On the fine Simpson moduli spaces of 1-dimensional sheaves supported on plane quartics

Oleksandr Iena
From the journal Open Mathematics

Abstract

A parametrization of the fine Simpson moduli spaces of 1-dimensional sheaves supported on plane quartics is given: we describe the gluing of the Brill-Noether loci described by Drézet and Maican, provide a common parameter space for these loci, and show that the Simpson moduli space M = M4m ± 1(ℙ2) is a blow-down of a blow-up of a projective bundle over a smooth moduli space of Kronecker modules. Two different proofs of this statement are given.

MSC 2010: 14D20

Introduction

Fix an algebraically closed field, char = 0. Let V be a 3-dimensional vector space over and let ℙ2 = ℙ V be the corresponding projective plane. Let P(m) = dm+c be a linear polynomial in m with integer coefficients, d > 0. Let M = Mdm+c = Mdm+c(ℙ2) be the Simpson moduli space (cf. [1]) of semi-stable sheaves on ℙ2 with Hilbert polynomial dm+c. As shown in [2], M is a projective irreducible locally factorial variety of dimension d2+1. In general, moduli space M parameterizes the s-equivalence classes, i. e., there is a bijection between the closed points of M and the s-equivalence classes of semistable sheaves on ℙ2 with Hilbert polynomial dm+c. This way different isomorphism classes of sheaves could be identified in the moduli space. However, if gcd(c, d) = 1, every semi-stable sheaf is stable, s-equivalence coincides with the notion of isomorphism, and M is a fine moduli space whose closed points are in bijection with the isomorphism classes of stable sheaves on ℙ2 with Hilbert polynomial dm+c. In this case there is a universal family of stable sheaves parameterized by M such that every family of (dm+c)-sheaves is obtained (up to a twist) as a pull-back of this universal family. As demonstrated in [2, Proposition 3.6], M is smooth in this case.

In [3] and [4] it was proved that Mdm+cMdm+c if and only if d = d′ and c = ± c′ mod d. Therefore, in order to understand, for fixed d, the Simpson moduli spaces Mdm+c it is enough to understand at most [d/2]+1 different moduli spaces.

For d ⩽ 3 the fine moduli spaces Mdm+c are completely understood. By [2, Théorème 5.1] Mdm+c ≅ ℙ(SdV*) for d = 1, and d = 2. For d = 3, M3m ± 1 is isomorphic to the universal cubic plane curve

{(C,p)P(S3V)×P2pC}.

These are the simplest and rather trivial examples of the Simpson moduli spaces of planar 1-dimensional sheaves. Each of them can be endowed with an open covering such that the coordinates in every open chart come from some globally defined objects, which can be called global coordinates (or moduli, using the original terminology of Riemann), and allow one to define the subvarieties of the moduli space in terms of equations in these coordinates.

Indeed, for ℙ(SdV*), which is constructed as Sd V*/GL1( ), every basis of Sd V* provides global homogeneous coordinates of ℙ(Sd V*). Being a universal planar curve, the moduli spaces M3m ± 1 have a nice description as a quotient (by a non-reductive group) of the variety of matrices

A=xypq,x,yH0(P2,OP2(1)),p,qH0(P2,OP2(2)),detA0,xy0,

which provides convenient global coordinates for M3m ± 1 and allows one to study the moduli spaces in more details (cf. [5]).

By [3] one has the isomorphisms M4m+bM4m−1 for d = 4 and odd b. In [6] a description of the moduli space M4m−1 is given in terms of two strata (Brill-Noether loci): an open stratum M0 and its closed complement M1 in codimension 2. The open stratum is naturally described as an open subvariety of a projective bundle 𝔹 → N associated to a vector bundle of rank 12 over a smooth 6-dimensional projective variety N. The closed stratum is the universal quartic planar curve. Each stratum is described as a geometric quotient of a set of morphisms of locally free sheaves modulo non-reductive algebraic groups. The morphisms come from the Beilinson’s resolutions and can be seen as coordinates (parameters, moduli) of the corresponding strata.

The main result of the paper

The aim of this paper is to “glue together” the parameterizations of the strata from [6] and to equip M := M4m−1, and hence all fine Simpson moduli spaces of 1-dimensional sheaves supported on plane quartics, with open charts parameterized by convenient coordinates.

The main result of this paper is an observation that every sheaf from M can be given as the cokernel of a morphism

2OP2(3)3OP2(2)OP2(2)3OP2(1),

which provides a common parameter space for the strata from [6] and gives a simple way to deform the sheaves from M0 to the ones from M1. This parametrization can be seen as a natural generalization of the parametrizations of the strata from [6]. The new parameter space is deduced from the closer understanding of the complement 𝔹′ of M0 in 𝔹. The way we obtain it immediately provides over 𝔹͠ :=Bl𝔹′ 𝔹 a family of stable sheaves on ℙ2 with Hilbert polynomial 4m − 1 and thus a map from 𝔹͠ to M. Under this map the exceptional divisor D of the blow-up 𝔹͠ → 𝔹 is a ℙ1-bundle over the closed stratum M1, which leads to the statement of Theorem 3.1 that M is a blow-down to M1 of the exceptional divisor D of the blow-up 𝔹͠. This result coincides with the statement of [7, Theorem 3.1], which appeared earlier. Our methods are, however, significantly different.

Theorem 3.1 can be also discovered by just looking at the geometric data involved and seeing the corresponding statement, which provides a very geometric proof. The exceptional divisor D can be naturally seen as a projective bundle over the closed stratum M1 with fibres being projective lines. The variety 𝔹͠ can be blown down along these fibres.

Structure of the paper

In Section 1 we review the description of the strata of M from [6] and give a description of the degenerations to the closed stratum. In Section 2 we present a geometric description of the fibres of the bundle 𝔹 → N and construct local charts around the closed subvariety 𝔹′ := 𝔹 ∖ M0. As a side remark we provide here a simple computation of the Poincaré polynomial of M that follows directly from [8] and [6]. The geometric description of the fibres of 𝔹 → N allows one to describe the blow up 𝔹͠ → 𝔹 geometrically and to see Theorem 3.1 in Section 3 by just looking at the geometric data involved. In Section 4 we construct a common parameter space for the sheaves in M and rigorously prove Theorem 3.1.

Some notations and conventions

Dealing with homomorphisms between direct sums of line bundles and identifying them with matrices, we consider the matrices acting on elements from the right. In particular, a section of a direct sum of line bundles 𝓔1⊕ ⋯ ⊕ 𝓔m is identified with the row-vector of sections of 𝓔i, i = 1,…, m.

1 M4m−1 as a union of two strata

As shown in [2] M = M4m−1 is a smooth projective variety of dimension 17. By [6] M is a disjoint union of two strata M1 and M0 such that M1 is a closed subvariety of M of codimension 2 and M0 is its open complement.

1.1 Closed stratum

The closed stratum M1 is a closed subvariety of M of codimension 2 given by the condition h0(𝓔) ≠ 0.

The sheaves from M1 possess a locally free resolution

02OP2(3)z1q1z2q2OP2(2)OP2E0,(1)

with linear independent linear forms z1 and z2 on ℙ2. M1 is a geometric quotient of the variety of injective matrices z1q1z2q2 as above by the non-reductive group

(Aut(2OP2(3))×Aut(OP2(2)OP2))/.

The points of M1 are the isomorphism classes of sheaves that are non-trivial extensions

0OCEO{p}0,(2)

where C is a plane quartic given by the determinant of z1q1z2q2 from (1) and pC a point on it given as the common zero set of z1 and z2.

This describes M1 as the universal plane quartic, the quotient map is given by

z1q1z2q2(C,p),C=Z(z1q2z2q1),p=Z(z1,z2).

M1 is smooth of dimension 15.

Let M11 be the closed subvariety of M1 defined by the condition that p is contained on a line L contained in C. Equivalently, a matrix from (1) represents a point in M11 if and only if it lies in the orbit of a matrix of the form z10z2q2. The dimension of M11 is 12.

Lemma 1.1

The sheaves in M11are non-trivial extensions

0OL(2)FOC0,

where Cis a cubic and L is a line.

Proof

Consider the isomorphism class of 𝓕 with resolution

02OP2(3)l0whOP2(2)OP2F0.(3)

This gives the commutative diagram with exact rows and columns.

Therefore, 𝓕 is an extension

0OL(2)FOC0,

which is nontrivial since 𝓕 is stable. This proves the required statement. □

Let M10 denote the open complement of M11 in M1.

1.2 Open stratum

The open stratum M0 is the complement of M1 given by the condition h0(𝓔) = 0, it consists of the isomorphism classes [𝓔A] of the cokernels 𝓔A of the injective morphisms

OP2(3)2OP2(2)A3OP2(1)(4)

such that the (2 × 2)-minors of the linear part z0z1z2w0w1w2of A=q0q1q2z0z1z2w0w1w2 are linear independent.

1.2.1 M0 as a geometric quotient

M0 is an open subvariety in the geometric quotient 𝔹 of the variety 𝕎s of stable matrices as in (4) (see [8, Proposition 7.7] for details) by the group

Aut(OP2(3)2OP2(2))×Aut(3OP2(1)).

Its complement in 𝔹 is a closed subvariety 𝔹′ corresponding to the matrices with zero determinant.

1.2.2 Extensions

If the maximal minors of the linear part of A corresponding to a point [𝓔A] in M0 have a linear common factor, say l, then det(A) = lh and 𝓔A is in this case a non-split extension

0OL(2)EAOC0,(5)

where L = Z(l), C′ = Z(h).

The subvariety M01 of such sheaves is closed in M0 and locally closed in M. Its boundary coincides with M11.

1.2.3 Twisted ideals of 3 points on a quartic

Let M00 denote the open complement of M01 in M0. In this case the maximal minors of the linear part of A are coprime, and the cokernel 𝓔A of (4) is a part of the exact sequence

0EAOC(1)OZ0,

where C is a planar quartic curve given by the determinant of A from (4) and Z is the zero dimensional subscheme of length 3 given by the maximal minors of the linear submatrix of A. Notice that in this case the subscheme Z does not lie on a line.

Proposition 1.2

1) Every sheaf in M1is a degeneration of sheaves from M00. This corresponds to a degeneration of ZC, where Z is a zero-dimensional scheme of length 3 not lying on a line and C is a quartic curve, to a flag ZC with Z contained in a line L that is not included in C. The limit corresponds to the point in M1described by the point (LC) ∖ Z on the quartic curve C.

2) The sheaves from M1given by pairs (C, p) such that p belongs to a line L contained in C, i. e., those from M11, are also degenerations of sheaves from M01. This corresponds to degenerations of extensions(5)without sections to extensions with sections.

The proof follows from the considerations below.

1.3.1 Degenerations along M00

Fix a curve C ⊆ ℙ2 of degree 4, C = Z(f), fΓ(ℙ2, 𝓞2(4)). Let ZC be a zero-dimensional scheme of length 3 contained in a line L = Z(l), lΓ(ℙ2, 𝓞2(1)). Let 𝓕 = 𝓘Z(1) be the twisted ideal sheaf of Z in C so that there is an exact sequence

0FOC(1)OZ0.

Lemma 1.3

In the notations as above, the twisted ideal sheaf 𝓕 = 𝓘Z(1) is semistable if and only if L is not contained in C.

Proof

Let us construct a locally free resolution of 𝓕. Let gΓ(ℙ2, 𝓞2(3)) such that 𝓞Z is given by the resolution

0OP2(4)lgOP2(3)OP2(1)glOP2OZ0.

Since Z = Z(l, g) is contained in C = Z(f), one concludes that f = lhwg for some wΓ(ℙ2, 𝓞2(1)) and hΓ(ℙ2, 𝓞2(3)). This gives the following commutative diagram with exact rows and columns.

Therefore, 𝓕 possesses a locally free resolution

02OP2(3)lgwhOP2(2)OP2F0.

In particular, if l and w are linear independent, which is true if and only if f is not divisible by l, this is a resolution of type (1), hence 𝓕 is a sheaf from M1.

If l and w are linear dependent, then without loss of generality we can assume that w = 0, which gives an extension

0OCFOL(2)0,C=Z(h),

and thus a destabilizing subsheaf 𝓞C of 𝓕. This concludes the proof. □

Let H(3, 4) be the flag Hilbert scheme of zero-dimensional schemes of length 3 on plane projective curves C ⊆ ℙ2 of degree 4. Let H′(3, 4) ⊆ H(3, 4) be the subscheme of those flags ZC such that Z lies on a linear component of C. Using the universal family on H(3, 4), one obtains a natural morphism

H(3,4)H(3,4)M,

whose image coincides with MM01.

Its restriction to the open subvariety H0(3, 4) of H(3, 4) of flags ZC ⊆ ℙ2 such that Z does not lie on a line gives an isomorphism

H0(3,4)M00.

Over M1 one gets one-dimensional fibres: over an isomorphism class in M1, which is uniquely defined by a point pC on a curve of degree 4, the fibre can be identified with the variety of lines through p that are not contained in C, i. e., with a projective line without up to 4 points.

Remark 1.4

Notice that the subvariety H′(3, 4) is a3-bundle overV* × ℙ S3V*, the fibre over the pair (L, C′) of a line L and a cubic curve Cis the Hilbert scheme L[3].

As shown in [9, Theorem 3.3 and Proposition 4.4], the blow-up of H(3,4) along H′(3, 4) can be blown down along the fibres L[3]to the blow-up M͠ := BlM1M.

Fig. 1

Moduli space M = M4m−1(ℙ2).

1.3.2 Degenerations along M01

For a fixed line L and a fixed cubic curve C′ one can compute Ext1(𝓞C, 𝓞L(−2)) ≅3. Therefore, using [10] one gets a projective bundle P over ℙ V* × ℙ S3V* with fibre ℙ2 and a universal family of extensions on it parameterizing the extensions

0OL(2)FOC0,LPV,CPS3V.

This provides a morphism PM and describes the degenerations of sheaves from M01 to sheaves in M11.

2 Description of 𝔹

𝔹 is a projective bundle associated to a vector bundle of rank 12 over the moduli space N = N(3; 2, 3) of stable (2 × 3) Kronecker modules, i. e., over the GIT-quotient of the space 𝕍s of stable (2 × 3)-matrices of linear forms on ℙ2 by Aut(2𝓞2(−2)) ×Aut(3𝓞2(−1)).

The projection 𝔹 → N is induced by

q0q1q2z0z1z2w0w1w2z0z1z2w0w1w2.

For more details see [8, Proposition 7.7].

2.1 The base N

The subvariety N′ ⊆ N corresponding to the matrices whose minors have a common linear factor is isomorphic to P2 = ℙ V*, the space of lines in ℙ2, such that a line corresponds to the common linear factor of the minors of the corresponding Kronecker module z0z1z2w0w1w2.

The blow up of N along N′ is isomorphic to the Hilbert scheme H=P2[3] of 3 points in ℙ2 (cf. [11, Théorème 4]). The exceptional divisor H′ ⊆ H is a ℙ3-bundle over N′, whose fibre over 〈l〉 ∈ P2 is the Hilbert scheme L[3] of 3 points on L = Z(l). The class in N of a Kronecker module z0z1z2w0w1w2 with coprime minors corresponds to the subscheme of 3 non-collinear points in ℙ2 defined by the minors of the matrix.

2.2 The fibres of 𝔹 → N

2.2.1 Fibres over N ∖ N′

A fibre over a point from NN′ can be seen as the space of plane quartics through the corresponding subscheme of 3 non-collinear points. Indeed, consider a point from NN′ given by a Kronecker module z0z1z2w0w1w2 with coprime minors d0, d1, d2. The fibre over such a point consists of the orbits of injective matrices

q0q1q2z0z1z2w0w1w2,q0,q1,q2S2V,

under the group action of

Aut(OP2(3)2OP2(2))×Aut(3OP2(1)).

If two matrices

q0q1q2z0z1z2w0w1w2,Q0Q1Q2z0z1z2w0w1w2

lie in the same orbit of the group action, then their determinants are equal up to a multiplication by a non-zero constant. Vice versa, if the determinants of two such matrices are equal, qQ = (q0Q0, q1Q1, q2Q2) lies in the syzygy module of d0d1d2, which is generated by the rows of z0z1z2w0w1w2 by Hilbert-Burch theorem. This implies that qQ is a combination of the rows and thus the matrices lie on the same orbit.

2.2.2 Fibres over N′

A fibre over 〈l〉 ∈ N′ can be seen as the join J(L*, ℙ S3V*) ≅ ℙ11 of L* ≅ ℙ H0(L, 𝓞L(1)) ≅ ℙ1 and the space of plane cubic curves ℙ(S3V*) ≅ ℙ9. To see this assume l = x0, i. e., 〈x0〉 is considered as the class of

x20x0x1x00.

Then the fibre over [x20x0x1x00] is given by the orbits of matrices

q0(x0,x1,x2)q1(x1,x2)q2(x1,x2)x20x0x1x00(6)

and can be identified with the projective space ℙ(2H0(L, 𝓞L(2))⊕ S2V*).

For a linear form w = w(x1, x2) such that q2(x1, x2) = q2(0, x2) + x1w, one can write

q1(x1,x2)=q1(x1,x2)x2w,q2(x1,x2)=q2(x2)+x1w,q2(x2)=q2(0,x2).

Abusing notations by renaming q1 and q2 into q1 and q2 respectively, rewrite the matrix (6) as

q0(x0,x1,x2)q1(x1,x2)x2wq2(x2)+x1wx20x0x1x00.

Its determinant equals

x0(x0q0(x0,x1,x2)+x1q1(x1,x2)+x2q2(x2)).

This allows to reinterpret the fibre as the projective space

P(H0(L,OL(1))S3V)J(L,PS3V).

J(L*, ℙ(S3V*)) ∖ L* is a rank 2 vector bundle over ℙ(S3V*), whose fibre over a cubic curve C′ ∈ ℙ S3V* is identified with the isomorphism classes of the extensions (5) from M01 with fixed L and C′. This corresponds to the projective plane joining C′ with L* inside the join J(L*, ℙ(S3V*)).

Fig. 2

The fibre of 𝔹 over LN′.

The points of J(L*, ℙ(S3V*)) ∖ L* parameterize the extensions (5) from M01 with fixed L.

2.3 Description of 𝔹′

𝔹′ is a union of lines L* from each fibre over N′ (as explained above). It is isomorphic to the tautological ℙ1-bundle over N′ = P2

{(L,p)P2×P2LP2,pL}.(7)

Equivalently (cf. [6, p. 36]), 𝔹′ is isomorphic to the projective bundle associated to the tangent bundle T P2. The fibre ℙ1 of 𝔹′ over, say, line L = Z(x0) ⊆ ℙ2 can be identified with the space of classes of matrices (4) with zero determinant

0x2wx1wx20x0x1x00,w=γx1+δx2,γ,δP1.(8)

2.4 Side remark: the Poincaré polynomial of M

The understanding of 𝔹 and 𝔹′ provides an easy way to compute the Poincaré polynomial P(M) of M. We present here the computation as a side remark. The following is a direct consequence of [8, Proposition 7.7], where a description of 𝔹 is given, and [6, p. 36], where the complement 𝔹′ is described. Our computation is slightly more straightforward than the ones from [9, Corollary 5.2] and [12, Theorem 5.2]. Notice that P(M) has been also computed using a torus action on M in [13, Theorem 1.1].

Recall that the Poincaré polynomial P(X)[t] ∈ ℤ[t] of a topological space X is defined by P(X)(t) = ∑i ⩾ 0 dim Hi(X, ℚ) ⋅ ti. Recall that the virtual Poincaré polynomial is a unique map Pv(⋅)(t) from algebraic varieties to ℤ[t] that gives a ring homomorphism from the Grothendieck ring of varieties over to ℤ[t] such that Pv(X)(t) = P(X)(t) for smooth projective varieties X. In particular, this means that Pv(X) = Pv(Y)+Pv(XY) if Y is a closed subvariety of X and Pv(X1 × X2) = Pv(X1) ⋅ Pv(X2). The latter property implies that the virtual Poincaré polynomial of a locally trivial fibration over X with fibre Y equals to the product Pv(X) ⋅ Pv(Y).

Proposition 2.1

The Poincaré polynomial of M equals 1+2t2+6t4+10t6+14t8+15t10+16t12+16t14+16t16+16t18+16t20+16t22+15t24+14t26+10t28+6t30+2t32}+t34.

Proof

Since M is a smooth projective variety, P(M) = Pv(M). Since M1 is a closed subvariety in M and M0 is its open complement, since 𝔹′ is a closed subvariety in 𝔹 and its complement 𝔹 ∖ 𝔹′ is isomorphic to M0, we obtain

Pv(M)=Pv(M0)+Pv(M1),Pv(B)=Pv(BB)+Pv(B),Pv(M0)=Pv(BB).

Therefore, Pv(M) = Pv(M0)+Pv(M1) = Pv(𝔹 ∖ 𝔹′)+Pv(M1) = Pv(𝔹) − Pv(𝔹′)+Pv(M1). Since 𝔹 is a projective bundle over N with fibre ℙ11, one gets Pv(𝔹) = Pv(N) ⋅ Pv(ℙ11). Similarly, since 𝔹′ is a ℙ1-bundle over N′ ≅ ℙ2 and the universal quartic M1 is a ℙ13-bundle over ℙ2, we obtain Pv(𝔹′) = Pv(ℙ2) ⋅ Pv(ℙ1) and Pv(M1) = Pv(ℙ2) ⋅ Pv(ℙ13). Therefore,

Pv(M)=Pv(N)Pv(P11)Pv(P2)Pv(P1)+Pv(P2)Pv(P13)

By [14, page 90] the (virtual) Poincaré polynomial of H is Pv(H) = P(H) = 1+2t2+5t4+6t6+5t8+2t10+t12. As H is a blow-up of N at N′ by [11, Théorème 4], one gets Pv(N) = Pv(NN′)+Pv(N′) = Pv(HH′)+Pv(N′) = Pv(H) − Pv(H′)+Pv(N′) = Pv(H) − Pv(ℙ2) ⋅ Pv(ℙ3)+Pv(ℙ2) because H′ is a ℙ3-bundle over N′. Using this and P(Pn)=1t2(n+1)1t2, we get the result. □

Lemma 2.2

LetLNbe the class of the Kronecker module

x20x0x1x00.

Then there is an open neighbourhood of L that can be identified with an open neighbourhoodUof zero in the affine space6via the map6UN,

(α,β,a,b,c,d)x2cx1x¯0x1x¯0+ax1+bx2dx2,

with0 = x0 + αx1 + βx2, which establishes a local section of the quotient 𝕍sN.

Proof

In some open neighbourhood U of zero in 6 the morphism

UVs,(α,β,a,b,c,d)x2cx1x¯0x1x¯0+ax1+bx2dx2

is well-defined. Notice that two Kronecker modules of the form

x2cx1x¯0x1x¯0+ax1+bx2dx2

can lie in the same orbit of the group action if and only if the matrices are equal. Therefore, the morphism

(α,β,a,b,c,d)x2cx1x¯0x1x¯0+ax1+bx2dx2

is injective. □

Remark 2.3

By abuse of notation we identifyUwith its image inN.

Lemma 2.4

Nis cut out inUby the equationsa = b = c = d = 0.

Proof

The maximal minors of x2cx1x¯0x1x¯0+ax1+bx2dx2 are

cdx1x2+x¯0(x¯0ax1bx2),dx22x¯0x1,x2(x¯0ax1bx2)cx12.

Clearly these minors have a common linear factor if a, b, c, d vanish. On the other hand the condition c = d = 0 is necessary to ensure the reducibility of these quadratic forms. If c = d = 0, the conditions a = b = 0 are necessary for the minors to have a common factor. □

Lemma 2.5

The restriction of 𝔹 toUis a trivial11-bundle. Identifying11with the projective space

P(S2V2Span(x12,x1x2,x22)),

i. e., a point in11is identified with the class of the triple of quadratic forms

(q0(x0,x1,x2),q1(x1,x2),q2(x1,x2)),

one can identifyU × ℙ11, and hence 𝔹|U, with the classes of matrices

q0(x0,x1,x2)q1(x1,x2)q2(x1,x2)x2cx1x¯0x1x¯0+ax1+bx2dx2.(9)

Assuming one of the coefficients ofq0, q1, q2equal to 1, we get local charts of the formU × 11and local sections of the quotient 𝕎s →𝔹.

Proof

It is enough to notice that as in (6) one can get rid of x0 in the expressions of q1 and q2.

Charts 𝔹(γ) and 𝔹(δ)

In order to get charts around [A] ∈ 𝔹′,

A=0x2wx1wx20x¯0x1x¯00,w=γx1+δx2,γ,δP1,

rewrite (9), similarly to what we already did with (6) in 2.2.2, in the form

q0(x0,x1,x2)q1(x1,x2)x2wq2(x2)+x1wx2cx1x¯0x1x¯0+ax1+bx2dx2.(10)

Putting γ = 1 or δ = 1, we get charts around 𝔹′, each isomorphic to U × × 10. Denote them by 𝔹(γ) and 𝔹(δ) respectively. Their coordinates are those of U together with δ respectively γ and the coefficients of qi, i = 0, 1, 2.

The equations of 𝔹′ are those of N′ in U, i. e., a = b = c = d = 0, and the conditions imposed by vanishing of q0, q1, q2.

Remark 2.6

Notice that these equations generate the ideal given by the vanishing of the determinant of(10)

3 Description of M

Consider the blow-up 𝔹͠ = Bl𝔹′𝔹. Let D denote its exceptional divisor.

Theorem 3.1

𝔹͠ is isomorphic to the blow-up ≔ BlM1M. The exceptional divisor ofcorresponds toDunder this isomorphism. The fibres of the morphismDM1over the point ofM1represented by a pointpon a quartic curveCis identified with the projective line of lines in2passing throughp.

3.1 A rather intuitive explanation

Before rigorously proving this, let us explain how to arrive to Theorem 3.1 and see it just by looking at the geometric data involved. What follows in not completely rigorous but provides, in our opinion, a nice geometric picture.

Blowing up 𝔹 along 𝔹′ substitutes 𝔹′ by the projective normal bundle of 𝔹′. So a point of 𝔹′ represented by a line LP2 and a point pL, which is encoded by some 〈w〉 ∈ ℙH0(L, 𝓞L(1)), is substituted by the projective space D(L,p) of the normal space T(L,p) 𝔹/T(L,p)𝔹′ to 𝔹′ at (L, p).

As 𝔹 is a projective bundle over N, and 𝔹′ is a ℙ1-bundle over N′, the normal space is a direct sum of the normal spaces along the base and along the fibre. Therefore, D(L,p) is the join of the corresponding projective spaces: of ℙ3 = L[3] (normal projective space to N′ in N at LN′) and ℙ9 = ℙ(S3V*) (normal projective space to L* in J(L*, ℙ(S3V*)) at pLL*; notice that the normal projective bundle of L*J(L*, ℙ(S3V*)), i. e., ℙ1 ⊆ ℙ11, is trivial).

The fibre J(L*, ℙ(S3V*)) of 𝔹 → N over LN′ is substituted under the blow-up by the fibre that consists of two components: the first component is the blow-up of J(L*, ℙ(S3V*)) along L*, the second one is a projective bundle over L* with the fibre ℙ13 = J( L[3], ℙ(S3V*) ), the components intersect along L* × ℙ(S3V*).

Fig. 3

The fibre of 𝔹͠ over LN′.

The space L[3] is naturally identified with the projective space of cubic forms on L* whereas ℙ(S3V*) is clearly the space of cubic curves on ℙ2.

Assume L = Z(x0) such that {x0, x1, x2} is a basis of V* = H0(ℙ2, 𝓞2(1)). Identifying x1 and x2 with their images in H0(L, 𝓞L(1)), {x1, x2} is a basis of H0(L, 𝓞L(1)).

We conclude that the join of L[3] and ℙ S3V* can be identified with the projective space corresponding to the vector space

{λx0h+λwgh(x0,x1,x2)S3V,g(x1,x2)H0(L,OL(3)),(λ,λ)2},

i. e., the space of planar quartic curves through the point p = Z(x0, w).

So the exceptional divisor of the blow-up Bl𝔹′𝔹 is a projective bundle with fibre over (L, p) being interpreted as the space of quartic curves through p. This way we obtain a map from the exceptional divisor to the universal quartic M1. Its fibre over a pair pC is identified with the space of lines LP2 through p, i. e., with a projective line. Contracting the exceptional divisor along these lines one gets M. The contraction is possible by [15, 16, 17], which can be seen as follows.

The fibre of DM1 over a pair pC may be identified with the fibre Bp over p ∈ ℙ2 of the map 𝔹′ → ℙ2 given by the projection to the second factor (cf. (7). Every two points (L, p) and (L′, p) of Bp ⊆ 𝔹′ are substituted by the projective spaces J( L[3], ℙ(S3V*)) and J(L[3], ℙ(S3V*)) respectively, each of which is naturally identified with the space of quartics through p. Assume without loss of generality p = 〈0, 0, 1〉.

The fibre Bp in this case is identified with the space of lines in ℙ2 through p, i. e., with the projective line in N′ = P2 = ℙV* consisting of classes of linear forms αx0 + βx1, 〈α, β〉 ∈ ℙ1. The fibre has a standard covering Bp,0 = {x0 + βx1}, Bp,1 = {αx0 + x1}, which is induced by the standard covering of P2. The elements of the fibre corresponding to the points of Bp,0 are the equivalence classes of matrices

A0=0x1x2x1x1x20x0+βx1x1(x0+βx1)0.

The elements of the fibre corresponding to the points of Bp,1 are the equivalence classes of matrices

A1=0x0x2x0x0x20αx0+x1x0(αx0+x1)0.

In this way, we have chosen, so to say, the normal forms for the representatives of the Bp. For β = α−1, i. e., on the intersection of Bp,0 and Bp,1, these matrices are equivalent. One computes that (gA1h = A0) for matrices

g=α3α2x0αx1α2x201000α,h=100α1α2000α1

with determinants

detg=α4,deth=α3.

Consider the automorphism WsξWs,AgAh. Then

det(ξ(A1+B1))=αdet(A1+B1).(11)

From (11) it follows that the restriction of the ideal sheaf of D to a fibre of the morphisms DM1 is 𝓞1(1). By [15, 16, 17], this means that one can blow down D in 𝔹͠ along the map DM1. This gives the blow down Bl𝔹′𝔹 → M that contracts the exceptional divisor of Bl𝔹′𝔹 along all lines Bp.

4 The main result

Now let us properly prove Theorem 3.1 by presenting here the main result of this paper.

4.1 Exceptional divisor D and quartic curves

Notice that the subvariety 𝕎′ in 𝕎s parameterizing 𝔹′ is given by the condition detA = 0. 𝔹′ can be seen as the indeterminacy locus of the rational map 𝔹 ⇢ℙ S4V*, [A] ↦ 〈det(A)〉. This way we realize 𝔹͠ as a subvariety in 𝔹 × ℙ S4V* (the closure of the graph of 𝔹 ⇢ ℙ S4V*) and obtain a morphism 𝔹͠ → ℙ S4V*.

Lemma 4.1

1. The restriction of 𝔹͠ → ℙ S4V* to Dmaps a point ofDlying over a pointp ∈ ℙ2 (via the mapD → 𝔹′ → ℙ2) to a quartic curve throughp, i. e., there is a morphismDM1 ⊆ ℙ2 × ℙ S4V*.

2. The fibreD(L,p) of D → 𝔹′ over (L, p) ∈ P2 × ℙ2, pL, is isomorphic via the map 𝔹͠ → ℙ S4V*to the linear subspace inSdV*of curves throughp.

3. The morphismDM1is a1-bundle overM1, its fibre over a point ofM1given by a pairpCcan be identified with the fibre of 𝔹′ → ℙ2overp.

Proof

1. Let [A] ∈ 𝔹′ with A as in (8) and let a0, a1, a2 be the rows of A. Let B be a tangent vector at A, which can be identified with a morphism of type (4). Let b0, b1, b2 be its rows. Then, since det A = 0,

det(A+tB)=fA,Btmod(t2),

for

fA,B=detb0a1a2+deta0b1a2+deta0a1b2.

Then fA,B is a non-zero quartic form if B is normal to 𝕎′. One computes

fA,B=x0i=02xib0iw(x1i=02xib1i+x2i=02xib2i)

and thus fA,B vanishes at p, which is the common zero point of x0 and w.

2. Since the map D(L,p) → ℙ S4V* is injective, it is enough to notice that, for a fixed A ∈ 𝕎 ′, every quartic form through p can be obtained by varying B. This gives a bijection and thus an isomorphism from D(L,p) to the space of quartics through p.

3. Follows from 1) and 2).

4.2 Local charts

Let us describe 𝔹͠ over 𝔹(δ) (cf. 2.5). Around points of D lying over [A] ∈ 𝔹(δ) there are 14 charts. For a fixed coordinate t of 𝔹(δ) different from α, β, γ, denote the corresponding chart of Bl𝔹′∩B(δ) 𝔹(δ) by 𝔹͠(t). Then 𝔹͠(t) can be identified with the variety of triples (A, t, B),

A=0x2(γx1+x2)x1(γx1+x2)x20x¯0x1x¯00,B=q0(x0,x1,x2)q1(x1,x2)q2(x2)0cx100ax1+bx2dx2,(12)

such that the coefficient of B corresponding to t equals 1 and A + tB belongs to 𝔹(δ). The blow-up map 𝔹͠(t) → 𝔹(t) is given under this identification by sending a triple (A, t, B) to A + tB.

4.3 Family of (4m − 1)-sheaves on 𝔹͠

Notice that the cokernel of (4) is isomorphic to the cokernel of

2OP2(3)3OP2(2)00000q0q1q20z0z1z20w0w1w21000OP2(2)3OP2(1).

Lemma 4.2

Fort ≠ 0 consider the matrix

000000wx2wx10x20x¯00x1x¯001000+t00000q0q1q20y0y1y20z0z1z20000(13)

as a morphism 2𝓞2(− 3)⊕ 3𝓞2(− 2) → 𝓞2(− 2)⊕ 3𝓞2(− 1). Then its cokernel is isomorphic to the cokernel of

x¯0x1y0+x2z0x1y1+x2z1x1y2+x2z2wq0q1q20x20x¯00x1x¯0000x2x1+t000000000y0y1y20z0z1z21000.(14)

Proof

Acting by the automorphisms of 2𝓞2(− 3)⊕ 3𝓞2(− 2) on the left and by the automorphisms of 𝓞2(− 2)⊕ 3𝓞2(− 1) on the right of (13), we transform this matrix as follows:

000000wx2wx10x20x¯00x1x¯001000+t00000q0q1q20y0y1y20z0z1z200000000w0wx2wx10x20x¯00x1x¯001000+t00000q0q1q20y0y1y20z0z1z200000000w0000x20x¯00x1x¯0010x2x1+t00000q0q1q20y0y1y20z0z1z20000x¯00x¯0x2x¯0x1w0000x20x¯00x1x¯0010x2x1+t00000q0q1q20y0y1y20z0z1z20000x¯0000w0000x20x¯00x1x¯0010x2x1+t0x1y0+x2z0x1y1+x2z1x1y2+x2z20q0q1q20y0y1y20z0z1z20000t1x¯0000t1w0000x20x¯00x1x¯0010x2x1+0x1y0+x2z0x1y1+x2z1x1y2+x2z20q0q1q20ty0ty1ty20tz0tz1tz20000x¯0000w0000x20x¯00x1x¯00t0x2x1+0x1y0+x2z0x1y1+x2z1x1y2+x2z20q0q1q20ty0ty1ty20tz0tz1tz20000=x¯0x1y0+x2z0x1y1+x2z1x1y2+x2z2wq0q1q20x20x¯00x1x¯0000x2x1+t000000000y0y1y20z0z1z21000,

which concludes the proof. □

Evaluating (14) at t = 0 gives

x¯0x1y0+x2z0x1y1+x2z1x1y2+x2z2wq0q1q20x20x¯00x1x¯0000x2x1.

Lemma 4.3

The isomorphism class of the cokernel 𝓕 of

2OP2(3)3OP2(2)x¯0p0p1p2wq0q1q20x20x¯00x1x¯0000x2x1OP2(2)3OP2(1)

is a sheaf fromM1with resolution

02OP2(3)x¯0gwhOP2(2)OP2F0,(15)

if0hwg ≠ 0 for g = 0p0 + x1p1 + x2p2, h = 0q0 + x1q1 + x2q2.

Proof

Consider the isomorphism class of 𝓕 with resolution (15). Then, using the Koszul resolution of 𝓞2, one concludes that the kernel of the composition of two surjective morphisms

OP2(2)3OP2(1)100x¯00x10x2OP2(2)OP2F

coincides with the image of

2OP2(3)3OP2(2)x¯0p0p1p2wq0q1q20x20x¯00x1x¯0000x2x1OP2(2)3OP2(1),

which concludes the proof. □

For A + tB with A and B as in (12) we obtain the morphism

2OP2(3)3OP2(2)OP2(2)3OP2(1),

given by the matrix

x¯00cx12+ax1x2+bx22dx22wq0q1q20x2tcx1x¯00x1x¯0+t(ax1+bx2)tdx2t0x2x1,

which defines by Lemma 4.3 a family of (4m − 1)-sheaves on 𝔹͠(t) and therefore a morphism 𝔹͠(t) → M. This morphism sends the point of the exceptional divisor represented by (A, 0, B) to the point given by the quartic curve C = Z(f),

f=x¯0(x¯0q0(x0,x1,x2)+x1q1(x1,x2)+x2q2(x2))w(cx13+ax12x2+bx1x22+dx23),

and the point p = Z(0, w) on C.

4.3.2 Gluing the morphisms 𝔹͠(t) → M

For different charts 𝔹͠(t) and 𝔹͠(t′) the corresponding morphisms agree on intersections. Therefore, we conclude that there exists a morphism 𝔹͠ → M. It is an isomorphism outside of D. As already mentioned in Lemma 4.1, the restriction of this morphism to D gives a ℙ1-bundle DM1.

Lemma 4.4

The map 𝔹͠ → Mis the blow-up BlM1MM.

Proof

By the universal property of blow-ups, there exists a unique morphism B~ϕ BlM1M over M, which maps D to the exceptional divisor E of BlM1M and is an isomorphism outside of D. This morphism must be surjective as its image is irreducible and contains an open set. Its fibres must be connected by the Zariski’s main theorem. Restricted to D we get a surjective morphism DE of ℙ1-bundles over M1. Over every point of M1 we have a surjective morphism ℙ1 → ℙ1 with connected fibres. The only connected subvarieties of ℙ1 are the subvarieties consisting of one point and ℙ1 itself. The latter can not be a fibre, since this would contradict the surjectivity. This implies that the map DE is a bijection. Therefore, ϕ is a bijective morphism and thus an isomorphism.

This concludes the proof of Theorem 3.1.

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