. The negativity lemma
We have the following negativity lemma for flips.
Lemma 2.21. Assume that X⇢X′ 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 X⇢X′ is a (KX+D)-flip and C⊂X 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 Xϕ⟵Wϕ′→X′ 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,
The last inequality is strict if and only if CW intersects F non-trivially, or, equivalently, C intersects the flipping locus non-trivially. □
3.
Factorize divisorial contractions to points
Let Y→X be a divisorial contraction between Q-factorial terminal threefolds that contracts a divisor E to a point. We construct the diagram
as follows: Let Z1→Y 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(Z1→Y). Thus, after finitely many (KZ1+HZ1+ϵEZ1)-flips Z1⇢...⇢Zk, the MMP ends with a divisorial contraction Zk→Y1 which contracts EZk, and Y1→X 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 Zi⇢Zi+1 is a KZi-flip or flop for all i and Zk→Y1 is a KZk-divisorial contraction. In particular, Y1, Z2, ..., Zk are all terminal.
Proof. Assume first that k=1. If Z1→Y1 is a KZ1-negative contraction, then we are done. Otherwise, Z1→Y1 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 Z1⇢Z2 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 Z1⇢Z2 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 Z2→Y1 is a KZ2-divisorial contraction, and for k>2, Z2⇢Z3 is a KZ2-flip. One can prove the statement by repeating this argument k−2 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 X↪An(x1,...,xn)/G=W/G be the embedding such that Y→X 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 Z1→Y→X can be viewed as a sequence of weighted blow-ups with respect to the embedding X↪W/G.
Let V be a suitable open set which contains P=CenterYF such that Z1→Y can be viewed as a weighted blow-up with respect to an embedding V↪An1(y′1,...,y′n1)/G′. For j=1, ..., n1, we define Dj⊂V to be the Weil divisor corresponding to y′j=0. Then Dj,X=ϕ∗Dj is a Weil divisor on X. Since X is Q-factorial, Dj,X corresponds to a G-semi-invariant function sj∈OW. We can consider the embedding
Then, Y→X 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 X↪ˉW is exactly what we need.
Now, let W′→W be the first weighted blow-up. We may assume P=CenterYF is the origin of Ui⊂W′. 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 X⊂W/G. Then f′4, ..., f′n define Y|Ui where f′i=(ϕ|−1Ui)∗(fi). Since Y has terminal singularities, the weighted embedding dimension of Y|Ui near P is less than 4. For 5≤j≤n, we may write f′j=ξjyj+f′j(y1,...y4) for some ξj which does not vanish on P. One can always assume that HY=(y3=0) and so i≠3.
Let f′j∘=f′j|yi=y3=0. Then f′4∘, ..., f′n∘ defines H∩E near P. If f′j∘ is irreducible as a G′-semi-invariant function, then we let η′j=f′j∘. Otherwise, let η′j be a G′-semi-invariant irreducible factor of f′j∘.
Lemma 3.2. Assume that Y→X can be viewed as a four-dimensional weighted blow-up. Then η′4=...=η′n=0 defines an irreducible component of HY∩E.
Proof. Since Y→X can be viewed as a four-dimensional weighted blow-up, we know that i≤4 and f′j∘=yj+f′j|y3=yi=0 for all j>4. Hence, we have η′j=f′j∘. One can see that the projection
is an isomorphism. Since η′4 is an irreducible function, it defines an irreducible curve. □
Notice that η′j is a polynomial in y1, ..., yn. There exists ηj∈OW such that η′j=(ϕ|−1Ui)∗(ηj). We assume that Y→X is a weighted blow-up with the weight 1r(a1,...,an) and Z1→Y is a weighted blow-up with the weight 1r′(a′1,...,a′n).
Lemma 3.3. Let Γ=(η′4=...=η′n=0) and assume that Γ is an irreducible and reduced curve. Then
Here, m is the integer in Lemma 2.1 corresponding to the weighted blow-up Y→X.
Proof. Since Γ⊂E and KZ1+HZ1 is numerically trivial over X, we only need to show that
We know that HZ1.ΓZ1=H.Γ−vF(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 Y⊂W′⊂PW(a1,...,an). Let Dj be the divisor on W′ which corresponds to η′j. 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
To compute F.ΓZ1, one writes
Since Z1→Y is a w-morphism, the integer λ in Section 2.2 is 1. Hence, we know that Fn=(−1)n−1r′n−1a′1...a′n. Now, vF(E)=a′ir′ and vF(HY)=a(Y,F)=1r′, so
□
Lemma 3.4. Notation and assumption as in Lemma 3.3. Assume that:
(ⅰ) For all 4≤j≤n, 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 j≠i, 3, δ4, ..., δn, then a3a′j≥aj.
Then KZ1.ΓZ1≤0.
Proof. Fix j≥4. From the construction and our assumption we know that that i, δ4, ..., δn are all distinct. One can see that rvE(ηj)=kjaδj and r′vF(η′j)≤kja′δj. Thus, we have a relation
One can always assume that if j>4, j≠i, 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
and
Since mai=r′, a3a2≥1a′2 and rvE(ηj)aδj≥r′vF(η′j)a′δj, we know that
So KZ1.ΓZ1≤0. □
Remark 3.5.
(1) From the construction we know that if j≥5, j≠i, then one can choose δj=j.
(2) If Y→X 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 i≤4, so η′4 is a two-variable irreducible function, hence there exists δ4≤4 such that yk4δ4∈η′4 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 j≠i, 3, δ4, ..., δn one has that aj≤ai, then condition (ⅱ) of Lemma 3.4 holds. Indeed, by Lemma 2.1 we know that Ui≅An/⟨τ,τ′⟩ where τ corresponds to the vector v=1ai(a1,...,ai−1,−r,ai+1,an). Let ˉv be the vector corresponding to the cyclic action near P∈Ui. Then v≡mˉv (mod Zn) and r′=mai. Since Z1→Y is a w-morphism, and since HY is defined by y3=0, we know that a′3=1 and ˉv≡a31r′(a′1,...,a′n) (mod Zn). One can see that a3a′j≡aj (mod r′). This implies that a3a′j≥aj since
Lemma 3.6. Assume that KZ1 is anti-nef over X and there exists u∈OX such that vE(u)<a(E,X)a(F,X)vF(u). Then Zi⇢Zi+1 is a KZi-flip or flop for all 1≤i≤k−1 and Zk→Y1 is a terminal divisorial contraction.
In particular, if there exists j≠i such that a3a′j>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 Z1→Y1 is a KZ1-trivial divisorial contraction. Notice that if CY⊂E is a curve which does not contain P, then KZ1.CZ1=KY.CY<0, hence the curve CZ1 is not contracted by Z1→Y1. Thus, Z1→Y1 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 u∈OX, one has that
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 Zk→Y1 is a terminal divisorial contraction.
Now, by Lemma 3.7, we know that
Consider u=xj. For j≠i we know that vE(xj)=ajr and
The inequality vE(u)≥a(E,X)a(F,X)vF(u) becomes
or, equivalently,
This is equivalent to
Hence, the condition a3a′j>aj implies that vE(u)<a(E,X)a(F,X)vF(u). □
Lemma 3.7. One has that
Proof. Since a(E,X,H)=0, we know that a(E,X)=vE(H)=a3r. Then
□
Lemma 3.8. Note that the assumption in Lemma 3.4 depends only on the first weighted blow-up Y→X. In other words, we can check whether the assumption holds or not by simply considering the embedding which defines the weighted blow-up Y→X instead of considering the (possibly) larger embedding which defines both Y→X and Z1→Y. Likewise, to apply Lemma 3.6, we can simply look at the embedding that defines Y→X, if condition a3a′j>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 j≠i, 3, δ4, ..., δn such that a3a′j>aj, or there exists j≥4 such that
(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 a3a′j>aj. In either case, Lemma 3.6 can be applied.
Notation 3.10. We say that a divisorial contraction Y→X is linked to another divisorial contraction Y1→X if the diagram
exists, where Z1→Y is a strict w-morphism over a non-Gorenstein point, Zk→Y1 is a divisorial contraction, and Zi⇢Zi+1 is a flip or a flop for all 1≤i≤k−1. We use the notation Y⇒XY1 if Y→X is linked to Y1→X.
Furthermore, if all Zi⇢Zi+1 are all flips, or k=1, then we say that Y is negatively linked to Y1, and use the notation Y−⇒XY1.
Remark 3.11. At this point, it is not clear why Z1→Y 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 Y→X and Y1→X, 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 Y⇒XY1.
(2) Assume that (Ξ−) or (Ξ′−) holds and (Θu) or (Θj) holds for some function u or index j. Then one has that Y−⇒XY1.
Lemma 3.13. Assume that
such that
(1) vE(g)=a1r+vE(p)=2a1r−1.
(2) i=1, a2+a4=a1 and a3=1.
Then Y−⇒XY1.
Proof. We know that a1>aj for j=2, ..., 4, so (Ξ′) holds. Consider the embedding
Then Y→X 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 a′5=rvE(g)>rvE(p)=a5, hence (Θ5) holds. Moreover, one can see that η′5=y5−p(y2,0,y4), so r′vF(η′5)=rvE(η5)=rvE(p(x2,0,x4)), hence
Thus, Y−⇒XY1. □
Lemma 3.14. Assume that Y→X and Y1→X are two divisorial contractions such that Y⇒XY1. Let E and F be the exceptional divisors of Y→X and Y1→X, respectively. Assume that there exists u∈OX 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
Since vE(u)≥1r and vF(˜u)>0, we know that vF(E)<1. Thus
where r′ is the Cartier index of CenterYF. Hence, a(F,X)<a(E,X) if a(E,X)>1. □
4.
Constructing links
The aim of this section is to prove the following proposition:
Proposition 4.1. Let X be a terminal threefold and Y→X, Y′→X be two different divisorial contractions to points over X. Then there exists Y1, ..., Yk, Y′1, ..., Y′k′ such that
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.
4.1. Divisorial contractions to cA/r points
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 Y→X is of type A1 with a>1, then Y−⇒XY1 for some Y1→X which is of type A1 with the discrepancy less than a.
(2) If Y→X is of type A1 with a=1 and b>r, then Y⇒XY1 where Y1 is an A1 type weighted blow-up with the weight 1r(b−r,c+r,1,r). One also has Y1⇒XY if we begin with Y1→X and interchange the role of x and y. Moreover, Y⧸−⇒XY1 if and only if η4=y.
(3) If Y→X is of type A2, then Y−⇒XY1 where Y1→X is a divisorial contraction of type A1.
Proof. Assume first that Y→X 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 Ux⊂Y 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
Since Ux is terminal, there exists a vector τ"=1b(b−δ,ϵ,1,δ) such that τ≡aτ" (mod Z4) and τ′≡λ′τ" (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
where g′ is the strict transform of g on Ux, since if zrpuq∈g′, then ap+q≥ak and
for δa≥r because δa≡r(mod b) and b>r. Thus,
hence (Θ4) holds, and there exists Y1→X such that Y⇒XY1.
We need to check whether (Ξ′−) holds or not. We have that f′4∘=η′4=y4+g′∘. One always has that
Now, g′∘=0 if and only if
and δa=r if and only if
Thus, (Ξ′−) holds if and only if g′∘≠0 or a do not divide r.
One can compute the discrepancy of Y1→X using Lemma 3.7. We know that a′i=b−δ and a3=a, so
If a=1, then r=δ, so a(F,X)=a(E,X)=1r. One can verify that Y1→X is the weighted blow-up with the weight 1r(b−r,c+r,1,r). In this case, Y−⇒XY1 if and only if g′∘≠0. Hence, Y⧸−⇒XY1 if and only if η4=y. This proves (2).
Now assume that a>1. We already know that δa≥r. If δa>r, then a(F,X)<a(E,X) and Y−⇒XY1, so (1) holds. Hence, one only needs to show that δa≠r. If δa=r, then
where λ′=b−aβr. One can see that b−β=(a−1)β+λ′aδ. On the other hand, since τ′≡λ′τ" (mod Zn), we know that b−β≡λ′δ (mod b). Hence, b divides
This is impossible since (a−1)(β+λ′δ) is a positive integer and is less than b.
Finally, assume that Y→X of type A2. In this case, one needs to look at the chart Ux⊂Y, 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
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,x∓y,z,u)=14(5,1,2,3). For both these two w-morphisms, one has that a2=2, a3=3 and a′2=2, so (Θ2) holds and (Ξ−) holds since a3a′2>a2. Thus, there exists Y1→X so that Y−⇒XY1. One can compute that the discrepancy of Y1→X is one, so Y1→X is of type A1. This proves (3). □
4.2. Divisorial contractions to cAx/r points
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 Y→X is of type Ax1 or Ax3. Then Y→X is the only divisorial contraction over X.
(2) Assume that Y→X is of type Ax2 or Ax4. Then there are exactly two divisorial contractions over X. Let Y1→X be another divisorial contraction. Then Y1→X has the same type of Y→X, and one has that Y−⇒XY1−⇒XY.
Proof. The number of divisorial contractions follows from [9, Section 7, 8]. So we can assume that Y→X is of type Ax2 or Ax4, and Lemma 3.13 implies that Y−⇒XY1. □
4.3. Divisorial contractions to cD points
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 Y→X and Y1→X are the two given divisorial contractions. It is enough to prove the following statements:
(ⅰ) If Y→X is of type D4, then Y1→X is of type D1 or D4.
(ⅱ) If Y→X is of type D2, then Y1→X is of type D1 or D5.
(ⅲ) If Y→X is of type D5, then Y1→X is not of type D3.
Let E and F be the exceptional divisors of Y→X and Y1→X, respectively. Then a(F,Y)<1, since otherwise a(F,X)>1. Thus, P=CenterFY is a non-Gorenstein point.
First, assume that Y→X 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, Y1→X is also of type D4 after a change of coordinates x↦x−p(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 Y1→X is of type D1.
Now, assume Y→X is of type D2. Then P is the origin of Uy⊂Y, 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 λ≠0 and k=b, then vF(u)=1. In this case, Y1→X is of type D5. Otherwise, vF(u)=2, and so Y1→X is of type D1.
Finally, assume that Y→X is of type D5. One can see that Y1→X cannot have type D3 since zb∈p(z,u). This finishes the proof. □
Proposition 4.5. Assume that there exist two different divisorial contractions with discrepancy one over X, say Y→X and Y1→X.
(1) If Y→X and Y1→X are both of type D4, then Y−⇒XY1−⇒XY.
(2) If Y→X is of type D3 and Y1→X is of type D1, then Y−⇒XY1. If Y1→X is of type D3, then there exists another divisorial contraction Y2→X which is of type D1 so that Y−⇒XY2−⇐XY1.
(3) If Y→X is of type D2 and Y1→X is of type D5, then Y−⇒XY1.
(4) If Y→X is of type D1 and Y1→X is not of type D3, then Y−⇒XY1.
Proof. Assume first that Y→X and Y1→X are both of type D4. Notice that, in this case, xpb(z,u)∈g(x,z,u). Thus, Y−⇒XY1 by Lemma 3.13.
Now assume that Y→X is of type D3. Consider the chart Ut⊂Y which is defined by
Notice that, using the notation in Section 3, we know that
η′5 can be y±μub if p(x,0,u)=−μ2u2b for some μ∈C, and otherwise η′5=f′5∘. One can see that η′4=η′5=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 r′vF(η′4)a′4=1. Thus, (Ξ−) holds and so there exists Y2→X such that Y−⇒XY2. One can compute that Y2→X is of type D1. If Y1→X 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 Y→X is of type D2 and Y1→X is of type D5. In this case, we consider the embedding corresponding to Y1→X. Under this embedding, Y→X is given by the weighted blow-up with the weight (b,b,1,1,b) and the chart Uy⊂Y is given by
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(b−1,1,1,b,2b) defines a w-morphism over Uy. One can see that (Θ5) holds. Moreover, since a′5=2b>b=a5, we know that (Ξ−) holds. Thus, Y−⇒XY1.
Finally, assume that Y→X is of type D1 and Y1→X is not of type D3. Let b and b1 be the integers in Table 4 corresponding to Y→X and Y1→X, respectively. First, we claim that b≤b1. Indeed, if Y1→X is of type D5, then zb1∈h(z,u), which implies that b1≥b+1. If Y1→X is of type D2 or D4, then the inequality b≤b1 follows from Corollary 2.17. Now, the origin of the chart Uu⊂Y is defined by
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 b≤b1, we know that 2k−b−1>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
Thus, (Ξ−) and (Θ2) holds and one has Y−⇒XY1. □
Now we study divisorial contractions of discrepancy greater than one. All such divisorial contractions are listed in Table 5.
Proposition 4.6. Assume that Y→X is a divisorial contraction with discrepancy a>1. Then there exists a divisorial contraction Y1→X such that Y⇒XY1, and a(F,X)<a where F=exc(Y1→X).
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 Y→X is of type D6. Consider the chart Ux⊂Y and take (y1,...,y4)=(x,y,z,u) with δ4=4.
(2) Assume that Y→X is of type D7. Consider the chart Ut⊂Y and take (y1,...,y5)=(y,u,z,x,t), δ4=4 and δ5=1 or 2.
(3) Assume that Y→X is of type D8 or D9. We consider the chart Ut⊂Y and take (y1,...,y5)=(x,y,z,u,t) with δ4=4 and δ5=2.
(4) Assume that Y→X is of type D10. We consider the chart Uy⊂Y and take (y1,...,y4)=(y,u,z,x) with δ4=4.
(5) Assume that Y→X is of type D11. We consider the chart Uy⊂Y and (y1,...,y4)=(y,z,u+λy,x) for some λ∈C with δ4=4.
(6) Assume that Y→X is of type D12. We consider the chart Uy⊂Y and (y1,...,y4)=(y,z,x+λy,u) for some λ∈C with δ4=4.
Now we know that there exists Y1→X so that Y⇒XY1. Then Y1→X is of one of types in Table 4 or Table 5. One can see that vF(z)=1 if Y1→X is of types D1–D5, D7–D9 or D11 and vF(u)=1 if Y1→X 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. □
4.4. Divisorial contractions to cD/r points with r>1
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 Y→X is of type D14 or if Y→X is of type D13 and both zu2 and z2u∉g(y,z,u), then there is only one w-morphism over X.
(2) If Y→X is of type D13 and zu2 or z2u∈g(y,z,u), then there are two or three w-morphisms over X. Say Y1→X, ..., Yk→X are other w-morphisms with k=1 or 2. Then Y−⇒XYi−⇒XY for all 1≤i≤k.
Proof. The statement about the number of w-morphisms follows from [9, Section 9]. Now we may assume that Y→X is of type D13 and zu2 or z2u∈g(y,z,u). The chart Uz⊂Y is defined by
with u2 or zu∈g′(y,z,u). We can take (y1,...,y4)=(z,x,u+λz,y) for some λ∈C 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 Y1→X such that Y−⇒XY1. One can compute that Y1→X 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 u↦u−z 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 Y2→X which is different to Y1, and we also have that Y−⇒XY2. This finishes the proof. □
Proposition 4.8. Assume that X has cD/2 singularities and Y→X is of type D15, D16 or D17.
(1) If Y→X is of type D15, then there is only one w-morphism over X.
(2) If Y→X is of type D17, then there exists exactly two w-morphisms over X. The other one, Y1→X, is of type D16, and one has that Y⇒XY1.
(3) If Y→X 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 Y→X is of type D17. Consider the chart Ut⊂Y 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 Y→X 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 z4∈g(y,z,u). Assume first that (b,c,d)=(1,1,4). Consider the chart Uu⊂Y 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 a′1=5>3=a1. Hence, there exists a w-morphism Y1→X so that Y−⇒XY1. If we interchange the roles of y and z, we can get another w-morphism Y2→X with Y−⇒XY2.
Now assume that (b,c,d)=(3,1,2). Consider the chart Uy⊂Y which is defined by
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, Y→X is of type D18–D22, and assume that there are two w-morphisms Y→X and Y1→X.
(1) Assume that Y→X is of type D18 and Y1→X is not of type D20. Then Y−⇒XY1.
(2) Assume that both Y→X and Y1→X are not of type D18. Then, one of the following holds:
(2–1) Y→X is of type D19 and Y1→X is of type D22. One has that Y−⇒XY1.
(2–2) Both Y→X and Y1→X are of type D21 and Y−⇒XY1−⇒XY.
(2–3) Both Y→X and Y1→X are of type D20 and there exists another w-morphism Y2→X which is of type D18 so that Y−⇒XY2−⇐XY1.
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 Y→X is a divisorial contraction with the discrepancy a2>1. Then there exists a divisorial contraction Y1→X such that Y⇒XY1, and a(F,X)<a2 where F=exc(Y1→X).
Proof. First, assume that Y→X is of type D23. Consider the chart Ux⊂Y which is defined by
We take (y1,...,y4)=(x,y,z,u) with δ4=2 or 4. One can see that (Ξ′) holds. Since a4=2 and a≥3, we know that (Θ4) holds. Thus, there exists Y1→X such that Y⇒XY1. 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
Assume that Y→X is of type D24. Consider the chart Ut⊂Y which is defined by
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(a′1,...,a′5) with a′2+a′4=b+4. It follows that a(a′2+a′4)>b+4=a2+a4, hence (Θj) holds for j=2 or 4. Thus, there exists Y1→X so that Y⇒XY1. One has that
Hence, Y1→X is a w-morphism.
Assume that Y→X is of type D25. The chart Uy⊂Y is given by
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(2b−1,2b+1,1,4b). Hence, (Θ2) holds. One can also compute that a(F,X)=12, hence there exists a w-morphism Y1→X such that Y⇒XY1.
Assume that Y→X is of type D26. The chart Uz⊂Y is a cA/4 point given by
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 b≥6. Hence, (Θ2) holds, and there exists Y1→X such that Y⇒XY1. One can compute that a(F,X)=12.
Assume that Y→X is of type D27. Consider the chart Ut⊂Y which is defined by
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 Y1→X which extracts F so that Y⇒XY1. One can compute that a(F,X)=12.
Finally, assume that Y→X is of type D28 or D29. The chart Ut⊂Y is defined by
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(a′1,...,a′5) with a′1+a′2=4b+2, a′2(2b+1−a)≡1 (mod 4b+2) and a′3=1. From the defining equation one can see that a′4>1, hence (Θ4) holds. Thus, there exists a divisorial contraction Y1→X which extracts F so that Y⇒XY1. We only need to show that a(F,X)<a2.
Assume that Y→X is of type D28. In this case, a′2 is the integer such that a′2(2b−1)≡1 (mod 4b+2). If b is odd, then a′2=b since
One can see that a′5≤2b. If b is even, then a′2=3b+1 since
Hence, a′1=b+1. Now, since xzb−1 or z2b∈p(x,z,u), we also have that a′5≤2b. In either case we have
Finally, assume that Y→X is of type D29. We want to show that a′5<4b+2. Then
and we can finish the proof. If zb∈p(x,z,u), then a′5≤b. If a′2<2b+1, then a′5≤4b. Assume that zb∉p(x,z,u) and a′2≥2b+1. Then a′1≤2b+1 and xzb−12∈p(x,z,u). Hence,
□
4.5. Divisorial contractions to cE points
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 Y→X and Y1→X. Let F=exc(Y1→X). Let P=CenterYF. One always has that a(F,Y)<1, so P is a non-Gorenstein point.
Lemma 4.11. Assume that both Y→X and Y1→X are of type E1–E13. Then Y→X is not of type E1 or E6.
Proof. Assume that Y→X is of type E1. Then the only non-Gorenstein point on Y is the origin of
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, Y→X is not of type E1.
Assume that Y→X 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 Y1→X should be of type E17. Nevertheless, in this case one can see that vF(σ)≤vE(σ) for all σ∈OX. 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
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 Y→X and Y1→X correspond to the same embedding X↪A4. Then, since vF(y)≤3, we know that Y1→X is of type E1–E5. However, in those cases one always has that vF(σ)≤vE(σ) for all σ∈OX. This contradicts Corollary 2.17. Thus, Y1→X corresponds to a different embedding.
Let Z→Y be a w-morphism over the origin of Uy. From the classification we know that Z→Y 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 ˉZ→Z be an economic resolution over those cyclic quotient points. Then F appears on ˉZ since a(F,Y)<1. Moreover, ˉZ→X can be viewed as a sequence of weighted blow-ups with respect to the embedding X↪A4(x,y,z,u). We write (x1,...,x4)=(x,y,z,u) and let X↪A4(x′1,...,x′4) be the embedding corresponding to the weighted blow-up Y1→X. One can always assume that x′4=x4=u since vE(u)=1. We write x′j=xj+qj. Since Y→X and Y1→X correspond to different embeddings, there exists j<4 such that qj≠0 and vF(x′j)>vF(xj)=vF(q). Since ˉZ→X can be viewed as a sequence of weighted blow-ups with respect to the embedding X↪A4(x,y,z,u), we know that the defining equation of ˉZ is of the form xj+qj+ˉh such that vF(x′j)=vF(ˉh). Hence, there is exactly one j such that qj≠0, 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 x′1=x1=x and x′2=x2=y. One can see that vF(x′2)=vF(y)≤3. So, Y1→X is of type E1–E5. In those cases, vF(x′j)≤2, so vF(xj)=vF(qj)=1 and vF(x′j)=2. Hence, Y1→X is of type E3–E5 and vF(y)=vF(x′2)≥2. Also, since vF(qj)=1, qj=λu for some λ∈C. Therefore vF(z)=vF(x′j−qj)=1. But, then
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(x′3)=1, then Y1→X is of type E1 or E2, and so vF(x′j)=2. However, we know that vF(p)≥2. This contradicts the assumption that vF(x′j)>vF(qj)=vF(p). Hence, vF(z)≥2 and so vF(p)≥3. Since
and vF(y)≤3 by the previous discussion, we know that vF(y)=3. Recall that we write
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 Y1→X and Y→X correspond to the same four-dimensional embedding, leading to a contradiction. □
Lemma 4.12. Assume that both Y→X and Y1→X are of type E1–E13. If P is the origin of Ux⊂Y, then Y→X is of type E2, E5 or E9, and Y1→X has the same type. One has that Y−⇒XY1−⇒XY.
Proof. This assumption implies that the origin of Ux is contained in Y, so Y→X is of type E2, E5 or E9 and P is a cyclic quotient point. If Y→X is a weight blow-up with the weight (b,c,d,1), then b=c+d and
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 a′b=a(F,Y). Since a(F,X)=a(F,Y)+vF(x′)=1, we know that vF(x′)=1−a′b. One can compute that
Since c′<b and a′c≡c′(mod b), we know that a′c−c′b≥0, 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 σ∈OX such that vF(σ)>vE(σ). This can only happen when
and in this case one can choose σ=x+p(z,u)∈OX. Now, Y1→X can be obtained by a weighted blow-up with respect to the embedding
and with the weight w1(σ,y,z,u)=(b1,c1,d1,1) where c1=c−(a′c−c′)b and d1=d−a′d−d′b. Since c1≤c, d1≤d 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 c≤c1 and d≤d1. Hence, c=c1 and d=d1 and so Y→X and Y1→X 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 Y−⇒XY1 by Lemma 3.13 and also Y1−⇒XY by the symmetry. □
Lemma 4.13. Assume that both Y→X and Y1→X are of type E1–E13. Then P is not the origin of Uy⊂Y.
Proof. By Lemma 4.11, we know that Y→X is not of type E1 or E6, hence Y→X is of type E4, E8 or E11 and the origin of Uy is a cyclic quotient point. We assume that Y→X 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 Y1→X as a weighted blow-up with respect to the embedding
and with the weight (b1,c1,d1,1), such that CenterY1E is the origin of U1,σ⊂Y1. 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 Y→X and Y1→X are of type E1–E13. Let X↪A4(x1,...,x4) be the embedding that corresponds to Y→X in Table 8. Then P is the origin of Ui for some i≤4.
Proof. Assume that P is not the origin of Ui for all i≤4. Then Y has a non-Gorenstein point on Ui∩Uj for some i≠j. In this case, Y→X is of type E7 or E10–E13. For simplicity we assume that i=1 and j=2. If Y→X 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 (x′1,x′3,x′4), where x′l is the strict transform of xl on U1.
Since F is a valuation of discrepancy less than 1 over P, we know that vF(x′1,x′3,x′4)=1d(a′1,a′3,a′4) with a′l<d for l=1, 3, and 4. One can compute that
and Y1→X can be obtained by the weighted blow-up with respect to the same embedding X↪A4(x1,...,x4) and with the weight vF. Nevertheless, one can easily see that vF(xl)≤vE(xl) for all 1≤l≤4. This contradicts Corollary 2.17. □
Proposition 4.15. Assume that both Y→X and Y1→X are both of type E1–E13. Then Y→X is of type E2, E5 or E9, and Y1→X has the same type. One has that Y−⇒XY1−⇒XY.
Proof. Let
be the embedding corresponding to Y→X, and
be the embedding corresponding to Y1→X. If CenterYF is the origin of Ux⊂Y or CenterY1E is the origin of Ux1⊂Y1, then the statement follows from Lemma 4.12. We do not consider these cases here. Then, Lemmas 16 and 17 imply that CenterYF=Uz⊂Y and CenterY1E=Uz1⊂Y1.
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 z≠z1. 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 z5∈f. 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 Y→X is of type E14–E18. Then there exists Y1→X which is of type E3 or E6 such that Y⇒XY1. Moreover, if Y→X is of type E15 or E17, then Y−⇒XY1.
Proof. Assume that Y→X is of type E14. Consider the chart
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
One can see that (Θ4) holds. Thus, there exists a divisorial contraction Y1→X so that Y⇒XY1 which extracts F. One can compute that vF(x,y,z,u)=(3,2,2,1), so Y1→X is of type E3.
Assume that Y→X is of type E15. Consider the chart
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 u′2∉p(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 Y1→X such that Y−⇒XY1. One also has vF(x,y,z,u)=(3,2,2,1), so Y1→X is of type E3.
Assume that Y→X is of type E16. Consider the chart
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 u′2∉p(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 Y1→X such that Y⇒XY1. One can compute that vF(x,y,z,u)=(3,2,2,1), so again Y1→X is of type E3.
Assume that Y→X is of type E17. Consider the chart
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 y′2u′2∈g′(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 Y1→X which is of type E3 such that Y−⇒XY1.
Finally, assume that Y→X is of type E18. Consider the chart
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
(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 Y1→X which extracts F so that Y⇒XY1. One can compute that vF(x,y,z,u)=(5,4,2,1), and so Y1→X 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 Y→X is a divisorial contraction with discrepancy a>1. Then there exists a w-morphism Y1→X such that Y⇒XY1.
Proof. Assume first that Y→X is of type E19. The chart Uy⊂Y is defined by
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
One can see that (Θ4) holds. Thus, there exists Y1→X such that Y⇒XY1. A direct computation shows that Y1→X is a w-morphism.
Assume that Y→X is of type E20. The chart Ut⊂Y is defined by
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 Y1→X such that Y⇒XY1. One can compute that Y1→X is a w-morphism.
Finally assume that Y→X is of type E21. The chart Uy⊂Y is defined by
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 Y1→X such that Y⇒XY1. One can compute that Y1→X is a w-morphism. □
4.6. Divisorial contractions to cE/2 points
Finally, we need to study divisorial contractions over cE/2 points. All such divisorial contractions are listed in Table 11.
Proposition 4.18. Let Y→X be a divisorial contraction.
(1) Assume that there are two w-morphisms over X. Then:
(1–1) If Y→X is of type E22 and there exists another w-morphism Y1→X, then Y1→X is of type E22 or E23 and Y⇒XY1.
(1–2) If Y→X is of type E23, then there are exactly two w-morphisms. The other one, Y1→X, is of type E22. Interchanging Y and Y1, we are back to Case (1–1).
(1–3) If Y→X 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) Y→X is not of type E25.
(2) Assume that Y→X is of type E26. Then there is a w-morphism Y1→X which is of type E22 such that Y−⇒XY1.
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 Y→X is of type E22. Let F=exc(Y1→X). The only non-Gorenstein point on Y is the origin of Uz, which is a cD/3 point defined by
One can see that
hence a(F,Y)=vF(z′)=13. Thus, F corresponds to a w-morphism over Uz. From Table 6 we know that
for some λ∈C, where (b,c)=(3,2) or (6,5). Now one can choose (y1,...,y4)=(y′,u′+λz′,u′+ξz′,x′) with δ4=4, where ξ∈C is a number so that u+ξz defines a Du Val section on X and ξ≠λ. Thus, (Ξ′) and (Θ2) hold and Y⇒XY1.
Now assume that Y→X is of type E24. The chart Ux⊂Y is defined by
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 Y1→X which extracts F such that Y−⇒XY1. One can compute that a(F,X)=12. Hence, Y1→X is also a w-morphism.
Finally, assume that Y→X is of type E26. Consider the chart Uy⊂Y which is defined by
One can take (y1,...,y4)=(y′,z′,u′,x′) with δ4=4. One can see that (Ξ′) holds. Now, let F be the exceptional divisor corresponding to the w-morphism over Ut. Then vF(y1,...,y4)=16(2,5,1,1). Hence, (Θ2) and (Ξ′−) hold. Thus, there exists a divisorial contraction Y1→X which extracts F so that Y−⇒XY1. One can compute that vF(x,y,z,u)=12(3,2,3,1), so Y1→X is of type E22. □
5.
Estimating depths
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→X is a divisorial contraction between terminal and Q-factorial threefolds.
(1–1) If Y→X is a divisorial contraction to a point, then
If Y→X is a divisorial contraction to a curve, then
(1-2) depGor(X)≥depGor(Y) and the inequality is strict if the non-isomorphic locus on X contains a Gorenstein singular point.
(2) Assume that X⇢X′ is a flip between terminal and Q-factorial threefolds.
(2–1)
(2–2) depGor(X)≤depGor(X′) and the inequality is strict if the non-isomorphic locus on X′ contains a Gorenstein singular point.
Corollary 5.2. Assume that
is a process of the minimal model program. Then:
(1) ρ(X0/Xk)≥gdep(Xk)−gdep(X0) and the equality holds if and only if Xi⇢Xi+1 is a strict w-morphism for all i.
(2) depGor(Xk)≥depGor(X0).
In particular, if X is a terminal Q-factorial threefold and W→X is a resolution of singularities, then ρ(W/X)≥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 ρ(X0/Xk)=k−m. On the other hand, we know that
It follows that gdep(Xk)≤gdep(X0)+k−2m, hence gdep(Xk)−gdep(X0)≤ρ(X0/Xk). Now gdep(Xk)−gdep(X0)=ρ(X0/Xk) if and only if m=0 and gdep(Xi+1)=gdep(Xi)+1 for all i, which is equivalent to Xi⇢Xi+1 being a strict w-morphism for all i.
Now assume that X is a terminal Q-factorial threefold and W→X is a resolution of singularities. We can run KW-MMP over X and the minimal model is X itself. Since gdep(W)=0, one has that ρ(W/X)≥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 S be a set consisting of birational maps between Q-factorial terminal threefolds. We say that (∗)S holds if, for all Z⇢V inside S, one has that:
(1) If Z→V is a divisorial contraction to a point, then
(2) If Z→V is a divisorial contraction to a smooth curve, then
(3) If Z⇢V is a flip, then
(4) If Z⇢V is a flop, then depGor(V)=depGor(Z).
Moreover, if there exists a Gorenstein singular point P∈V such that P is not contained in the isomorphic locus of Z⇢V, then depGor(V)>depGor(Z) unless V⇢Z is a flop.
We say that (∗)(1)S holds if statement (1) is true, but statements (2) and (3) are unknown.
If V⇢Z is a flip or a divisorial contraction, we denote the condition (∗)V⇢Z=(∗)S where S is the set containing only one element V⇢Z.
It is easy to see that if Y→X is a divisorial contraction, then (∗)Y→X holds if and only if the inequalities in Proposition 5.1 (1) hold. Likewise, if X⇢X′ is a flip, then (∗)X⇢X′ holds if and only if the inequalities in Proposition 5.1 (2) hold.
Remark 5.4. If Z⇢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∈Z≥0, we denote
Lemma 5.6. Assume that Y→X and Y1→X are two divisorial contractions between terminal threefolds, such that Y⇒XY1. If (∗)Sgdep(Y)−1 holds, then gdep(Y1)≤gdep(Y). Moreover, gdep(Y1)=gdep(Y) if and only if Y1⇒XY.
Proof. We have a diagram
such that Z1→Y is a strict w-morphism and Zi⇢Zi+1 is a flip or a flop for all 1≤i≤k−1. Since gdep(Z1)=gdep(Y)−1 and (∗)Sgdep(Y)−1 holds, we know that gdep(Z2)≤gdep(Y)−1. Repeating this argument k−2 times, one can say that gdep(Zk)≤gdep(Y)−1. Again, since (∗)Sgdep(Y)−1 holds, we know that gdep(Y1)≤gdep(Zk)+1=gdep(Y).
Now, gdep(Y1)=gdep(Y) if and only if all the inequalities above are equalities. This is equivalent to Zk→Y1 being a strict w-morphism and Zi⇢Zi+1 being a flop for all i=1, ..., k−1, or k=1. In other words, we also have Y1⇒XY. □
Corollary 5.7. Assume that Y→X is a strict w-morphism over P∈X and Y1→X is another divisorial contraction over P. If (∗)Sgdep(Y) holds, then there exist divisorial contractions Y1→X, ..., Yk→X such that
Proof. By Proposition 4.1, we know that there exists Y1, ..., Yl, Y′1, ..., Y′l′ such that
One can apply Lemma 5.6 to the sequence Y⇒XY′1⇒X...⇒XY′l′ and conclude that gdep(Y′i)≤gdep(Y) for all i. Since Y′i→X∈Sgdep(Y) for all i=1, ..., l′, one has gdep(Y′i)≥gdep(X)−1=gdep(Y). Thus, gdep(Y′i)=gdep(Y) and Lemma 5.6 says that one has
Now we can take k=l+l′−1 and let Yi=Y′l′−i+l for l<i≤k. □
Corollary 5.8. Assume that (∗)Sn−2 holds. Assume that P∈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 Y1→X in Theorem 2.18 is a strict w-morphism.
Proof. Propositions 4.2 and 4.3 say that if Y→X and Y1→X are two different w-morphisms over P, then there exists Y2→X, ..., Yk→X such that
We can assume that Y→X is a strict w-morphism, so gdep(Y)=n−1. Lemma 5.6 implies that Y1 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→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 Y1→X over P induces a diagram in Theorem 2.18. Hence, we can choose Y1→X to be a strict w-morphism. □
Convention 5.9. Let DV be the set of symbols
One can define an ordering on DV by
Given ◻∈DV, define
and
Here, GE(P∈X) denotes the type of the general elephant near P. That is, the type of a general Du Val section H∈|−KX| near an analytic neighborhood of P∈X.
Convention 5.10. Let T be a set of terminal threefolds. We say that the condition (Π)T holds if for all X∈T and for all strict w-morphisms Y→X over non-Gorenstein points of X, one has that dep(Y)=dep(X)−1.
Remark 5.11.
(1) Assume that Y→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∈|−KX| near P, then HY∈|−KY| and HY→H is a partial resolution by [3, Lemma 2.7].
(2) One has that (Π)TA1 always holds since if GE(P∈X)=A1 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→X over P and Y is smooth over X. Hence,
and gdep(Y)=dep(Y)=0 over X. Thus, Y→X is a strict w-morphism and also dep(Y)=dep(X)−1.
Lemma 5.12. Assume that Y→X is a strict w-morphism over a non-Gorenstein point P. If (∗)Sgdep(Y) holds and (Π)T◻ holds for all ◻<GE(P∈X), then dep(Y)=dep(X)−1.
Proof. By the definition we know that dep(Y)≥dep(X)−1. Assume that dep(Y)>dep(X)−1. Then there exists Y1→X such that dep(Y1)=dep(X)−1<dep(Y). Corollary 5.7 says that there exist divisorial contractions Y2→X, ..., Yk→X such that Y1⇒XY2⇒X...⇒XYk⇒XY. From Remark 5.11 (1) we know that Yi∈T◻ for some ◻<GE(P∈X) for all i, hence, if
is the induced diagram of Yi⇒XYi+1, then dep(Zi,1)=dep(Yi)−1. By Lemma 2.20 we know that
for all i. Hence, dep(Y1)≤dep(Y). This leads to a contradiction as we assume that dep(Y)>dep(Y1). □
Lemma 5.13. Fix an integer n and assume that (∗)Sn−1 holds. Then (Π)Tn holds.
Proof. We need to show that for all X∈Tn and for all strict w-morphism Y→X over a non-Gorenstein point, one has that dep(Y)=dep(X)−1. By Remark 5.11 (2) we know that (Π)TA1 holds, hence (Π)TAm,n and (Π)TD4,n hold for all m∈N by Lemma 5.12 and by induction on m. Then, one can prove that (Π)TDm,n and (Π)TE6,n hold by again applying Lemma 5.12 and by induction on m. Statements (Π)TE7,n and (Π)TE8,n can be proved in the same way. □
Lemma 5.14. Fix an integer n and assume that (∗)Sn−1 holds. Assume that we have a diagram
such that gdep(X)=n, Y1→X is a strict w-morphism, Yi⇢Yi+1 is a flip or a flop for i=1, ..., k−1, and Yk→X′ is a divisorial contraction. Then,
(1) depGor(X)=depGor(Y1)≤depGor(X′) and gdep(X′)≤gdep(X).
(2) (∗)Yi⇢Yi+1 holds for i=1, ..., k−1.
(3) (∗)Yk→X′ holds.
Moreover, if the non-isomorphic locus of X⇢X′ on X′ contains a Gorenstein singular point, then depGor(X)<depGor(X′).
Proof. Since (∗)Sn−1 holds, we know that (Π)Tn holds by Lemma 5.13. Hence, depGor(Y1)=depGor(X). We know that gdep(Y)=n−1. Since (∗)Sn−1 holds, one can prove that gdep(Yi)≤gdep(Y1)≤n−1 for all i and hence (∗)Yi⇢Yi+1 and (∗)Yk→X′ hold. Thus,
and
Now, if the non-isomorphic locus of X⇢X′ on X′ contains a Gorenstein singular point, then either the non-isomorphic locus of Yi⇢Yi+1 on Yi+1 contains a Gorenstein singular point or the non-isomorphic locus of Yk→X′ on X′ contains a Gorenstein singular point. Hence, at least one of the above inequalities is strict. Thus, one has depGor(X)<depGor(X′). □
Lemma 5.15. Assume that (∗)Sn−1 holds. Then (∗)(1)Sn holds.
Proof. Let Y→X be a divisorial contraction to a point which belongs to Sn. We know that gdep(Y)=n. Assume first that Y→X is a strict w-morphism over a point P∈X. Notice that dep(Y)≥dep(X)−1 by Lemma 2.20. If P is a non-Gorenstein point, then depGor(Y)=depGor(X) by Lemma 5.13, hence (∗)Y→X holds. If P is a Gorenstein point, then
Moreover, since dep(P∈X)=0, dep(Y)−dep(X)≥0, hence
Thus, (∗)Y→X holds.
In general, by Corollary 5.7 there exist Y1→X, ..., Yk→X such that Yk→X is a strict w-morphism and one has Y⇒XY1⇒X...⇒XYk. By induction on k we may assume that (∗)Yi→X holds for all i (notice that gdep(Yi)≤gdep(Y)=n for all i by Lemma 5.6). Now we have a diagram
By Lemma 5.14 we know that depGor(Y)≤depGor(Y1) and (∗)Zi⇢Zi+1, (∗)Zk→Y1 hold. Since (∗)Y1→X holds, we know that depGor(X)≥depGor(Y1)≥depGor(Y) and
Moreover, if Y→X is a divisorial contraction to a Gorenstein point, then Y1→X is also a divisorial contraction to a Gorenstein point. Hence, depGor(Y1)<depGor(X) and we also have depGor(Y)<depGor(X). □
Proof of Proposition 5.1. We need to say that (∗)Sn holds for all n and we will prove this by induction on n. If n=0, then S0 consists only smooth blow-downs and smooth flops. One can see that (∗)S0 holds. In general, assume that (∗)Sn−1 holds. By Lemma 5.15 we know that (∗)(1)Sn holds. Hence, it is enough to show that, given a flip X⇢X′ or a divisorial contraction to a curve X→V such that gdep(X)=n, (∗)X⇢X′ or (∗)X→V holds.
If X→V is a smooth blow-down, then depGor(X)=depGor(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 depGor(X)≤depGor(X′) and (∗)Yi⇢Yi+1, (∗)Yk→X′ hold. One has that
Moreover, if X⇢X′ is a flip, then either one of Yi⇢Yi+1 is a flip or Yk→X′ is a divisorial contraction to a curve by [3, Remark 3.4]. This implies that either gdep(Y1)>gdep(Yk) or gdep(Yk)≥gdep(X′). If X→V is a divisorial contraction to a curve, then Yk→X′ is a divisorial contraction to a curve, hence one always has that gdep(Yk)≥gdep(X′). In conclusion, we have
If X⇢X′ is a flip, then one can see that (∗)X⇢X′ holds. Now assume that X→V is a divisorial contraction to a curve. Then X′→V is a divisorial contraction to a point. We know that (∗)X′→V holds since (∗)(1)Sn holds and gdep(X′)≤n by Lemma 5.14. One can see that depGor(X)≤depGor(X′)≤depGor(V) and
Thus, (∗)X→V holds. □
6.
Comparing different feasible resolutions
In this section, we describe the difference between two different feasible resolutions of a terminal Q-factorial threefold. The final result is the diagrams in Theorem 1.1.
6.1. General discussion
Lemma 6.1. Assume that X is a terminal threefold and Y→X, Y1→X are two different strict w-morphisms over P∈X such that Y⇒XY1. Let
be the corresponding diagram. Then one of the following holds:
(1) Y−⇒XY1−⇒XY and k=1.
(2) k=2 and Z1⇢Z2 is a smooth flop.
(3) k=2, Z1⇢Z2 is a singular flop, P∈X is of type cA/r and both Y→X and Y1→X are of type A1 in Table 2.
Proof. Since Y→X and Y1→X are both strict w-morphisms, we know that gdep(Y)=gdep(Y1). Hence, Zi⇢Zi+1 can not be a flip. Thus, if Y−⇒XY1, then k=1.
Now assume that Y⧸−⇒XY1. 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→X and Y1→X are both of type A1. Since Y⧸−⇒XY1, we know that f∘4=η4=y is irreducible. Hence, there is only one KZ1-trivial curve on Z1 and so k=2.
(ⅱ) P is not of type cAx/r since one always has that Y−⇒XY1 if Y→X and Y1→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→X is of type D17 in Table 6.
We know that Z1→Y is a w-morphism over the origin of
and we take (y1,...,y5)=(y′,u′,y′+z′,x′,t′). One can see that there are two curves contained in y3=y5=0, namely Γ1=(x′2+g′(0,u′,0)=y′=y′+z′=u′=0) and Γ2=(x′2+g′(−y′,−y′2,0)=u′+y′2=y′+z′=t′=0). We know that (Θ1) holds. When computing the intersection number for Γ2 one can take δ4=4 and δ5=2, hence (Ξ−) holds in this case. This implies that the proper transform Γ2,Z1 of Γ2 on Z1 is a KZ1-trivial curve and is not contained in exc(Z1→Y). Thus, Zi⇢Zi+1 is a flip along Γ2,Zi for some i, and hence gdep(Y1)<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→X is of type E14, E16, or E18 in Table 9. In those cases, Y→X are five-dimensional weighted blow-ups. Let Γ⊂Y be a curve contained in exc(Y→X) such that the proper transform ΓZ1 of Γ on Z1 is a possibly KZ1-trivial curve. From the proof of Proposition 4.16, one can see that:
(ⅵ–ⅰ) Y→X is of type E14. In this case, Z1→Y is a w-morphism over the origin of
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. We also know that (Θ4) holds. If both δ4 and δ5≠4, then (Ξ−) holds, which implies that Y−⇒XY1. This contradicts our assumption. Hence, δ5=4. Now, Γ=(x′2+y′3=z′=u′=t′=0).
(ⅵ–ⅱ)Y→X is of type E16. In this case Z1→Y is a w-morphism over the origin of
As in the proof of Proposition 4.16, we choose (y1,...,y5)=(y′,z′,u′,x′,t′) with δ4=4 and δ5=1 or 2. We know that (Θ2) holds. Hence, if δ5=1, then (Ξ−) holds, and so Y−⇒XY1. This leads to a contradiction. Thus, δ5=2 and one has Γ=(x′2+y′3=z′=u′=t′=0).
(ⅵ–ⅲ) Y→X is of type E18. In this case, Z1→Y is a w-morphism over the origin of
We choose (y1,...,y5)=(y′,z′,u′,x′,t′) with δ4=4 and δ5=1. We know that (Θ1) holds. If z′3∈p, then we can choose δ5=2. Then, since (Θ1) holds, we know that (Ξ−) holds, and so Y−⇒XY1. This contradicts our assumption. Hence, z′3∉p. In this case, u divides p and X has cE8 singularities, so z′5∈g′. One can see that Γ=(x′2+z′5=y′=u′=t′=0).
Now the origin of Ut is a cyclic quotient point and the w-morphism over this point is a weighted blow-up with weights vF(x′,y′,z′,u′,t′)=15(3,2,6,1,4), 14(3,2,5,1,3) and 17(5,10,2,1,6), respectively. An easy computation shows that ΓZ1 does not pass through any singular point of Z1, hence Z1⇢Z2 is a smooth flop. Since there is only one KZ1-trivial curve, we know that k=2.
(ⅶ) If P is of type cE/2, then by Proposition 4.18 we know that Y→X is of type E22 or E23 in Table 11. Moreover, if Y→X is of type E23, then Y1→X is of type E22. Thus, interchanging Y and Y1 if necessary, we can always assume that Y→X is of type E22. As in the proof of Proposition 4.18, we know that Z1→Y is a w-morphism over the origin of Uz⊂Y, which is a cD/3 point defined by
The only possible KZ1-trivial curve ΓZ1 is the lifting of the curve Γ=(x′2+y′3=z′=u′=0) on Z1. Moreover, Z1→Y is defined by a weighted blow-up with the weight
for some λ∈C, where (b,c)=(3,2) or (6,5). If (b,c)=(6,5), then one can see that (Ξ−) holds by considering the function y, hence Y−⇒XY1, which contradicts our assumption. Hence, (b,c)=(3,2). In this case, an easy computation shows that ΓZ1 does not pass through any singular point of Z1, so Z1⇢Z2 is a smooth flop. Since there is only one KZ1-trivial curve, we know that k=2.
□
We need to construct a factorization of the flop in Lemma 6.1 (3). Assume that
is a cA/r singularity with f(z,u)=zrk+g(z,u) and Y→X, Y1→X are two different strict w-morphisms with the factorization
such that Z1⇢Z2 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. Z1⇢Z2 is a flop over
where f1=f(zu1r,u)/uk. Moreover, Z1=Bl(ˉx,ˉu)V and Z2=Bl(ˉy,ˉu)V.
Proof. We may assume that Y→X is a weighted blow-up with the weight w(x,y,z,u)=1r(b,c,1,r) with b>r. The chart Uu⊂Y is defined by
and the chart Ux⊂Y is defined by
for some f2. As in the proof of Proposition 4.2, we know that Z1→Y is a weighted blow-up over the origin of Ux with the weight w′(x2,y2,z2,u2)=1b(b−r,rk,1,r). The flopping curve Γ of Z1→Z2 is the strict transform of the curve ΓY⊂Y such that ΓY|Uu=(y1=z1=u1=0) and ΓY|Ux=(x2=y2=z2=0). One can see that Γ intersects exc(Z1→Y) at the origin of U′u⊂Z1, which is defined by
It is easy to see that Γ is contained in U′u∪Uu, and on U′u we know that Γ is defined by (x′=z′=0).
We have the following change of coordinates formula:
Also,
One can see that
It follows that
If we choose an isomorphism
then Uu∪U′u=Bl(x′,u1)V, where
by noticing that
Now we can choose (ˉx,ˉy,ˉz,ˉu)=(x′,y1,z1,u1).
Finally, we know that Y1→X is a weighted blow-up with the weight 1r(b−r,c+r,1,r) and Z2→Y1 is a w-morphism over the origin of U1,y⊂Y1. Hence, the local picture of Z2→V can be obtained by a similar computation, but interchanging the role of x and y. One then has that Z2=Bl(ˉy,ˉu)V. □
6.2. Explicit factorization of flops
In this subsection, we assume that
with f(z,u)=zrk+g(z,u) for some k>1. Let Z1=Bl(x,u)V and Z2=Bl(y,u)V such that Z1⇢Z2 is a Q-factorial terminal flop. Let w be the weight w(z,u)=1r(1,r) and m=w(f(z,u)). Then m≤k. Let U1,x⊂Z1 be the chart
and U1,u⊂Z1 be the chart
with the relations u=u1x, x=x1u and f1=f(z,xu1). Similarly, let U2,y⊂Z2 be the chart
and U2,u⊂Z2 be the chart
with the relations u=u2y, y=y2u and f2=f(z,yu2).
Lemma 6.4. Let ϕ:V′→V be a strict w-morphism. Then:
(1) The chart U′u⊂V′ is Q-factorial.
(2) If m=k then U′z⊂V′ is smooth. Otherwise, U′z contains exactly one non-Q-factorial cA point which is defined by xy+uf"(z,u)=0 where f"=f(z1r,zu)/zm. 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′→V is a weighted blow-up with the weight 1r(b,c,1,r) with b+c=r(m+1). Statement (3) follows from direct computations. One can compute that
with f′=f(zu1r,u)/um. By [18, 2.2.7] we know that U′u is Q-factorial if and only if f′ is irreducible as a Z/rZ-invariant function. If U′u is not Q-factorial, then f′=f′1f′2 for some non-unit Z/rZ-invariant functions f′1 and f′2. Write
Let
Notice that if f′=∑(i,j)σi,jzriuj, then j+m−i≥0 if σi,j≠0 because f′=f(zu1r,u)/um. Hence, we have the relation m1+m2≤m. Now let
Then f1 and f2 can be viewed as Z/rZ-invariant functions on V such that ϕ∗fi=umif′i for i=1, 2. This means that f=um−m1−m2f1f2 is not irreducible. Nevertheless, the chart U1,u⊂Z1 is defined by x1y+f(z,u)=0 and has Q-factorial singularities. This leads to a contradiction. Thus, U′u is 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-Q-factorial cA singularity. □
From the construction in [19], for any strict w-morphism V′→V we have the following diagram
where V"i→V′ is a Q-factorization and V"i⇢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⇢V"2 is a flop over the singularity xy+uf"(z,u)=0.
First we discuss the factorization of V"1⇢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-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.
∙ ˉU"1=U′x=(y"1+u"1f"1(x"1,z"1,u"1)=0) ⊂A4(x"1,y"1,z"1,u"1)/1b(b−r,c,1,r)
with x=x"1br, y=y"1x"1cr, z=z"1x"11r, u=u"1x"1 and f"1=f(z"1x"11r,u"1x"1)/x"1m.
∙ ˉU"2=U′y=(x"2+u"2f"2(y"2,z"2,u"2)=0) ⊂A4(x"2,y"2,z"2,u"2)/1c(b,c−r,1,r)
with x=x"2y"2br, y=y"2cr, z=z"2y"21r, and .
with , , , and .
If , define . In this case this chart is smooth. When we define
with , , , and .
with , , , and .
On the other hand, we will see later that it is enough to assume that is a divisorial contraction over the origin of . This means that is covered by five affine charts , , , and where the latter four charts correspond to the weighted blow-up . Notice that and are the same divisor since and are isomorphic in codimension one. One can see that is a weighted blow-up with the weight . Again, we use the following notation:
with , , , and .
with , , , and .
with , , , and .
with , , , and .
with , , , and .
Lemma 6.5. Assume that is a divisorial contraction over the origin of . Then:
(1) .
(2) All the singular points on the non-isomorphic loci of on both and are cyclic quotient points.
(3) The flip is of type IA in the convention of [20, Theorem 2.2].
Proof. It is easy to see that for , and if . Since is smooth, the only singular point contained in the non-isomorphic locus of on is the origin of . This point is a cyclic quotient point of index , so it has generalized depth .
On the other hand, singular points on the non-isomorphic locus of on are origins of and . They are cyclic quotient points of indices and , respectively. One can then see that
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 -type since it comes from the factorization of a flop over a 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 is a flop, then it is a Gorenstein flop.
(2) If is a flip, or is a flop and is a flip, then the flip is of type IA.
Proof. Let be the flipping curve. Since is a type IA flip, there is exactly one non-Gorenstein point which is contained in , and this point is a point. From the construction, we know that is a -morphism over this point. Also, there exists a Du Val section such that . We know that by [3, Lemma 2.7 (2)]. Hence, all non-Gorenstein point of is contained in . Now assume that contains a non-Gorenstein point. Then intersects non-trivially. Since , we know that , hence is a -negative curve and so is a flip. Thus, if is a flop, then it is a Gorenstein flop.
If is a flip, then passes through a non-Gorenstein point since by [21, Theorem 0]. Since passes through exactly one non-Gorenstein point, passes through exactly one non-Gorenstein point and this point is contained in . Since is a -morphism over a point, an easy computation shows that contains only singularities. Also, we know that is a Du Val section which does not contain . Thus, is also a flip of type IA by the classification [20, Theorem 2.2].
Now assume that is a flop and is a flip. Then the flipping curve of is contained in since all -negative curves over are contained in . Since is a flop, we know that contains only singularities [17, Theorem 2.18] and is also a Du Val section. If , then is a flip of type IA by the classification from [20, Theorem 2.2].
Thus, we only need to prove that . Assume that . Then, since does not intersect (otherwise is a -negative curve and then is not a flop), we know that does not intersect the flopping curve . Let be a curve which intersects non-trivially. Then for some since the both curves are contracted by . Hence, for all divisors such that , we know that . On the other hand, is a -anti-flip since is not contained in . By Corollary 2.22 we know that , hence . Now we know that and is not numerically equivalent to a multiple of , hence we may write for some . Since
and , we know that . Hence, is not contained in the boundary of the relative effective cone . Thus, cannot be the flipping curve of . This leads to a contradiction. □
Lemma 6.7. Assume that is a three-dimensional terminal -factorial flip which satisfies conditions (1)–(3) of Lemma 6.5. Then, the factorization in Theorem 2.18 for is one of the following diagrams:
(1)
where is a flip which also satisfies conditions (1)–(3) of Lemma 6.5 and is a strict -morphism.
(2)
where is a smooth flop and is a flip which also satisfies conditions (1)–(3) of Lemma 6.5 and is a strict -morphism.
(3)
where is a smooth flop and where is a smooth curve contained in the smooth locus of .
Proof. We have the factorization
such that is a flip or a flop and is a flip for all . One has that
If , then and is a flop. By [3, Remark 3.4] we know that is a divisorial contraction to a curve. Since singular points on the non-isomorphic locus of 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 is a divisorial contraction to a curve contained in the smooth locus. Now, we also have that , hence is smooth over and so is also a smooth curve.
Now assume that . Then , hence either or and is a flop. Also, is a -morphism. Since singular points on the non-isomorphic locus of are all cyclic quotient points, singular points on the exceptional divisor of 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 are all cyclic quotient points for , ..., . If 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 is a flip for or . 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 is a three-dimensional terminal -factorial flip which satisfies conditions (1)–(3) of Lemma 6.5. Then we have a factorization
where and are feasible resolutions, where is a smooth curve, and is a sequence of smooth flops.
Proof. For convenience we denote the diagram
by .
We know that the factorization of is of the form (1)–(3) in Lemma 6.7. Notice that, since the only singular points on the non-isomorphic locus of are cyclic quotient points, the feasible resolutions and are uniquely determined. Hence, is also a feasible resolution of . If the factorization of is of type (3) in Lemma 6.7, then the non-isomorphic locus of contains no singular points. This means that induces a smooth blow-up on , and induces a smooth flop . Thus, exists.
Now, if the factorization of is of type (1), then is a feasible resolution of . Since , we may utilize induction on and assume that exists, and then can be induced by . If the factorization of is of type (2), then again by induction we may assume that
exists. Since is a strict -morphism, we know that . Also, since is a smooth flop, it induces a smooth flop . If we let and let be the composition , then we get the diagram . □
Definition 6.9. Let be a birational map between smooth threefolds. We say that is of type if it is a composition of smooth flops. We say that is of type if there exists the diagram
such that:
(1) for some smooth curve for , .
(2) is a composition of smooth flops for , .
(3) has the factorization
such that is a birational map of type for some .
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), is of type . In both diagram (2) and (3), is of type .
Definition 6.11. Let be a birational map between smooth threefolds. We say that has an -type factorization if there exists birational maps between smooth threefolds
such that is of type for some .
Proposition 6.12. Assume that is a -factorial terminal threefold and , are two different feasible resolutions. Then the birational map has an -type factorization.
Proof. First, notice that if , then has either a cyclic quotient point of index 2, or a point defined by for or 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 . Let (resp. ) be the strict -morphism which is the first factor of (resp. ). If , then and are two different feasible resolutions of . In this case, the statement can be proved by induction on . Thus, we may assume that .
Since both and are strict -morphisms, by Corollary 5.7 there exists a sequence of strict -morphisms , , ..., such that for , ..., . For each , let be a feasible resolution. Then is also a feasible resolution of and it is enough to prove that our statement holds for and , for all , ..., . Thus, we may assume that .
We have the diagram
Lemma 6.1 says that there are three possibilities. If , then and are two different feasible resolutions of . Since , again by induction on we know that has an -type factorization. Assume that and is a smooth flop. Then it induces a smooth flop where is a feasible resolution of for , . Also, we know that and are two feasible resolutions of , and and are two feasible resolutions of . Again, by induction on , we know that both and have -type factorizations, hence does as well.
Finally, assume that we are in the case of Lemma 6.1 (3), namely that has a singularity and is a singular flop. By Lemma 6.3 we know that is a flop over
We have the factorization of the flop
We use the notation at the beginning of this subsection. Assume that . Then we can choose to be the weighted blow-up with the weight . In this case, is a divisorial contraction over the origin of for , since both and . When , let be the weighted blow-up with the weight . Then, is a divisorial contraction over the origin of , but is a divisorial contraction over the origin of . Since and by Remark 6.2, we know that for some unit . If we let , then we can write the defining equation of as where . One has that , and under this notation one also has that is a divisorial contraction over the origin of . In conclusion, Corollary 6.8 holds for both and .
Let be a feasible resolution. If , then one can see that is of type . Assume that is a flop. Notice that is a -morphism since where . One has that
where the second equality follows from Corollary 5.8. Now we know that is a flop over with . By induction on , we may assume that has an -type factorization where is a feasible resolution corresponding to the diagram in Corollary 6.8. One can see that can be connected by a diagram of the form for some . Finally, we know that and (resp. and ) are feasible resolutions of (resp. ). Again, by induction on , we may assume that and have -type factorizations. Hence, has an -type factorization. □
Remark 6.13. Assume that is a singularity. Then there exists feasible resolutions and such that the birational map is connected by for .
7.
Minimal resolutions of threefolds
In this section, we prove our main theorems. First, we recall some definitions which are defined in the introduction section.
Definition 7.1. Let be a projective variety. We say that a resolution of singularities is a P-minimal resolution if for any smooth model one has that .
Definition 7.2. Let 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 such that and are P-minimal resolutions.
Proposition 7.3. Assume that is a threefold. Then, is a P-minimal resolution if and only if is a feasible resolution of a terminalization of . In particular, if is a terminal and -factorial threefold, then P-minimal resolutions of coincide with feasible resolutions.
Proof. Let be a resolution of singularities and let be the -MMP over . Then is a terminalization of . We know that by Corollary 5.2. Assume first that is -minimal. Let be a feasible resolution of , then . Since is also a smooth resolution of , the inequality is an equality. Therefore, , which implies that is a sequence of strict -morphisms by Corollary 5.2, or, equivalently, is a feasible resolution.
Conversely, assume that is not -minimal, but it is a feasible resolution of some which is a terminalization of . There exists a -minimal resolution such that . From the above argument, there exists a terminalization of such that is a feasible resolution. Hence, . However, since terminalizations are connected by flops [23, Theorem 1] and flops do not change singularities by [17, Theorem 2.4], we know that . This means that , so can not be a feasible resolution of . This is a contradiction. □
Proof of Theorem 1.1. Let be a threefold and , be two P-minimal resolutions. By Proposition 7.3, we know that (resp. ) is a feasible resolution of a terminalization (resp. ). If , then and are connected by flops [23, Theorem 1], hence is connected by P-desingularizations of terminal -factorial flops.
Now assume that . Then and are two different feasible resolutions of . The first two paragraphs in the proof of Proposition 6.12 and Lemma 6.7 imply that can be also connected by P-desingularizations of terminal -factorial flops. Moreover, Proposition 6.12 says that those P-desingularizations of flops can be factorized into compositions of diagrams of the form . This finishes the proof. □
Remark 7.4. Assume that is a terminal -factorial threefold and , are two different P-minimal resolutions. We know that and can be connected by P-desingularizations of flops. Let be a P-desingularization of a flop which appears in the factorization of . Then, from the construction we know that .
Now we compare an arbitrary resolution of singularities to a P-minimal resolution.
Definition 7.5. Let be a birational map where is a smooth threefold and is a terminal threefold. We say that the birational map has a bfw-factorization if can be factorized into a composition of smooth blow-downs, P-desingularizations of flops, and strict -morphisms.
Remark 7.6. If is a strict -morphism and is a smooth blow-down or a P-desingularization of a flop, then on the indeterminacy locus of is disjoint to the indeterminacy locus of since the former one lies on the smooth locus of and the latter one is a singular point. Hence, there exists where is a smooth blow-down or a P-desingularization of a flop, and is a strict -morphism. In other words, has a bfw-factorization if and only if there exists a birational map which is a composition of smooth blow-downs and P-desingularization of flops, where is a feasible resolution of .
Proposition 7.7. Assume that a birational map has a bfw-factorization where is a smooth threefold and is a terminal threefold.
(1) If is a flop, then there is a birational map which has a bfw-factorization.
(2) If is a strict -morphism, then there exists a birational map which also has a bfw-factorization.
(3) If is a flip or a divisorial contraction, then the induced birational map has a bfw-factorization.
Proof. Assume first that is a flop. By Remark 7.6 we know that there exists a bfw-map , where is a feasible resolution of . Let be a feasible resolution of . Then is a composition of strict -morphisms and the induced birational map is a P-desingularization of the flop . It follows that the composition
is a bfw-map. This proves (1).
We will prove (2) and (3) by induction on . If , then is smooth. In this case, there is no strict -morphism or flip . Assume that is a divisorial contraction. If is smooth, then it is a smooth blow-down by [22, Theorem 3.3, Corollary 3.4], and if is singular, then should be a strict -morphism since in this case is terminal -factorial and is a P-minimal resolution of . Now we may assume that and statements (2) and (3) hold for threefolds with generalized depth less then .
Let
be a sequence of birational maps so that is a smooth blow-down, a P-desingularization of a flop, or a strict -morphism for all . By Remark 7.6 we can assume that is a strict -morphism. Now, given a strict -morphism , if , then there is nothing to prove. Otherwise, by Corollary 5.7 there exists a sequence of strict -morphisms , ..., such that
For each , one has the factorization
such that is a composition of flops, is a strict -morphism, and
By the induction hypothesis, we know that if there exists a bfw-map , then there exists a bfw-map . Now one can prove statement (2) by induction on .
Assume that is a flip. Then we have a factorization
as in Theorem 2.18. Since is a strict -morphism by Corollary 5.8, there exists a bfw-map . Since
the induction hypothesis implies that there exists a bfw-map .
Finally, assume that is a divisorial contraction. If it is a smooth blow-down or a strict -morphism, then there is nothing to prove. Otherwise, there exists a diagram
such that is a strict -morphism and is a flip or a flop for all . One has that
hence there exists a bfw-map . If is a divisorial contraction to a curve, then is a divisorial contraction to a curve as in Theorem 2.18. In this case, we also have , so there exists a bfw-map . If is a divisorial contraction to a point, then the discrepancy of is less than the discrepancy of unless is a -morphism. Also, when is a -morphism, we know that by Lemma 5.6. Thus, we can prove statement (3) by induction on the generalized depth and the discrepancy of over . □
One can easily see the following corollary:
Corollary 7.8. Assume that is a smooth threefold and 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 -morphisms.
Proof of Theorem 1.2. By Corollary 7.8 and Remark 7.6 we know that there exists a feasible resolution such that is a composition of smooth blow-downs and P-desingularizations of flops, where is a minimal model of over . By Proposition 7.3 we know that is also a P-minimal resolution of , hence the birational map is connected by P-desingularizations of flops. Thus, the composition is connected by smooth blow-downs and P-desingularizations of flops. □
Proof of Corollary 1.3. Let
be a sequence of smooth blow-downs and P-desingularization of flops as in Theorem 1.2. We only need to show that if is a P-desingularization of a flop , then for all , ..., .
By [24, Lemma 2.12] we know that for all . Since and have the same analytic singularities [17, Theorem 2.4], there exists a feasible resolution such that for all . Now, and are two different P-minimal resolutions of , 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 . Hence, for all , ..., . □
8.
Further discussion
This section is dedicated to exploring minimal resolutions for singularities in higher dimensions and the potential applications of our main theorems.
8.1. Higher-dimensional minimal resolutions
In three dimensions, -minimal resolutions appear to be a viable generalization of minimal resolutions for surfaces. However, in higher dimensions, -minimal resolutions are not good enough. For example, let be a smooth flip (eg. a standard flip [1, Section 11.3]). Then, and are both -minimal resolutions of the underlying space, but is better than . It is reasonable to assume that is a minimal resolution, while is not. Inspired by Corollary 1.3, we define a new kind of minimal resolution:
Definition 8.1. Let be a projective variety over complex numbers. We say that a resolution of singularities is a -minimal resolution if for any smooth model one has that for all .
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 over the complex numbers, one has that:
(1) B-minimal resolutions of exist.
(2) Two different B-minimal resolutions are connected by desingularizations of -factorial terminal flops.
(3) If is a B-minimal resolution and is an arbitrary resolution of singularities, then can be connected by smooth blow-downs, smooth flips, and desingularizations of -factorial terminal flops.
8.2. The strong factorization theorem
Let
where if is connected by P-desingularizations of -factorial terminal flops. For , we say that if there exist and so that and is a smooth blow-down. Then, Theorems 1.1 and 1.2 imply the following.
Corollary 8.2. Given a threefold , let
Then has a unique minimal element.
In other words, if we consider the resolution of singularities inside , then there is a unique minimal resolution, which behaves similarly to the minimal resolution of a surface.
As a consequence, inside the space the following strong factorization theorem holds.
Theorem 8.3 (Strong factorization theorem for ). Assume that and are smooth threefolds which are birational to each other. Then there exists a smooth threefold such that inside one has for , .
Proof. Let be a common resolution. Then for , . Since the minimal element of is itself, one has that for , . □
8.3. Essential valuations
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 be a projective threefold over the complex numbers. We say that a divisorial valuation over is an almost essential valuation if for any P-minimal resolution one has that is an irreducible component of the exceptional locus of .
This name comes from the "essential valuation" in the theories of arc spaces.
Definition 8.5. Let be a variety. We say that a divisorial valuation over is an essential valuation if for any resolution of singularities one has that is an irreducible component of the exceptional locus of .
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 for some . There is exactly one -morphism over the singular point, which is obtained by blowing-up the origin. There is only one singular point on , which is defined by . Blowing-up the singular point more times, we get a resolution of singularities . From the construction we know that is a unique feasible resolution of . Since is terminal and -factorial, is the unique P-minimal resolution of . Hence, almost essential valuations of are those divisorial valuations which appear on . One can compute that such that . On the other hand, by [25, Lemma 15] we know that essential valuations of are and . Hence, , ..., 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 . The set of almost essential valuations carries more information of the singularity.
8.4. Derived categories
Let be a smooth variety. The bounded derived category of coherent sheaves of , denoted by , is an interesting subject of investigation. One possible method to study is to construct a semi-orthogonal decomposition of (refer to [26] for more information). Orlov [27] proved that a smooth blow-down yields a semi-orthogonal decomposition. In particular, if is a smooth surface, then the -MMP is a series of smooth blow-downs, thereby resulting in a semi-orthogonal decomposition of .
Now assume that is a smooth threefold and let
be the process of -minimal model program. According to Corollary 7.8 and Remark 7.6, can be factored into a composition of smooth blow-downs and P-desingularizations of flops, where is a P-minimal resolution of . If every P-desingularizations of flops that appears in the factorization is a smooth flop, then the sequence induces a semi-orthogonal decomposition of since smooth flops are derived equivalent [28].
Example 8.7. Let be the flip which is a quotient of an Atiyah flop by an -action [16, Example 2.7]. Then is smooth and has a singular point. Let be the smooth resolution obtained by blowing-up the singular point. Then is a sequence of MMP.
The factorization of the flip is exactly diagram (3) in Lemma 6.7, namely the diagram
where is a smooth flop and is a blow-down of a smooth curve. We know that there exists an equivalence of category and a semi-orthogonal decomposition . Hence, is a semi-orthogonal decomposition.
In general, a P-desingularization of a flop may not be derived equivalent since and may not be isomorphic in codimension one. Nevertheless, due to the symmetry between and , one might expect that a semi-orthogonal decomposition on will result in a semi-orthogonal decomposition on . It still hopeful that our approach will be effective for all smooth threefolds.
Use of AI tools declaration
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
Acknowledgments
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.
Conflict of interest
The authors declare there is no conflicts of interest.