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On blowup of secant varieties of curves

  • In this paper, we show that for a nonsingular projective curve and a positive integer k, the k-th secant bundle is the blowup of the k-th secant variety along the (k1)-th secant variety. This answers a question raised in the recent paper of the authors on secant varieties of curves.

    Citation: Lawrence Ein, Wenbo Niu, Jinhyung Park. On blowup of secant varieties of curves[J]. Electronic Research Archive, 2021, 29(6): 3649-3654. doi: 10.3934/era.2021055

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  • In this paper, we show that for a nonsingular projective curve and a positive integer k, the k-th secant bundle is the blowup of the k-th secant variety along the (k1)-th secant variety. This answers a question raised in the recent paper of the authors on secant varieties of curves.



    Throughout the paper, we work over an algebraically closed field k of characteristic zero. Let C be a nonsingular projective curve of genus g0, and L be a very ample line bundle on C. The complete linear system |L| embeds C into a projective space Pr:=P(H0(C,L)). For an integer k0, the k-th secant variety

    Σk=Σk(C,L)Pr

    of C in Pr is the Zariski closure of the union of (k+1)-secant k-planes to C.

    Assume that degL2g+2k+1. Then the k-th secant variety Σk can be defined by using the secant sheaf Ek+1,L and the secant bundle Bk(L) as follows. Denote by Cm the m-th symmetric product of C. Let

    σk+1:Ck×CCk+1

    be the morphism sending (ξ,x) to ξ+x, and p:Ck×CC the projection to C. The secant sheaf Ek+1,L on Ck+1 associated to L is defined by

    Ek+1,L:=σk+1,pL,

    which is a locally free sheaf of rank k+1. Notice that the fiber of Ek+1,L over ξCk+1 can be identified with H0(ξ,L|ξ). The secant bundle of k-planes over Ck+1 is

    Bk(L):=P(Ek+1,L)

    equipped with the natural projection πk:Bk(L)Ck+1. We say that a line bundle L on a variety X separates m+1 points if the natural restriction map H0(X,L)H0(ξ,L|ξ) is surjective for any effective zero-cycle ξX with length(ξ)=m+1. Notice that a line bundle L is globally generated if and only if L separates 1 point, and L is very ample if and only if L separates 2 points. Since degL2g+k, it follows from Riemann–Roch that L separates k+1 points. Then the tautological bundle OBk(L)(1) is globally generated. We have natural identifications

    H0(Bk(L),OBk(L)(1))=H0(Ck+1,Ek+1,)=H0(C,L),

    and therefore, the complete linear system |OBk(L)(1)| induces a morphism

    βk:Bk(L)Pr=P(H0(C,L)).

    The k-th secant variety Σk=Σk(C,L) of C in Pr can be defined to be the image βk(Bk(L)). Bertram proved that βk:Bk(L)Σk is a resolution of singularities (see [1,Section 1]).

    It is clear that there are natural inclusions

    C=Σ0Σ1Σk1ΣkPr.

    The preimage of Σk1 under the morphism βk is actually a divisor on Bk(L). Thus there exits a natural morphism from Bk(L) to the blowup of Σk along Σk1. Vermeire proved that B1(L) is indeed the blowup of Σ1 along Σ0=C ([3,Theorem 3.9]). In the recent work [2], we showed that Bk(L) is the normalization of the blowup of Σk along Σk1 ([2,Proposition 5.13]), and raised the problem asking whether Bk(L) is indeed the blowup itself ([2,Problem 6.1]). The purpose of this paper is to give an affirmative answer to this problem by proving the following:

    Theorem 1.1. Let C be a nonsingular projective curve of genus g, and L be a line bundle on C. If degL2g+2k+1 for an integer k1, then the morphism βk:Bk(L)Σk(C,L) is the blowup of Σk(C,L) along Σk1(C,L).

    To prove the theorem, we utilize several line bundles defined on symmetric products of the curve. Let us recall the definitions here and refer the reader to [2] for further details. Let

    Ck+1=C××Ck+1times

    be the (k+1)-fold ordinary product of the curve C, and pi:Ck+1C be the projection to the i-th component. The symmetric group Sk+1 acts on p1Lpk+1L in a natural way: a permutation μSk sends a local section s1sk+1 to sμ(1)sμ(k+1). Then p1Lpk+1L is invariant under the action of Sk+1, so it descends to a line bundle Tk+1(L) on the symmetric product Ck+1 via the quotient map q:Ck+1Ck+1. We have qTk+1(L)=p1Lpk+1L. Define a divisor δk+1 on Ck+1 such that the associated line bundle OCk+1(δk+1)=det(σk+1,(OCk×C)). Let

    Ak+1,L:=Tk+1(L)(2δk+1)

    be a line bundle on Ck+1. When k=0, we use the convention that T1(L)=E1,L=L and δ1=0.

    The main ingredient in the proof of Theorem 1.1 is to study the positivity of the line bundle Ak+1,L. Some partial results and their geometric consequences have been discussed in [2,Lemma 5.12 and Proposition 5.13]. Along this direction, we establish the following proposition to give a full picture in a general result describing the positivity of the line bundle Ak+1,L. This may be of independent interest.

    Proposition 1.2. Let C be a nonsingular projective curve of genus g, and L be a line bundle on C. If degL2g+2k+ for integers k,0, then the line bundle Ak+1,L on Ck+1 separates +1 points.

    In particular, if degL2g+2k, then Ak+1,L is globally generated, and if degL2g+2k+1, then Ak+1,L is very ample.

    In this section, we prove Theorem 1.1. We begin with showing Proposition 1.2.

    Proof of Proposition 1.2. We proceed by induction on k and . If k=0, then A1,L=L and degL2g+. It immediately follows from Riemann–Roch that L separates +1 points. If =0, then degL2g+2k. By [2,Lemma 5.12], Ak+1,L separates 1 point.

    Assume that k1 and 1. Let z be a length +1 zero-dimensional subscheme of Ck+1. We aim to show that the natural restriction map

    rz,k+1,L:H0(Ck+1,Ak+1,L)H0(z,Ak+1,L|z)

    is surjective. We can choose a point pC such that Xp contains a point in the support of z, where Xp is the divisor on Ck+1 defined by the image of the morphism CkCk+1 sending ξ to ξ+p. Let y:=zXp be the scheme-theoretic intersection, and Ix:=(Iz:IXp), which defines a subscheme x of z in Ck+1, where Iz and IXp are ideal sheaves of z and Xp in Ck+1, respectively. We have the following commutative diagram

    rz,k+1,L:H0(Ck+1,Ak+1,L)H0(z,Ak+1,L|z)

    where all rows and columns are short exact sequences. By tensoring with Ak+1,L and taking the global sections of last two rows, we obtain the commutative diagram with exact sequences

    rz,k+1,L:H0(Ck+1,Ak+1,L)H0(z,Ak+1,L|z)

    in which we use the fact that H1(Ak+1,L(Xp))=0 (see the proof of [2,Lemma 5.12]). Note that Ak+1,L(Xp)=Ak+1,L(p) and Ak+1,L|XpAk,L(2p), where we identify Xp=Ck.

    Since length(y)length(z)=+1 and degL(2p)2g+2(k1)+, the induction hypothesis on k implies that ry,k,L(2p) is surjective. On the other hand, if x=, which means that z is a subscheme of Xp, then trivially rx,k+1,L(p) is surjective. Otherwise, suppose that x. By the choice of Xp, we know that y is not empty, and therefore, we have length(x)length(z)1=. Now, degL(p)2g+2k+(1), so the induction hypothesis on implies that L(p) separates points. In particular, rx,k+1,L(p) is surjective. Hence rz,k+1,L is surjective as desired.

    Lemma 2.1. Let φ:XY be a finite surjective morphism between two varieties. If φ1(q) is scheme theoretically a reduced point for each closed point qY, then φ is an isomorphism.

    Proof. Note that φ is proper, injective, and unramifield. Then it is indeed a classical result that φ is an isomorphism. Here we give a short proof for reader's convenience. The problem is local. We may assume that X=SpecB and Y=SpecA for some rings A,B. We may regard A as a subring of B. For any qY, let p:=φ1(q)X. It is enough to show that the localizations A:=Amq and B:=Bmp are isomorphic. Let mq,mp be the maximal ideals of the local rings A,B, respectively. The assumption says that mqB=mp. We have

    B/AAA/mq=B/(mqB+A)=B/(mp+A)=0.

    By Nakayama lemma, we obtain B/A=0.

    We keep using the notations used in the introduction. Recall that C is a nonsingular projective curve of genus g0, and L is a very ample line bundle on C. Consider ξkCk and xC, and let ξ:=ξk+xCk+1. The divisor ξk spans a k-secant (k1)-plane P(H0(ξk,L|ξk)) to C in P(H0(C,L)), and it is naturally embedded in the (k+1)-secant k-plane P(H0(ξ,L|ξ)) spanned by ξ. This observation naturally induces a morphism

    αk,1:Bk1(L)×CBk(L).

    To see it in details, we refer to [1,p.432,line –5]. We define the relative secant variety Z=Zk1 of (k1)-planes in Bk(L) to be the image of the morphism αk,1. The relative secant variety Z is a divisor in the secant bundle Bk(L), and it is the preimage of (k1)-th secant variety Σk1 under the morphism βk. It plays the role of transferring the codimension two situation (Σk,Σk1) into the codimension one situation (Bk(L),Z). We collect several properties of Z.

    Proposition 2.2. ([2,Proposition 3.15,Theorem 5.2,and Proposition 5.13]) Recall the situation described in the diagram

    αk,1:Bk1(L)×CBk(L).

    Let H be the pull back of a hyperplane divisor of Pr by βk, and let IΣk1|Σk be the ideal sheaf on Σk defining the subvariety Σk1. Then one has

    1. OBk(L)((k+1)HZ)=πkAk+1,L.

    2. Riβk,OBk(L)(Z)={IΣk1|Σkifi=00ifi>0.

    3. IΣk1|ΣkOBk(L)=OBk(L)(Z).

    As a direct consequence of the above proposition, we have an identification

    H0(Ck+1,Ak+1,L)=H0(Σk,IΣk1|Σk(k+1)).

    We are now ready to give the proof of Theorem 1.1.

    Proof of Theorem 1.1. Let

    b:˜Σk:=BlΣk1ΣkΣk

    be the blowup of Σk along Σk1 with exceptional divisor E. As IΣk1|ΣkOBk(L)=OBk(L)(Z) (see Proposition 2.2), there exists a morphism α from Bk(L) to the blowup ˜Σk fitting into the following commutative diagram

    b:˜Σk:=BlΣk1ΣkΣk

    We shall show that α is an isomorphism.

    Write V:=H0(Σk,IΣk1|Σk(k+1)). As proved in [2,Theorem 5.2], IΣk1|Σk(k+1) is globally generated by V. This particularly implies that on the blowup ˜Σk one has a surjective morphism VO˜ΣkbOΣk(k+1)(E), which induces a morphism

    γ:˜ΣkP(V).

    On the other hand, one has an identification V=H0(Ck+1,Ak+1,L) by Proposition 2.2. Recall from Proposition 1.2 that Ak+1,L is very ample. So the complete linear system |V|=|Ak+1,L| on Ck+1 induces an embedding

    ψ:Ck+1P(V).

    Also note that α(bOΣk(k+1)(E))=βkOΣk(k+1)(Z)=πkAk+1,L by Proposition 2.2. Hence we obtain the following commutative diagram

    ψ:Ck+1P(V).

    Take an arbitrary closed point x˜Σk, and consider its image x:=b(x) on Σk. There is a nonnegative integer mk such that xΣmΣm1Σk. In addition, the point x uniquely determines a degree m+1 divisor ξm+1,x on C in such a way that ξm+1,x=ΛC, where Λ is a unique (m+1)-secant m-plane to C with xΛ (see [2,Definition 3.12]). By [2,Proposition 3.13], β1k(x)Ckm and πk(β1k(x))=ξm+1,x+CkmCk+1. Consider also x:=γ(x) which lies in the image of ψ. As ψ is an embedding, we may think x as a point of Ck+1. Now, through forming fiber products, we see scheme-theoretically

    α1(x)π1k(x)β1k(x).

    However, the restriction of the morphism πk on β1k(x) gives an embedding of Ckm into Ck+1. This suggests that π1k(x)β1k(x) is indeed a single reduced point, and so is α1(x). Finally by Lemma 2.1, α is an isomorphism as desired.



    [1] A. Bertram, Moduli of rank-2 vector bundles, theta divisors, and the geometry of curves in projective space, J. Differential Geom., 35, (1992), 429–469.
    [2] Singularities and syzygies of secant varieties of nonsingular projective curves. Invent. Math. (2020) 222: 615-665.
    [3] Some results on secant varieties leading to a geometric flip construction. Compositio Math. (2001) 125: 263-282.
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