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

Oleksandr Iena

doi:10.1515/math-2018-0003

Article Open Access

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

Oleksandr Iena

Published/Copyright: February 23, 2018

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$Open Mathematics$

From the journal Open Mathematics Volume 16 Issue 1

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 = M_{4m ± 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.

Keywords: Simpson moduli spaces; 1-dimensional sheaves; Blow-up; Blow-down; Quotients by non-reductive groups

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 = M_dm+c = M_dm+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 M_dm+c ≅ M_d′m+c′ if and only if d = d′ and c = ± c′ mod d. Therefore, in order to understand, for fixed d, the Simpson moduli spaces M_dm+c it is enough to understand at most [d/2]+1 different moduli spaces.

For d ⩽ 3 the fine moduli spaces M_dm+c are completely understood. By [2, Théorème 5.1] M_dm+c ≅ ℙ(S^dV^*) for d = 1, and d = 2. For d = 3, M_{3m ± 1} is isomorphic to the universal cubic plane curve

${(C,p)∈P(S3V∗)×P2∣p∈C}.$

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(S^dV^*), which is constructed as S^d V^*/GL₁( ), every basis of S^d V^* provides global homogeneous coordinates of ℙ(S^d V^*). Being a universal planar curve, the moduli spaces M_{3m ± 1} have a nice description as a quotient (by a non-reductive group) of the variety of matrices

$A=xypq,x,y∈H0(P2,OP2(1)),p,q∈H0(P2,OP2(2)),detA≠0,x∧y≠0,$

which provides convenient global coordinates for M_{3m ± 1} and allows one to study the moduli spaces in more details (cf. [5]).

By [3] one has the isomorphisms M_4m+b ≅ M_4m−1 for d = 4 and odd b. In [6] a description of the moduli space M_4m−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 := M_4m−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 :=Bl_𝔹′ 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⊕ ⋯ ⊕ 𝓔_m is identified with the row-vector of sections of 𝓔_i, i = 1,…, m.

1 M_4m−1 as a union of two strata

As shown in [2] M = M_4m−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

$0→2OP2(−3)→z1q1z2q2OP2(−2)⊕OP2→E→0,$ (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

$0→OC→E→O{p}→0,$ (2)

where C is a plane quartic given by the determinant of $z1q1z2q2$ from (1) and p ∈ C 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

$z1q1z2q2↦(C,p),C=Z(z1q2−z2q1),p=Z(z1,z2).$

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

$0→OL(−2)→F→OC′→0,$

where C′ is a cubic and L is a line.

Proof

Consider the isomorphism class of 𝓕 with resolution

$0→2OP2(−3)→l0whOP2(−2)⊕OP2→F→0.$ (3)

This gives the commutative diagram with exact rows and columns.

Therefore, 𝓕 is an extension

$0→OL(−2)→F→OC′→0,$

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 [𝓔_A] of the cokernels 𝓔_A 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 𝕎^s 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 [𝓔_A] in M₀ have a linear common factor, say l, then det(A) = l ⋅ h and 𝓔_A is in this case a non-split extension

$0→OL(−2)→EA→OC′→0,$ (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 𝓔_A of (4) is a part of the exact sequence

$0→EA→OC(1)→OZ→0,$

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.

1.3 Degenerations to the closed stratum

Proposition 1.2

1) Every sheaf in M₁is a degeneration of sheaves from M₀₀. This corresponds to a degeneration of Z ⊆ C, where Z is a zero-dimensional scheme of length 3 not lying on a line and C is a quartic curve, to a flag Z ⊆ C with Z contained in a line L that is not included in C. The limit corresponds to the point in M₁described by the point (L ∩ C) ∖ 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), f ∈ Γ(ℙ₂, 𝓞_ℙ₂(4)). Let Z ⊆ C be a zero-dimensional scheme of length 3 contained in a line L = Z(l), l ∈ Γ(ℙ₂, 𝓞_ℙ₂(1)). Let 𝓕 = 𝓘_Z(1) be the twisted ideal sheaf of Z in C so that there is an exact sequence

$0→F→OC(1)→OZ→0.$

Lemma 1.3

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

Proof

Let us construct a locally free resolution of 𝓕. Let g ∈ Γ(ℙ₂, 𝓞_ℙ₂(3)) such that 𝓞_Z is given by the resolution

$0→OP2(−4)→lgOP2(−3)⊕OP2(−1)→−glOP2→OZ→0.$

Since Z = Z(l, g) is contained in C = Z(f), one concludes that f = lh − wg for some w ∈ Γ(ℙ₂, 𝓞_ℙ₂(1)) and h ∈ Γ(ℙ₂, 𝓞_ℙ₂(3)). This gives the following commutative diagram with exact rows and columns.

Therefore, 𝓕 possesses a locally free resolution

$0→2OP2(−3)→lgwhOP2(−2)⊕OP2→F→0.$

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

$0→OC′→F→OL(−2)→0,C′=Z(h),$

and thus a destabilizing subsheaf 𝓞_C′ 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 Z ⊆ C 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 M ∖ M₀₁.

Its restriction to the open subvariety H₀(3, 4) of H(3, 4) of flags Z ⊆ C ⊆ 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 p ∈ C 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 a ℙ₃-bundle over ℙ V^* × ℙ S³V^*, the fibre over the pair (L, C′) of a line L and a cubic curve C′ is 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 := Bl_M₁M.

$Fig. 1 Moduli space M = M4m−1(ℙ2).$

Fig. 1

Moduli space M = M_4m−1(ℙ₂).

1.3.2 Degenerations along M₀₁

For a fixed line L and a fixed cubic curve C′ one can compute Ext¹(𝓞_C′, 𝓞_L(−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

$0→OL(−2)→F→OC′→0,L∈PV∗,C′∈PS3V∗.$

This provides a morphism P → M 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 𝕍^s of stable (2 × 3)-matrices of linear forms on ℙ₂ by Aut(2𝓞_ℙ₂(−2)) ×Aut(3𝓞_ℙ₂(−1)).

The projection 𝔹 → N is induced by

$q0q1q2z0z1z2w0w1w2↦z0z1z2w0w1w2.$

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 N ∖ N′ can be seen as the space of plane quartics through the corresponding subscheme of 3 non-collinear points. Indeed, consider a point from N ∖ N′ 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

$q0q1q2z0z1z2w0w1w2,q0,q1,q2∈S2V∗,$

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, q − Q = (q₀ − Q₀, q₁ − Q₁, q₂ − Q₂) lies in the syzygy module of $d0d1d2,$ which is generated by the rows of $z0z1z2w0w1w2$ by Hilbert-Burch theorem. This implies that q − Q 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, 𝓞_L(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

$−x20x0x1−x00.$

Then the fibre over $[−x20x0x1−x00]$ is given by the orbits of matrices

$q0(x0,x1,x2)q1(x1,x2)q2(x1,x2)−x20x0x1−x00$ (6)

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

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

$q1(x1,x2)=q1′(x1,x2)−x2⋅w,q2(x1,x2)=q2′(x2)+x1⋅w,q2′(x2)=q2(0,x2).$

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

$q0(x0,x1,x2)q1(x1,x2)−x2⋅wq2(x2)+x1⋅w−x20x0x1−x00.$

Its determinant equals

$x0⋅(x0⋅q0(x0,x1,x2)+x1⋅q1(x1,x2)+x2⋅q2(x2)).$

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^*)).

$Fig. 2 The fibre of 𝔹 over L ∈ N′.$

Fig. 2

The fibre of 𝔹 over L ∈ N′.

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′ = $P2∗$

${(L,p)∈P2∗×P2∣L∈P2∗,p∈L}.$ (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

$0−x2⋅wx1⋅w−x20x0x1−x00,w=γx1+δx2,〈γ,δ〉∈P1.$ (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 P_v(⋅)(t) from algebraic varieties to ℤ[t] that gives a ring homomorphism from the Grothendieck ring of varieties over to ℤ[t] such that P_v(X)(t) = P(X)(t) for smooth projective varieties X. In particular, this means that P_v(X) = P_v(Y)+P_v(X ∖ Y) if Y is a closed subvariety of X and P_v(X₁ × X₂) = P_v(X₁) ⋅ P_v(X₂). The latter property implies that the virtual Poincaré polynomial of a locally trivial fibration over X with fibre Y equals to the product P_v(X) ⋅ P_v(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) = P_v(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

$Pv(M)=Pv(M0)+Pv(M1),Pv(B)=Pv(B∖B′)+Pv(B′),Pv(M0)=Pv(B∖B′).$

Therefore, P_v(M) = P_v(M₀)+P_v(M₁) = P_v(𝔹 ∖ 𝔹′)+P_v(M₁) = P_v(𝔹) − P_v(𝔹′)+P_v(M₁). Since 𝔹 is a projective bundle over N with fibre ℙ₁₁, one gets P_v(𝔹) = P_v(N) ⋅ P_v(ℙ₁₁). Similarly, since 𝔹′ is a ℙ₁-bundle over N′ ≅ ℙ₂ and the universal quartic M₁ is a ℙ₁₃-bundle over ℙ₂, we obtain P_v(𝔹′) = P_v(ℙ₂) ⋅ P_v(ℙ₁) and P_v(M₁) = P_v(ℙ₂) ⋅ P_v(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 P_v(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 P_v(N) = P_v(N ∖ N′)+P_v(N′) = P_v(H ∖ H′)+P_v(N′) = P_v(H) − P_v(H′)+P_v(N′) = P_v(H) − P_v(ℙ₂) ⋅ P_v(ℙ₃)+P_v(ℙ₂) because H′ is a ℙ₃-bundle over N′. Using this and $P(Pn)=1−t2(n+1)1−t2,$ we get the result. □

2.5 Local charts around 𝔹′

Lemma 2.2

LetL ∈ N′ be the class of the Kronecker module

$−x20x0x1−x00.$

Then there is an open neighbourhood of L that can be identified with an open neighbourhoodUof zero in the affine space⁶via the map⁶ ⊇ U → N,

$(α,β,a,b,c,d)↦−x2cx1x¯0x1−x¯0+ax1+bx2dx2,$

withx̄₀ = x₀ + αx₁ + βx₂, which establishes a local section of the quotient 𝕍^s → N.

Proof

In some open neighbourhood U of zero in ⁶ the morphism

$U→Vs,(α,β,a,b,c,d)↦−x2cx1x¯0x1−x¯0+ax1+bx2dx2$

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

$−x2cx1x¯0x1−x¯0+ax1+bx2dx2$

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

$(α,β,a,b,c,d)↦−x2cx1x¯0x1−x¯0+ax1+bx2dx2$

is injective. □

Remark 2.3

By abuse of notation we identifyUwith its image inN.

Lemma 2.4

N′ is cut out inUby the equationsa = b = c = d = 0.

Proof

The maximal minors of $−x2cx1x¯0x1−x¯0+ax1+bx2dx2$ are

$cdx1x2+x¯0(x¯0−ax1−bx2),−dx22−x¯0x1,x2(x¯0−ax1−bx2)−cx12.$

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 trivial ℙ₁₁-bundle. Identifying ℙ₁₁with the projective space

$P(S2V∗⊕2Span(x12,x1x2,x22)),$

i. e., a point in ℙ₁₁is 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

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

Assuming one of the coefficients ofq₀, q₁, q₂equal to 1, we get local charts of the formU × ¹¹and local sections of the quotient 𝕎^s →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′,

$A=0−x2⋅wx1⋅w−x20x¯0x1−x¯00,w=γx1+δx2,〈γ,δ〉∈P1,$

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

$q0(x0,x1,x2)q1(x1,x2)−x2⋅wq2(x2)+x1⋅w−x2cx1x¯0x1−x¯0+ax1+bx2dx2.$ (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 q_i, 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 = Bl_𝔹′𝔹. Let D denote its exceptional divisor.

Theorem 3.1

𝔹͠ is isomorphic to the blow-upM͠ ≔ Bl_M₁M. The exceptional divisor ofM͠corresponds toDunder this isomorphism. The fibres of the morphismD → M₁over the point ofM₁represented by a pointpon a quartic curveCis identified with the projective line of lines in ℙ₂passing 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 L ∈ $P2∗$ and a point p ∈ L, which is encoded by some 〈w〉 ∈ ℙH⁰(L, 𝓞_L(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 L ∈ N′) and ℙ₉ = ℙ(S³V^*) (normal projective space to L^* in J(L^*, ℙ(S³V^*)) at p ∈ L ≅ L^*; 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 L ∈ N′ 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^*).

$Fig. 3 The fibre of 𝔹͠ over L ∈ N′.$

Fig. 3

The fibre of 𝔹͠ over L ∈ N′.

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, 𝓞_ℙ₂(1)). Identifying x₁ and x₂ with their images in H⁰(L, 𝓞_L(1)), {x₁, x₂} is a basis of H⁰(L, 𝓞_L(1)).

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

${λ⋅x0h+λ′⋅wg∣h(x0,x1,x2)∈S3V∗,g(x1,x2)∈H0(L,OL(3)),(λ,λ′)∈2},$

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

So the exceptional divisor of the blow-up Bl_𝔹′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 p ∈ C is identified with the space of lines L ∈ $P2∗$ 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 D → M₁ over a pair p ∈ C 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

$A0=0x1⋅x2−x1⋅x1−x20x0+βx1x1−(x0+βx1)0.$

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

$A1=0x0⋅x2−x0⋅x0−x20αx0+x1x0−(αx0+x1)0.$

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

$g=α3α2x0−αx1α2x201000−α,h=100α−1−α−2000α−1$

with determinants

$detg=−α4,deth=−α−3.$

Consider the automorphism $Ws→ξWs,A↦$ gAh. Then

$det(ξ(A1+B1))=α⋅det(A1+B1).$ (11)

From (11) it follows that the restriction of the ideal sheaf of D to a fibre of the morphisms D → M₁ is 𝓞_ℙ₁(1). By [15, 16, 17], this means that one can blow down D in 𝔹͠ along the map D → M₁. This gives the blow down Bl_𝔹′𝔹 → M that contracts the exceptional divisor of Bl_𝔹′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 𝕎^s 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

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 morphismD → M₁ ⊆ ℙ₂ × ℙ S⁴V^*.
The fibreD_(L,p) of D → 𝔹′ over (L, p) ∈ $P2∗$ × ℙ₂, p ∈ L, is isomorphic via the map 𝔹͠ → ℙ S⁴V^*to the linear subspace in ℙS^dV^*of curves throughp.
The morphismD → M₁is a ℙ₁-bundle overM₁, its fibre over a point ofM₁given by a pairp ∈ Ccan be identified with the fibre of 𝔹′ → ℙ₂overp.

Proof

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,
$det(A+tB)=fA,B⋅tmod(t2),$
for
$fA,B=detb0a1a2+deta0b1a2+deta0a1b2.$
Then f_A,B is a non-zero quartic form if B is normal to 𝕎′. One computes
$fA,B=x0∑i=02xib0i−w(x1∑i=02xib1i+x2∑i=02xib2i)$
and thus f_A,B vanishes at p, which is the common zero point of x₀ and w.
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.
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 Bl_{𝔹′∩B(δ)} 𝔹(δ) by 𝔹͠(t). Then 𝔹͠(t) can be identified with the variety of triples (A, t, B),

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

such that the coefficient of B corresponding to t equals 1 and A + t ⋅ B 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 + t ⋅ B.

4.3 Family of (4m − 1)-sheaves on 𝔹͠

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

$2OP2(−3)⊕3OP2(−2)→00000q0q1q20z0z1z20w0w1w21000OP2(−2)⊕3OP2(−1).$

4.3.1 Local construction

Lemma 4.2

Fort ≠ 0 consider the matrix

$000000−wx2wx10−x20x¯00x1−x¯001000+t⋅00000q0q1q20y0y1y20z0z1z20000$ (13)

as a morphism 2𝓞_ℙ₂(− 3)⊕ 3𝓞_ℙ₂(− 2) → 𝓞_ℙ₂(− 2)⊕ 3𝓞_ℙ₂(− 1). Then its cokernel is isomorphic to the cokernel of

$x¯0x1y0+x2z0x1y1+x2z1x1y2+x2z2wq0q1q20−x20x¯00x1−x¯0000x2−x1+t000000000y0y1y20z0z1z21000.$ (14)

Proof

Acting by the automorphisms of 2𝓞_ℙ₂(− 3)⊕ 3𝓞_ℙ₂(− 2) on the left and by the automorphisms of 𝓞_ℙ₂(− 2)⊕ 3𝓞_ℙ₂(− 1) on the right of (13), we transform this matrix as follows:

$000000−wx2wx10−x20x¯00x1−x¯001000+t⋅00000q0q1q20y0y1y20z0z1z20000∼0000w0−wx2wx10−x20x¯00x1−x¯001000+t⋅00000q0q1q20y0y1y20z0z1z20000∼0000w0000−x20x¯00x1−x¯0010x2−x1+t⋅00000q0q1q20y0y1y20z0z1z20000∼x¯00x¯0x2−x¯0x1w0000−x20x¯00x1−x¯0010x2−x1+t⋅00000q0q1q20y0y1y20z0z1z20000∼x¯0000w0000−x20x¯00x1−x¯0010x2−x1+t⋅0x1y0+x2z0x1y1+x2z1x1y2+x2z20q0q1q20y0y1y20z0z1z20000∼t−1x¯0000t−1w0000−x20x¯00x1−x¯0010x2−x1+0x1y0+x2z0x1y1+x2z1x1y2+x2z20q0q1q20ty0ty1ty20tz0tz1tz20000∼x¯0000w0000−x20x¯00x1−x¯00t0x2−x1+0x1y0+x2z0x1y1+x2z1x1y2+x2z20q0q1q20ty0ty1ty20tz0tz1tz20000=x¯0x1y0+x2z0x1y1+x2z1x1y2+x2z2wq0q1q20−x20x¯00x1−x¯0000x2−x1+t000000000y0y1y20z0z1z21000,$

which concludes the proof. □

Evaluating (14) at t = 0 gives

$x¯0x1y0+x2z0x1y1+x2z1x1y2+x2z2wq0q1q20−x20x¯00x1−x¯0000x2−x1.$

Lemma 4.3

The isomorphism class of the cokernel 𝓕 of

$2OP2(−3)⊕3OP2(−2)→x¯0p0p1p2wq0q1q20−x20x¯00x1−x¯0000x2−x1OP2(−2)⊕3OP2(−1)$

is a sheaf fromM₁with resolution

$0→2OP2(−3)→x¯0gwhOP2(−2)⊕OP2→F→0,$ (15)

ifx̄₀h − wg ≠ 0 for g = x̄₀p₀ + x₁p₁ + x₂p₂, h = x̄₀q₀ + x₁q₁ + x₂q₂.

Proof

Consider the isomorphism class of 𝓕 with resolution (15). Then, using the Koszul resolution of 𝓞_ℙ₂, one concludes that the kernel of the composition of two surjective morphisms

$OP2(−2)⊕3OP2(−1)→100x¯00x10x2OP2(−2)⊕OP2→F$

coincides with the image of

$2OP2(−3)⊕3OP2(−2)→x¯0p0p1p2wq0q1q20−x20x¯00x1−x¯0000x2−x1OP2(−2)⊕3OP2(−1),$

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

$x¯00cx12+ax1x2+bx22dx22wq0q1q20−x2tcx1x¯00x1−x¯0+t(ax1+bx2)tdx2t0x2−x1,$

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(x̄₀, 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 D → M₁.

Lemma 4.4

The map 𝔹͠ → Mis the blow-up Bl_M₁M → M.

Proof

By the universal property of blow-ups, there exists a unique morphism $B~→ϕ$ Bl_M₁M over M, which maps D to the exceptional divisor E of Bl_M₁M 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 D → E 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 D → E 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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Received: 2017-08-28

Accepted: 2017-12-22

Published Online: 2018-02-23

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Articles in the same Issue

https://doi.org/10.1515/math-2018-0003

Keywords for this article

Simpson moduli spaces; 1-dimensional sheaves; Blow-up; Blow-down; Quotients by non-reductive groups

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

Article

Abstract

Introduction

The main result of the paper

Structure of the paper

Some notations and conventions

1 M4m−1 as a union of two strata

1.1 Closed stratum

Lemma 1.1

Proof

1.2 Open stratum

1.2.1 M0 as a geometric quotient

1.2.2 Extensions

1.2.3 Twisted ideals of 3 points on a quartic

1.3 Degenerations to the closed stratum

Proposition 1.2

1.3.1 Degenerations along M00

Lemma 1.3

Proof

Remark 1.4

1.3.2 Degenerations along M01

2 Description of 𝔹

2.1 The base N

2.2 The fibres of 𝔹 → N

2.2.1 Fibres over N ∖ N′

2.2.2 Fibres over N′

2.3 Description of 𝔹′

2.4 Side remark: the Poincaré polynomial of M

Proposition 2.1

Proof

2.5 Local charts around 𝔹′

Lemma 2.2

Proof

Remark 2.3

Lemma 2.4

Proof

Lemma 2.5

Proof

Charts 𝔹(γ) and 𝔹(δ)

Remark 2.6

3 Description of M

Theorem 3.1

3.1 A rather intuitive explanation

4 The main result

4.1 Exceptional divisor D and quartic curves

Lemma 4.1

Proof

4.2 Local charts

4.3 Family of (4m − 1)-sheaves on 𝔹͠

4.3.1 Local construction

Lemma 4.2

Proof

Lemma 4.3

Proof

4.3.2 Gluing the morphisms 𝔹͠(t) → M

Lemma 4.4

Proof

References

Articles in the same Issue

Articles in the same Issue

Articles in the same Issue

1 M_4m−1 as a union of two strata

1.2.1 M₀ as a geometric quotient

1.3.1 Degenerations along M₀₀

1.3.2 Degenerations along M₀₁