Moduli of stable sheaves supported on curves of genus three contained in a quadric surface

TL;DR

This paper investigates the moduli space of stable sheaves supported on genus three curves within a smooth quadric surface, demonstrating its rationality and computing its Betti numbers through a detailed geometric analysis.

Contribution

It provides a comprehensive description of the moduli space, including its rationality, Betti numbers, classification of sheaves, and a global geometric construction involving flips and blow-downs.

Findings

The moduli space is rational.

Betti numbers are explicitly computed.

Stable sheaves are classified via resolutions and extensions.

Abstract

We study the moduli space of stable sheaves of Euler characteristic 1 supported on curves of arithmetic genus 3 contained in a smooth quadric surface. We show that this moduli space is rational. We compute its Betti numbers by studying the variation of the moduli spaces of alpha-semi-stable pairs. We classify the stable sheaves using locally free resolutions or extensions. We give a global description: the moduli space is obtained from a certain flag Hilbert scheme by performing two flips followed by a blow-down.

Equations

$$P_{\mathcal{F}}(m,n)=rm+sn+t,$$

$$p(\mathcal{F})=\frac{t}{r+s}.$$

$$0⟶\mathcal{O}(-1,-3)\oplus \mathcal{O}(0,-3)\oplus \mathcal{O}(-1,-2)\stackrel{\phi }{⟶}\mathcal{O}(0,-2)\oplus \mathcal{O}(0,-2)\oplus \mathcal{O}⟶\mathcal{F}⟶0,$$

$$0⟶\mathcal{O}(-2,-2)\oplus \mathcal{O}(-1,-3)\stackrel{\phi }{⟶}\mathcal{O}(-1,-2)\oplus \mathcal{O}(0,1)⟶\mathcal{F}⟶0,$$

$$0⟶\mathcal{O}(-2,-1)\oplus \mathcal{O}(-1,-4)\stackrel{\phi }{⟶}\mathcal{O}(-1,-1)\oplus \mathcal{O}⟶\mathcal{F}⟶0,$$

$$0⟶{\mathcal{O}}_Q⟶\mathcal{F}⟶{\mathcal{O}}_L(1,0)⟶0$$

$$0⟶{\mathcal{O}}_Q(p)⟶\mathcal{F}⟶{\mathcal{O}}_L⟶0,$$

$$P(X)(\xi )=\sum _{i\ge 0}{\dim }_{\mathbb{Q}}{H}^i(X,\mathbb{Q}){\xi }^{i/2}.$$

$$\begin{gathered} {\xi }^{17}+3{\xi }^{16}+8{\xi }^{15}+16{\xi }^{14}+21{\xi }^{13}+23{\xi }^{12}+24{\xi }^{11}+24{\xi }^{10}+24{\xi }^9 \\ +24{\xi }^8+24{\xi }^7+24{\xi }^6+23{\xi }^5+21{\xi }^4+16{\xi }^3+8{\xi }^2+3\xi +1. \end{gathered}$$

$$n_{\beta }(X)=(-1{)}^{\dim M(rm+sn+1)}e(M(rm+sn+1)).$$

$$N_{\beta }(X)=\sum _{k|\beta }\frac{n_{\beta /k}(X)}{k^3}.$$

$$N_{(4,2)}(X)=$$

$$=$$

$$H^2(\mathcal{F}(-1,-1))\otimes \mathcal{O}(-1,-1)=E_1^{-2,2}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$E_1^{-1,j}=H^j(\mathcal{F}(0,-1))\otimes \mathcal{O}(0,-1)\oplus H^j(\mathcal{F}(-1,0))\otimes \mathcal{O}(-1,0).$$

$$0⟶\mathcal{K}er({\theta }_1)\stackrel{{\theta }_5}{⟶}\mathcal{C}oker({\theta }_4)⟶\mathcal{F}⟶\mathcal{K}er({\theta }_2)/\mathcal{I}m({\theta }_1)⟶0.$$

$$0⟶2\mathcal{O}(-2,-2)\stackrel{{\zeta }^T}{⟶}\mathcal{O}(-1,-2)\oplus \mathcal{O}(-2,-1)\oplus \mathcal{O}(-1,-1)⟶{\mathcal{I}}_Z⟶0,$$

$$0⟶\mathcal{O}(-2,-4)⟶\mathcal{O}(-1,-3)\oplus \mathcal{O}(0,-3)\oplus \mathcal{O}(-1,-2)\stackrel{\zeta }{⟶}2\mathcal{O}(0,-2)⟶{\mathcal{E}xt}^2({\mathcal{O}}_Z,\mathcal{O})⟶0.$$

$$0⟶{\mathcal{I}}_Z⟶\mathcal{O}⟶{\mathcal{O}}_Z⟶0$$

$$0\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$0⟶\mathcal{O}⟶{\mathcal{I}}_Z(1,1)⟶\mathcal{C}oker({\theta }_1)⟶0,$$

$$0⟶2\mathcal{O}(-1,-1)⟶\mathcal{O}(0,-1)\oplus \mathcal{O}(-1,0)⟶\mathcal{C}oker({\theta }_1)⟶0$$

$$0⟶2\mathcal{O}(-1,-1)⟶\mathcal{O}(0,-1)\oplus \mathcal{O}(-1,0)\oplus \mathcal{O}⟶{\mathcal{I}}_Z(1,1)⟶0.$$

$$0⟶{\mathcal{O}}_C⟶\mathcal{F}⟶{\mathcal{E}xt}_{{\mathcal{O}}_S}^2({\mathcal{O}}_Z,{\mathcal{O}}_S)⟶0.$$

$$0⟶{\mathcal{O}}_C⟶\mathcal{F}⟶\mathcal{Z}⟶0$$

$$0⟶{\mathcal{H}om}_{{\mathcal{O}}_C}(\mathcal{F},{\mathcal{O}}_C)⟶{\mathcal{O}}_C⟶{\mathcal{E}xt}_{{\mathcal{O}}_C}^1(\mathcal{Z},{\mathcal{O}}_C)⟶0.$$

$$0⟶\mathcal{H}om(\mathcal{F},{\omega }_C)⟶{\omega }_C⟶{\mathcal{O}}_Z⟶0.$$

$$0⟶{\omega }_S⟶{\omega }_S\otimes \mathcal{O}(C)⟶{\omega }_S\otimes \mathcal{O}(C){|}_C\simeq {\omega }_C⟶0.$$

$$0⟶\mathcal{H}om(\mathcal{F},{\omega }_C)⟶{\mathcal{E}xt}^1(\mathcal{F},{\omega }_S)⟶{\mathcal{E}xt}^1(\mathcal{F},{\omega }_S\otimes \mathcal{O}(C)).$$

$${\mathcal{E}xt}^1({\mathcal{O}}_Z,{\omega }_S)=\{0\},{\mathcal{E}xt}^1({\omega }_C,{\omega }_S)\simeq {\mathcal{O}}_C,{\mathcal{E}xt}^2({\omega }_C,{\omega }_S)=\{0\}.$$

$${\theta }_2:\mathcal{O}(0,-1)\oplus 3\mathcal{O}(-1,0)⟶d\mathcal{O}$$

Full text

1. Introduction

Let $\mathbb{P}^{1}$ be the projective line over $\mathbb{C}$ and consider the surface $\mathbb{P}^{1}\times\mathbb{P}^{1}$ with fixed polarization $\mathcal{O}(1,1)=\mathcal{O}_{\mathbb{P}^{1}}(1)\otimes\mathcal{O}_{\mathbb{P}^{1}}(1)$. For a coherent algebraic sheaf ${\mathcal{F}}$ on $\mathbb{P}^{1}\times\mathbb{P}^{1}$, with support of dimension $1$, the Euler characteristic $\chi({\mathcal{F}}(m,n))$ is a polynomial expression in $m$, $n$, of the form

[TABLE]

where $r$, $s$, $t$ are integers depending only on ${\mathcal{F}}$. This is the Hilbert polynomial of ${\mathcal{F}}$. The slope of ${\mathcal{F}}$ is

[TABLE]

Let $\operatorname{M}(P)$ be the coarse moduli space of S-equivalence classes of sheaves on $\mathbb{P}^{1}\times\mathbb{P}^{1}$ that are semi-stable with respect to the fixed polarization and that have Hilbert polynomial $P$. We recall that ${\mathcal{F}}$ is semi-stable, respectively, stable, if it is pure and for any proper subsheaf ${\mathcal{F}}^{\prime}\subset{\mathcal{F}}$ we have $\operatorname{p}({\mathcal{F}}^{\prime})\leq\operatorname{p}({\mathcal{F}})$, respectively, $\operatorname{p}({\mathcal{F}}^{\prime})<\operatorname{p}({\mathcal{F}})$. According to [10], $\operatorname{M}(P)$ is projective, irreducible, and smooth at points given by stable sheaves. Its dimension is $2rs+1$ if $r>0$ and $s>0$. The spaces $\operatorname{M}(rm+n+1)$, $\operatorname{M}(2m+2n+1)$ and $\operatorname{M}(2m+2n+2)$ were studied in [1]. In fact, it is not difficult to see that $\operatorname{M}(rm+n+1)$ consists of the structure sheaves of curves of degree $(1,r)$, so it is isomorphic to $\mathbb{P}^{2r+1}$. The space $\operatorname{M}(3m+2n+1)$ was studied in [4] and [12]. We refer to the introductory section of [12] for more background information.

This paper is concerned with the study of $\mathbf{M}=\operatorname{M}(4m+2n+1)$. The closed points of $\mathbf{M}$ are in a bijective correspondence with the isomorphism classes $[{\mathcal{F}}]$ of stable sheaves ${\mathcal{F}}$ supported on curves of degree $(2,4)$ and satisfying the condition $\chi({\mathcal{F}})=1$. As already mentioned, $\mathbf{M}$ is a smooth irreducible projective variety of dimension $17$. For any $t\in\mathbb{Z}$, twisting by $\mathcal{O}(t,t)$ gives an isomorphism $\mathbf{M}\simeq\operatorname{M}(4m+2n+6t+1)$. According to [12, Corollary 1], $\mathbf{M}\simeq\operatorname{M}(4m+2n-1)$. In the following theorem we classify the sheaves in $\mathbf{M}$.

Theorem 1.1**.**

The variety $\mathbf{M}$ can be decomposed into an open subset $\mathbf{M}_{0}$, two closed irreducible subsets $\mathbf{M}_{2}$, $\mathbf{M}_{2}^{\prime}$, each of codimension $2$, a locally closed irreducible subset $\mathbf{M}_{3}$ of codimension $3$, and a locally closed irreducible subset $\mathbf{M}_{4}$ of codimension $4$. These subsets are defined as follows: $\mathbf{M}_{0}$ is the set of sheaves ${\mathcal{F}}$ having a resolution of the form

[TABLE]

where the entries $\varphi_{12}$ and $\varphi_{22}$ are linearly independent and the maximal minors of the matrix $(\varphi_{ij})_{i=1,2,j=1,2,3}$, describing the corestriction of $\varphi$ to the first two summands, have no common factor; $\mathbf{M}_{2}$ is the set of sheaves ${\mathcal{F}}$ having a resolution of the form

[TABLE]

with $\varphi_{11}\neq 0$, $\varphi_{12}\neq 0$; $\mathbf{M}_{2}^{\prime}$ is the set of sheaves ${\mathcal{F}}$ having a resolution of the form

[TABLE]

with $\varphi_{11}\neq 0$, $\varphi_{12}\neq 0$; $\mathbf{M}_{4}$ is the set of extensions of the form

[TABLE]

satisfying the condition $\operatorname{H}^{0}({\mathcal{F}})\simeq\mathbb{C}$, where $Q\subset\mathbb{P}^{1}\times\mathbb{P}^{1}$ is a quintic curve of degree $(2,3)$ and $L\subset\mathbb{P}^{1}\times\mathbb{P}^{1}$ is a line of degree $(0,1)$; $\mathbf{M}_{3}$ is the set of extensions of the form

[TABLE]

where $\mathcal{O}_{Q}(p)$ is a non-split extension of $\mathbb{C}_{p}$ by $\mathcal{O}_{Q}$ for a point $p\in Q$, and satisfying the condition $\operatorname{H}^{0}({\mathcal{F}})\simeq\mathbb{C}$.

Moreover, $\mathbf{M}_{2}$ is the Brill-Noether locus of sheaves for which $\operatorname{H}^{1}({\mathcal{F}})\neq\{0\}$.

The proof of Theorem 1.1, given in Section 3, relies on the Beilinson spectral sequence, which we recall in Section 2. The varieties $X$ that appear in this paper have no odd homology, so we can define the Poincaré polynomial

[TABLE]

Theorem 1.2**.**

The Euler characteristic of $\mathbf{M}$ is $288$. The Poincaré polynomial of $\mathbf{M}$ is

[TABLE]

The proof of this theorem rests on the wall-crossing method of Choi and Chung [3]. In Section 4 we investigate how the moduli spaces $\operatorname{M}^{\alpha}(4m+2n+1)$ of $\alpha$-semi-stable pairs with Hilbert polynomial $4m+2n+1$ change as the parameter $\alpha$ varies. In Theorem 4.11 we find that $\operatorname{M}^{\alpha}(4m+2n+1)$ are related by two explicitly described flipping diagrams. Combining this with Proposition 4.12, yields a global description: $\mathbf{M}$ is obtained from the flag Hilbert scheme of three points on curves of degree $(2,4)$ in $\mathbb{P}^{1}\times\mathbb{P}^{1}$ by performing two flips followed by a blow-down with center the Brill-Noether locus $\mathbf{M}_{2}$.

The total space $X$ of $\omega_{\mathbb{P}^{1}\times\mathbb{P}^{1}}$ is a Calabi-Yau threefold. For a homology class $\beta=(r,s)\in\operatorname{H}_{2}(\mathbb{P}^{1}\times\mathbb{P}^{1})\subset\operatorname{H}_{2}(X)$ let $N_{\beta}(X)$ be the genus zero Gromov-Witten invariant of $X$ and let $n_{\beta}(X)$ be the genus zero Gopakumar-Vafa invariant of $X$, as introduced in [8]. It was noticed in [4] that, up to sign, the latter is the Euler characteristic of a moduli space:

[TABLE]

In [8] Katz conjectured the relation

[TABLE]

For $\beta=(4,2)$, this conjecture reads

[TABLE]

2. Preliminaries

Our main technical tool in Section 3 will be the Beilinson spectral sequence. Let ${\mathcal{F}}$ be a coherent sheaf on $\mathbb{P}^{1}\times\mathbb{P}^{1}$. According to [2, Lemma 1], we have a spectral sequence converging to ${\mathcal{F}}$, whose first level $E_{1}$ has display diagram

[TABLE]

where $E_{1}^{ij}=\{0\}$ if $i\notin\{-2,-1,0\}$ or if $j\notin\{0,1,2\}$ and

[TABLE]

If ${\mathcal{F}}$ has support of dimension $1$, then the first row of (1) vanishes and the convergence of the spectral sequence forces $\theta_{2}$ to be surjective and yields the exact sequence

[TABLE]

An application of the Beilinson spectral sequence is the following lemma that will be used in Section 3.

Lemma 2.1**.**

Let $Z\subset\mathbb{P}^{1}\times\mathbb{P}^{1}$ be a zero-dimensional subscheme of length $3$ that is not contained in a line of degree $(1,0)$ or $(0,1)$. Then the ideal of $Z$ has resolution

[TABLE]

where the maximal minors of $\zeta$ have no common factor. The dual of the structure sheaf of $Z$ has resolution

[TABLE]

Proof.

We apply the spectral sequence (1) to the sheaf ${\mathcal{F}}={\mathcal{I}}_{Z}(1,1)$. By hypothesis, $\operatorname{H}^{0}({\mathcal{I}}_{Z}(1,0))=\{0\}$ and $\operatorname{H}^{0}({\mathcal{I}}_{Z}(0,1))=\{0\}$ hence, from (2), we obtain the vanishing of $E_{1}^{-1,0}$. Since $\operatorname{H}^{0}({\mathcal{I}}_{Z})=\{0\}$, also $E_{1}^{-2,0}$ vanishes. From the short exact sequence

[TABLE]

we obtain the vanishing of $\operatorname{H}^{2}({\mathcal{I}}_{Z})$. Analogously, $\operatorname{H}^{2}({\mathcal{I}}_{Z}(1,0))$, $\operatorname{H}^{2}({\mathcal{I}}_{Z}(0,1))$ and $\operatorname{H}^{2}({\mathcal{I}}_{Z}(1,1))$ vanish. The first row of (1) vanishes. Denote $d=\dim_{\mathbb{C}}\operatorname{H}^{1}({\mathcal{I}}_{Z}(1,1))$. Display diagram (1) now takes the simplified form

[TABLE]

From the convergence of the spectral sequence we see that $\theta_{2}$ is surjective. There is no surjective morphism $\theta_{2}\colon\mathcal{O}(0,-1)\oplus\mathcal{O}(-1,0)\to d\mathcal{O}$ for $d\geq 1$, hence $d=0$. Thus, ${\mathcal{K}er}(\theta_{1})$ is a subsheaf of $\mathcal{O}$. We claim that ${\mathcal{K}er}(\theta_{1})=\{0\}$. Indeed, if ${\mathcal{K}er}(\theta_{1})$ were non-zero, then $\mathcal{O}/{\mathcal{K}er}(\theta_{1})$ would be a torsion subsheaf of ${\mathcal{I}}_{Z}(1,1)$. Combining the exact sequences

[TABLE]

[TABLE]

yields the resolution

[TABLE]

Applying ${\mathcal{H}om}(-,\mathcal{O}(-1,-3))$, we obtain resolution (4). If the maximal minors of the matrix representing $\zeta$ had a common factor $f$, then the reduced support of ${\mathcal{C}oker}(\zeta)$ would contain the curve $\{f=0\}$. But this is impossible because ${\mathcal{E}xt}^{2}(\mathcal{O}_{Z},\mathcal{O})$ has support of dimension zero. ∎

Lemma 2.2**.**

Let $S$ be a smooth projective surface and let $C\subset S$ be a locally Cohen-Macaulay curve. Let ${\mathcal{Z}}$ be a coherent sheaf on $S$ with support of dimension zero. Let ${\mathcal{F}}$ be an extension of ${\mathcal{Z}}$ by $\mathcal{O}_{C}$ without zero-dimensional torsion. Then ${\mathcal{F}}$ is uniquely determined up to isomorphism, meaning that if ${\mathcal{F}}^{\prime}$ is another extension of ${\mathcal{Z}}$ by $\mathcal{O}_{C}$ without zero-dimensional torsion, then ${\mathcal{F}}^{\prime}\simeq{\mathcal{F}}$. Moreover, ${\mathcal{Z}}\simeq{\mathcal{E}xt}^{2}_{\mathcal{O}_{S}}(\mathcal{O}_{Z},\mathcal{O}_{S})$ for a subscheme $Z\subset C$ of dimension zero, so we have the exact sequence

[TABLE]

Proof.

This lemma is a direct consequence of [13, Proposition B.5]. Indeed, given an exact sequence

[TABLE]

in which ${\mathcal{F}}$ has no zero-dimensional torsion, then the pair $(\mathcal{O}_{C},{\mathcal{F}})$ is a stable pair supported on $C$, in the sense of [13]. By [13, Lemma B.2], we have ${\mathcal{E}xt}_{\mathcal{O}_{C}}^{1}({\mathcal{F}},\mathcal{O}_{C})=\{0\}$. Applying ${\mathcal{H}om}_{\mathcal{O}_{C}}(-,\mathcal{O}_{C})$ to (6), yields the exact sequence

[TABLE]

Thus, ${\mathcal{E}xt}^{1}_{\mathcal{O}_{C}}({\mathcal{Z}},\mathcal{O}_{C})$ is the structure sheaf $\mathcal{O}_{Z}$ of a zero-dimensional subscheme $Z\subset C$. Under the bijection of [13, Proposition B.5] between stable pairs supported on $C$ and zero-dimensional subschemes of $C$, the pair $(\mathcal{O}_{C},{\mathcal{F}})$ corresponds to $Z$, so it is uniquely determined, up to isomorphism. Tensoring (7) with the dualising line bundle $\omega_{C}$ on $C$, yields the exact sequence

[TABLE]

We claim that ${\mathcal{H}om}({\mathcal{F}},\omega_{C})\simeq{\mathcal{E}xt}^{1}({\mathcal{F}},\omega_{S})$. This follows by applying ${\mathcal{H}om}({\mathcal{F}},-)$ to the exact sequence

[TABLE]

We obtain the exact sequence

[TABLE]

The last morphism is locally multiplication with an equation $f$ defining $C$. But $C=\operatorname{supp}({\mathcal{F}})$, hence $f$ annihilates ${\mathcal{F}}$, and hence $f$ annihilates ${\mathcal{E}xt}^{1}({\mathcal{F}},\omega_{S})$. This proves the claim. According to [11, Remark 4], ${\mathcal{E}xt}^{1}({\mathcal{E}xt^{1}}({\mathcal{F}},\omega_{S}),\omega_{S})\simeq{\mathcal{F}}$. Clearly,

[TABLE]

Applying ${\mathcal{H}om}(-,\omega_{S})$ to (8) yields extension (5). Comparing with (6), we see that ${\mathcal{Z}}\simeq{\mathcal{E}xt}^{2}(\mathcal{O}_{Z},\mathcal{O}_{S})$. ∎

Crucial for our classification of semi-stable sheaves is the following vanishing result that should be compared with [12, Proposition 4]. We fix vector spaces $V_{1}$ and $V_{2}$ over $\mathbb{C}$ of dimension $2$ and we identify $\mathbb{P}^{1}\times\mathbb{P}^{1}$ with $\mathbb{P}(V_{1})\times\mathbb{P}(V_{2})$. Let $\{x,y\}$ be a basis of $V_{1}^{*}$ and let $\{z,w\}$ be a basis of $V_{2}^{*}$. A morphism $\mathcal{O}(i,j)\to\mathcal{O}(k,l)$ will be represented by a form in $\operatorname{S}^{k-i}V_{1}^{*}\otimes\operatorname{S}^{l-j}V_{2}^{*}$.

Proposition 2.3**.**

Assume that the sheaf ${\mathcal{F}}$ gives a point in $\mathbf{M}$.

1. (i)

We have $\operatorname{H}^{0}({\mathcal{F}}(-1,-1))=\{0\}$ and $\operatorname{H}^{0}({\mathcal{F}}(-1,0))=\{0\}$. 2. (ii)

If ${\mathcal{F}}$ satisfies the vanishing condition $\operatorname{H}^{0}({\mathcal{F}}(0,-1))=\{0\}$, then $\operatorname{H}^{1}({\mathcal{F}})=\{0\}$.

Proof.

(i) The vanishing of $\operatorname{H}^{0}({\mathcal{F}}(-1,-1))$ follows from [12, Proposition 2(i)]. To prove the vanishing of $\operatorname{H}^{0}({\mathcal{F}}(-1,0))$ we can argue as in the proof of [12, Proposition 3].

(ii) Assume now that $\operatorname{H}^{0}({\mathcal{F}}(0,-1))=\{0\}$. From (2) and part (i) of the proposition, we deduce that $E_{1}^{-1,1}\simeq\mathcal{O}(0,-1)\oplus 3\mathcal{O}(-1,0)$. Denote $d=\dim_{\mathbb{C}}\operatorname{H}^{1}({\mathcal{F}})$. There is no surjective morphism

[TABLE]

for $d\geq 4$, hence $d\leq 3$. Assume that $d=3$. The maximal minors for a matrix representation of $\theta_{2}$ have no common factor, otherwise $\theta_{2}$ would not be surjective. Thus, ${\mathcal{K}er}(\theta_{2})\simeq\mathcal{O}(-3,-1)$, hence $\theta_{1}=0$, and hence, from the exact sequence (3), we obtain a surjective morphism ${\mathcal{F}}\to\mathcal{O}(-3,-1)$. This is absurd. Thus, the case when $d=3$ is unfeasible.

Consider now the case when $d=2$. If $\theta_{2}$ is represented by a matrix of the form

[TABLE]

then ${\mathcal{K}er}(\theta_{2})\simeq\mathcal{O}(0,-1)\oplus\mathcal{O}(-3,0)$, hence $\mathcal{O}(-3,0)$ is a direct summand of ${\mathcal{K}er}(\theta_{2})/{\mathcal{I}m}(\theta_{1})$, and hence, from the exact sequence (3), we obtain a surjective morphism ${\mathcal{F}}\to\mathcal{O}(-3,0)$. This is absurd. If $\theta_{2}$ is represented by a matrix of the form

[TABLE]

then ${\mathcal{K}er}(\theta_{2})\simeq\mathcal{O}(-2,-1)\oplus\mathcal{O}(-1,0)$, hence $\mathcal{O}(-2,-1)$ is a direct summand of ${\mathcal{K}er}(\theta_{2})/{\mathcal{I}m}(\theta_{1})$, and hence we obtain a surjective morphism ${\mathcal{F}}\to\mathcal{O}(-2,-1)$. This is absurd. If $\theta_{2}$ is represented by a matrix of the form

[TABLE]

then ${\mathcal{K}er}(\theta_{2})\simeq\mathcal{O}(-1,-1)\oplus\mathcal{O}(-2,0)$ and we obtain a surjective morphism ${\mathcal{F}}\to\mathcal{O}(-2,0)$. This is absurd. We claim that, if $\theta_{2}$ is not of the form $A$, $B$ or $C$, then $\theta_{2}$ is represented by a matrix of the form

[TABLE]

with $v\neq 0$. Indeed, since $\theta_{2}\nsim A$ and $\theta_{2}\nsim B$, we may write

[TABLE]

with $u\neq 0$, $v\neq 0$. Since $\theta_{2}\nsim B$, $v_{1}$ and $v_{2}$ cannot be both zero. If $v_{1}$ and $v_{2}$ are linearly independent, then $\theta_{2}\sim D$. If $v_{1}$ and $v_{2}$ span a one-dimensional vector space, then, since $\theta_{2}\nsim B$, we may write

[TABLE]

Since $\theta_{2}\nsim C$, we have $u_{1}\neq 0$, forcing $\theta_{2}\sim D$. In the case when $\theta_{2}=D$, it is easy to see that the morphism

[TABLE]

is represented by a matrix of the form

[TABLE]

hence ${\mathcal{K}er}(\theta_{1})\simeq 3\mathcal{O}(-1,-1)$, and hence ${\mathcal{C}oker}(\theta_{5})$ has Hilbert polynomial $3m+3n+3$. But then, in view of the exact sequence (3), ${\mathcal{C}oker}(\theta_{5})$ is a destabilizing subsheaf of ${\mathcal{F}}$. Thus, the case when $d=2$ is also unfeasible.

It remains to examine the case when $d=1$. Recall that $\theta_{2}$ is surjective, hence it can have two possible forms. Firstly, if

[TABLE]

then ${\mathcal{K}er}(\theta_{2})\simeq\mathcal{O}(0,-1)\oplus\mathcal{O}(-2,0)\oplus\mathcal{O}(-1,0)$ and we obtain a surjective morphism ${\mathcal{F}}\to\mathcal{O}(-2,0)$, which is absurd. The second form is

[TABLE]

If $\theta_{1}$ is represented by a matrix having two zero columns, then ${\mathcal{K}er}(\theta_{1})\simeq 2\mathcal{O}(-1,-1)$, hence ${\mathcal{C}oker}(\theta_{5})$ has Hilbert polynomial $2m+2n+2$, and hence ${\mathcal{C}oker}(\theta_{5})$ is a destabilizing subsheaf of ${\mathcal{F}}$. Thus, we may write

[TABLE]

hence ${\mathcal{K}er}(\theta_{1})\simeq\mathcal{O}(-1,-2)\oplus\mathcal{O}(-1,-1)$, and hence ${\mathcal{C}oker}(\theta_{5})$ has Hilbert polynomial $3m+2n+2$. But then ${\mathcal{C}oker}(\theta_{5})$ is a destabilizing subsheaf of ${\mathcal{F}}$. We deduce that the case when $d=1$ is also unfeasible. ∎

3. Classification of sheaves

We begin our classification of semi-stable sheaves by examining the Brill-Noether locus of sheaves that do not satisfy the first vanishing condition in Proposition 2.3(ii).

Proposition 3.1**.**

The sheaves ${\mathcal{F}}$ in $\mathbf{M}$ satisfying the condition $\operatorname{H}^{0}({\mathcal{F}}(0,-1))\neq\{0\}$ are precisely the non-split extension sheaves of the form

[TABLE]

where $C\subset\mathbb{P}^{1}\times\mathbb{P}^{1}$ is a curve of degree $(2,4)$ and $p$ is a point on $C$. Moreover, the sheaves from (9) are precisely the sheaves ${\mathcal{F}}$ having a resolution of the form

[TABLE]

with $\varphi_{11}\neq 0$, $\varphi_{12}\neq 0$. Let $\mathbf{M}_{2}\subset\mathbf{M}$ be the subset of sheaves ${\mathcal{F}}$ from (9). Then $\mathbf{M}_{2}$ is closed, irreducible, of codimension $2$, and is isomorphic to the universal curve of degree $(2,4)$ in $\mathbb{P}^{1}\times\mathbb{P}^{1}$. Thus, $\mathbf{M}_{2}$ is a fiber bundle with fiber $\mathbb{P}^{13}$ and base $\mathbb{P}^{1}\times\mathbb{P}^{1}$.

Proof.

Let ${\mathcal{F}}$ give a point in $\mathbf{M}$ and satisfy $\operatorname{H}^{0}({\mathcal{F}}(0,-1))\neq\{0\}$. As in the proof of [12, Proposition 2], there is an injective morphism $\mathcal{O}_{C}\to{\mathcal{F}}(0,-1)$ for a curve $C$ of degree $(s,r)$, $0\leq s\leq 2$, $0\leq r\leq 4$, $1\leq r+s\leq 6$. From the stability of ${\mathcal{F}}$ we have the inequality

[TABLE]

which has the unique solution $(s,r)=(2,4)$. We obtain extension (9). Conversely, let ${\mathcal{F}}$ be given by the non-split extension (9). As in the proof of [12, Proposition 3], we can show that $\mathcal{O}_{C}(0,1)$ is stable, from which it immediately follows that ${\mathcal{F}}$ gives a point in $\mathbf{M}$ and that $\operatorname{H}^{0}({\mathcal{F}}(0,-1))\neq\{0\}$. Choose $\varphi_{11}\in V_{1}^{*}\otimes\mathbb{C}$ and $\varphi_{12}\in\mathbb{C}\otimes V_{2}^{*}$ defining $p$. Since $p\in C$, we can find $\varphi_{21}\in\operatorname{S}^{2}V_{1}^{*}\otimes\operatorname{S}^{3}V_{2}^{*}$ and $\varphi_{22}\in V_{1}^{*}\otimes\operatorname{S}^{4}V_{2}^{*}$ such that the polynomial $\varphi_{11}\varphi_{22}-\varphi_{12}\varphi_{21}$ defines $C$. Consider the morphism

[TABLE]

[TABLE]

From the snake lemma we see that ${\mathcal{C}oker}(\varphi)$ is an extension of $\mathbb{C}_{p}$ by $\mathcal{O}_{C}(0,1)$. Since ${\mathcal{C}oker}(\varphi)$ has no zero-dimensional torsion, we can apply Lemma 2.2 to deduce that ${\mathcal{F}}\simeq{\mathcal{C}oker}(\varphi)$. Thus, $[{\mathcal{F}}]\in\mathbf{M}_{2}$ if and only if ${\mathcal{F}}$ has resolution (10). ∎

In the remaining part of this section we will assume that ${\mathcal{F}}$ satisfies both vanishing conditions from Proposition 2.3(ii). The exact sequence (3) takes the form

[TABLE]

where

[TABLE]

Proposition 3.2**.**

Assume that $[{\mathcal{F}}]\in\mathbf{M}$ and that $\operatorname{H}^{0}({\mathcal{F}}(0,-1))=\{0\}$. Assume that the maximal minors of $\theta_{1}$ have a common factor. Then ${\mathcal{F}}$ is an extension of the form

[TABLE]

for a quintic curve $Q\subset\mathbb{P}^{1}\times\mathbb{P}^{1}$ of degree $(2,3)$ and a line $L\subset\mathbb{P}^{1}\times\mathbb{P}^{1}$ of degree $(0,1)$, or is an extension of the form

[TABLE]

where $\mathcal{O}_{Q}(p)$ is a non-split extension of $\mathbb{C}_{p}$ by $\mathcal{O}_{Q}$ for a point $p\in Q$.

Conversely, any extension ${\mathcal{F}}$ as in (12) or (13) satisfying the condition $\operatorname{H}^{0}({\mathcal{F}})\simeq\mathbb{C}$ is semi-stable. Let $\mathbf{M}_{4}\subset\mathbf{M}$ be the subset of sheaves ${\mathcal{F}}$ as in (12) satisfying the condition $\operatorname{H}^{0}({\mathcal{F}})\simeq\mathbb{C}$. Let $\mathbf{M}_{3}\subset\mathbf{M}$ be the subset of sheaves ${\mathcal{F}}$ as in (13) satisfying the condition $\operatorname{H}^{0}({\mathcal{F}})\simeq\mathbb{C}$. Then $\mathbf{M}_{3}$ and $\mathbf{M}_{4}$ are locally closed, irreducible subsets, of codimension $3$, respectively, $4$.

Proof.

Let $\eta_{i}$ be the maximal minor of a matrix representing $\theta_{1}$ obtained by deleting column $i$. Denote $g=\gcd(\eta_{1},\ldots,\eta_{5})$. Let $(s,r)=(2,4)-\deg(g)$. It is easy to check that the sequence

[TABLE]

[TABLE]

is exact. From (11) we see that ${\mathcal{C}oker}(\theta_{5})$ is a subsheaf of ${\mathcal{F}}$, hence we have the inequality

[TABLE]

forcing $(s,r)=(2,3)$ or $(s,r)=(2,2)$. If $(s,r)=(2,2)$, then $P_{{\mathcal{C}oker}(\theta_{1})}=2m+1$ and ${\mathcal{C}oker}(\theta_{1})$ is semi-stable, which follows from the semi-stability of ${\mathcal{F}}$. But, according to [1, Proposition 10], $\operatorname{M}(2m+1)=\emptyset$. This contradiction shows that $(s,r)\neq(2,2)$, hence $(s,r)=(2,3)$. From (11) we obtain the extension

[TABLE]

If ${\mathcal{C}oker}(\theta_{1})$ has no zero-dimensional torsion, we obtain extension (12). Otherwise, the zero-dimensional torsion has length $1$, its pull-back in ${\mathcal{F}}$ is a semi-stable sheaf $\mathcal{O}_{Q}(p)$, and we obtain extension (13).

Conversely, let ${\mathcal{F}}$ be an extension as in (12) satisfying $\operatorname{H}^{0}({\mathcal{F}})\simeq\mathbb{C}$. Assume that ${\mathcal{F}}$ had a destabilizing subsheaf ${\mathcal{F}}^{\prime}$. Let ${\mathcal{G}}$ be the image of ${\mathcal{F}}^{\prime}$ in $\mathcal{O}_{L}(1,0)$. According to [12, Proposition 1], $\mathcal{O}_{Q}$ is stable, hence $\chi({\mathcal{F}}^{\prime}\cap\mathcal{O}_{Q})\leq-1$. Since $\chi({\mathcal{F}}^{\prime})\geq 1$, we see that $\chi({\mathcal{G}})\geq 2$, hence ${\mathcal{G}}=\mathcal{O}_{L}(1,0)$ and $\mathcal{O}_{Q}\nsubseteq{\mathcal{F}}^{\prime}$. Thus $\operatorname{H}^{0}({\mathcal{F}}^{\prime}\cap\mathcal{O}_{Q})=\{0\}$, hence the map $\operatorname{H}^{0}({\mathcal{F}}^{\prime})\to\operatorname{H}^{0}(\mathcal{O}_{L}(1,0))$ is injective. But this map factors through $\operatorname{H}^{0}({\mathcal{F}})\to\operatorname{H}^{0}(\mathcal{O}_{L}(1,0))$, which, by hypothesis, is the zero map. We deduce that $\operatorname{H}^{0}({\mathcal{F}}^{\prime})=\{0\}$, which yields a contradiction. Thus, there is no destabilizing subsheaf. The same argument applies for extensions (13) satisfying $\operatorname{H}^{0}({\mathcal{F}})\simeq\mathbb{C}$.

By Serre duality

[TABLE]

From the short exact sequence

[TABLE]

we obtain the long exact sequence

[TABLE]

Thus $\operatorname{Ext}^{1}(\mathcal{O}_{L}(1,0),\mathcal{O}_{Q})\simeq\mathbb{C}^{2}$, hence $\mathbf{M}_{4}$ is isomorphic to an open subset of a $\mathbb{P}^{1}$-bundle over $\mathbb{P}^{11}\times\mathbb{P}^{1}$. By Serre duality we have

[TABLE]

Using Lemma 2.2, it is easy to see that the sheaves $\mathcal{O}_{Q}(p)$ are precisely the sheaves having a resolution of the form

[TABLE]

where $\varphi_{11}\neq 0$, $\varphi_{12}\neq 0$ (cf. Proposition 3.1). From resolution (14) we obtain the long exact sequence

[TABLE]

Thus, $\operatorname{Ext}^{1}(\mathcal{O}_{L},\mathcal{O}_{Q}(p))\simeq\mathbb{C}^{2}$, hence $\mathbf{M}_{3}$ has dimension $14$. The other claims about $\mathbf{M}_{3}$ are obvious. ∎

Lemma 3.3**.**

Assume that $[{\mathcal{F}}]\in\mathbf{M}$ and $\operatorname{H}^{0}({\mathcal{F}}(0,-1))=\{0\}$. Assume that the maximal minors of $\theta_{1}$ have no common factor. Then ${\mathcal{K}er}(\theta_{1})\simeq\mathcal{O}(-2,-4)$ and ${\mathcal{C}oker}(\theta_{1})\simeq{\mathcal{E}xt}^{2}(\mathcal{O}_{Z},\mathcal{O})$ with $Z$ described below. We have an extension

[TABLE]

where $C$ is a curve of degree $(2,4)$ and $Z\subset C$ is a subscheme of dimension zero and length $3$. Moreover, $Z$ is not contained in a line of degree $(0,1)$.

Proof.

The fact that ${\mathcal{K}er}(\theta_{1})\simeq\mathcal{O}(-2,-4)$ is well-known. The Hilbert polynomial of ${\mathcal{C}oker}(\theta_{1})$ is $3$, hence ${\mathcal{C}oker}(\theta_{1})$ has dimension zero and length $3$. From (11), we obtain the exact sequence

[TABLE]

We can now apply Lemma 2.2 to obtain the extension (15) and the isomorphism ${\mathcal{C}oker}(\theta_{1})\simeq{\mathcal{E}xt}^{2}(\mathcal{O}_{Z},\mathcal{O})$.

Assume that $Z$ is contained in a line $L$ of degree $(0,1)$. Then $\mathcal{O}_{Z}\simeq{\mathcal{E}xt}^{2}(\mathcal{O}_{Z},\mathcal{O})$. Choose $\varphi_{11}\in\mathbb{C}\otimes V_{2}^{*}$ defining $L$. Choose $\varphi_{12}\in\operatorname{S}^{3}V_{1}^{*}\otimes\mathbb{C}$ such that $\varphi_{11}$ and $\varphi_{12}$ define $Z$. If $L\nsubseteq C$, then $L.C=2$, which contradicts the fact that $Z\subset L\cap C$. Thus $L\subset C$, so there is $\varphi_{22}\in\operatorname{S}^{2}V_{1}^{*}\otimes\operatorname{S}^{3}V_{2}^{*}$ such that $\varphi_{11}\varphi_{22}$ is a defining polynomial of $C$. Consider the exact sequence

[TABLE]

[TABLE]

Then ${\mathcal{F}}^{\prime}$ is an extension of $\mathcal{O}_{Z}$ by $\mathcal{O}_{C}$ without zero-dimensional torsion. Since, from the exact sequence (15), ${\mathcal{F}}$ is also an extension of $\mathcal{O}_{Z}$ by $\mathcal{O}_{C}$ without zero-dimensional torsion, we can apply Lemma 2.2 to deduce that ${\mathcal{F}}\simeq{\mathcal{F}}^{\prime}$. We obtain a contradiction from the isomorphisms $\mathbb{C}\simeq\operatorname{H}^{0}({\mathcal{F}})\simeq\operatorname{H}^{0}({\mathcal{F}}^{\prime})\simeq\mathbb{C}^{3}$. ∎

Proposition 3.4**.**

Let $\mathbf{M}_{0}\subset\mathbf{M}$ be the subset of sheaves ${\mathcal{F}}$ for which $\operatorname{H}^{0}({\mathcal{F}}(0,-1))$ $=\{0\}$, ${\mathcal{K}er}(\theta_{1})\simeq\mathcal{O}(-2,-4)$ and $\operatorname{supp}({\mathcal{C}oker}(\theta_{1}))$ is not contained in a line of degree $(1,0)$ or $(0,1)$. Then $\mathbf{M}_{0}$ is open and can be described as the subset of sheaves ${\mathcal{F}}$ having a resolution of the form

[TABLE]

where $\varphi_{12}$ and $\varphi_{22}$ are linearly independent and the maximal minors of the matrix $(\varphi_{ij})_{i=1,2,j=1,2,3}$ have no common factor.

Proof.

Let ${\mathcal{F}}$ give a point in $\mathbf{M}_{0}$. Let $Z$ and $C$ be as in Lemma 3.3. By hypothesis $Z$ is not contained in a line of degree $(1,0)$ or $(0,1)$, hence ${\mathcal{E}xt}^{2}(\mathcal{O}_{Z},\mathcal{O})\simeq{\mathcal{C}oker}(\zeta)$ as in (4). Let $\zeta_{1}$, $\zeta_{2}$, $\zeta_{3}$ be the maximal minors of $\zeta$. They are the defining polynomials of $Z$, hence we can find $\varphi_{31}\in V_{1}^{*}\otimes\operatorname{S}^{3}V_{2}^{*}$, $\varphi_{32}\in\mathbb{C}\otimes\operatorname{S}^{3}V_{2}^{*}$, $\varphi_{33}\in V_{1}^{*}\otimes\operatorname{S}^{2}V_{2}^{*}$ such that $\zeta_{1}\varphi_{31}-\zeta_{2}\varphi_{32}+\zeta_{3}\varphi_{33}$ is the polynomial defining $C$. Let

[TABLE]

Then ${\mathcal{C}oker}(\varphi)$ is an extension of ${\mathcal{E}xt}^{2}(\mathcal{O}_{Z},\mathcal{O})$ by $\mathcal{O}_{C}$ without zero-dimensional torsion and, by Lemma 3.3, the same is true of ${\mathcal{F}}$. From Lemma 2.2 we deduce that ${\mathcal{F}}\simeq{\mathcal{C}oker}(\varphi)$. By Proposition 2.3, $\operatorname{H}^{0}({\mathcal{F}})\simeq\mathbb{C}$, hence the map $\operatorname{H}^{1}(\mathcal{O}(0,-3))\to\operatorname{H}^{1}(2\mathcal{O}(0,-2))$ is injective, which is equivalent to saying that $\varphi_{12}$ and $\varphi_{22}$ are linearly independent. We have shown that ${\mathcal{F}}$ has resolution (16).

Conversely, assume that ${\mathcal{F}}$ has resolution (16). Then $\operatorname{H}^{0}({\mathcal{F}})\simeq\mathbb{C}$ because $\varphi_{12}$ and $\varphi_{22}$ are linearly independent. From the snake lemma we see that ${\mathcal{F}}$ is an extension of ${\mathcal{E}xt}^{2}(\mathcal{O}_{Z},\mathcal{O})$ by $\mathcal{O}_{C}$, where $Z$ is the zero-dimensional scheme of length $3$ given by the maximal minors of the matrix obtained by deleting the third row of $\varphi$, and $C$ is the curve of degree $(2,4)$ defined by $\det(\varphi)$. Thus, $\operatorname{H}^{0}({\mathcal{F}})$ generates $\mathcal{O}_{C}$. We will show that ${\mathcal{F}}$ is semi-stable. Assume that ${\mathcal{F}}$ had a destabilizing subsheaf ${\mathcal{F}}^{\prime}$. Then $\chi({\mathcal{F}}^{\prime})>0$ and $\chi({\mathcal{F}}^{\prime})\leq\dim_{\mathbb{C}}\operatorname{H}^{0}({\mathcal{F}})=1$, hence $\chi({\mathcal{F}}^{\prime})=1$, forcing $\operatorname{H}^{0}({\mathcal{F}}^{\prime})\simeq\mathbb{C}$. Thus $\operatorname{H}^{0}({\mathcal{F}}^{\prime})=\operatorname{H}^{0}({\mathcal{F}})$, hence $\mathcal{O}_{C}\subset{\mathcal{F}}^{\prime}$, and hence ${\mathcal{F}}^{\prime}$ has multiplicity $6$. There are no destabilising subsheaves of ${\mathcal{F}}$ of multiplicity $6$. Thus, ${\mathcal{F}}$ gives a point in $\mathbf{M}$. Since $\varphi_{12}$ and $\varphi_{22}$ are linearly independent, we have $\operatorname{H}^{0}({\mathcal{F}}(0,-1))=\{0\}$. Since $\operatorname{H}^{0}({\mathcal{F}})$ generates $\mathcal{O}_{C}$, ${\mathcal{K}er}(\theta_{1})\simeq\mathcal{O}(-2,-4)$ and ${\mathcal{C}oker}(\theta_{1})\simeq{\mathcal{E}xt}^{2}(\mathcal{O}_{Z},\mathcal{O})$. Note that $Z$ is not contained in a line of degree $(1,0)$ or $(0,1)$. In conclusion, ${\mathcal{F}}$ gives a point in $\mathbf{M}_{0}$. ∎

Proposition 3.5**.**

The variety $\mathbf{M}$ is rational.

Proof.

By Lemma 2.1, Lemma 2.2, Lemma 3.3 and Proposition 3.4, the open subset of $\mathbf{M}_{0}$, given by the condition that $Z$ consist of three distinct points, is a $\mathbb{P}^{11}$-bundle over an open subset of $\operatorname{Hilb}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(3)$, so it is rational. ∎

Proposition 3.6**.**

Let ${\mathcal{F}}$ be an extension as in (15) without zero-dimensional torsion, for a curve $C$ of degree $(2,4)$ and a subscheme $Z\subset C$ that is the intersection of two curves of degree $(1,0)$, respectively, $(0,3)$. Then ${\mathcal{F}}$ gives a point in $\mathbf{M}$. Let $\mathbf{M}_{2}^{\prime}\subset\mathbf{M}$ be the subset of such sheaves ${\mathcal{F}}$. Then $\mathbf{M}_{2}^{\prime}$ is closed, irreducible, of codimension $2$, and can be described as the set of sheaves ${\mathcal{F}}$ having a resolution of the form

[TABLE]

with $\varphi_{11}\neq 0$, $\varphi_{12}\neq 0$.

Proof.

Note that $\mathcal{O}_{Z}\simeq{\mathcal{E}xt}^{2}(\mathcal{O}_{Z},\mathcal{O})$. Let ${\mathcal{F}}$ be an extension of $\mathcal{O}_{Z}$ by $\mathcal{O}_{C}$ without zero-dimensional torsion. Let $\varphi_{11}\in V_{1}^{*}\otimes\mathbb{C}$ and $\varphi_{12}\in\mathbb{C}\otimes\operatorname{S}^{3}V_{2}^{*}$ be the defining polynomials of $Z$. We can find $\varphi_{21}\in\operatorname{S}^{2}V_{1}^{*}\otimes V_{2}^{*}$ and $\varphi_{22}\in V_{1}^{*}\otimes\operatorname{S}^{4}V_{2}^{*}$ such that $\varphi_{11}\varphi_{22}-\varphi_{12}\varphi_{21}$ is the defining polynomial of $C$. Then the cokernel of $\varphi=(\varphi_{ij})_{1\leq i,j\leq 2}$ is an extension of $\mathcal{O}_{Z}$ by $\mathcal{O}_{C}$ without zero-dimensional torsion, hence, by Lemma 2.2, ${\mathcal{F}}\simeq{\mathcal{C}oker}(\varphi)$. Conversely, arguing as in Proposition 3.4, we can show that any sheaf of the form ${\mathcal{C}oker}(\varphi)$, with $\varphi$ as in (17), is semi-stable. ∎

Proof of Theorem 1.1. By Propositions 3.1, 3.2, 3.4 and 3.6, $\mathbf{M}$ is the union of the subvarieties $\mathbf{M}_{0}$, $\mathbf{M}_{2}$, $\mathbf{M}_{2}^{\prime}$, $\mathbf{M}_{3}$, $\mathbf{M}_{4}$. For $[{\mathcal{F}}]\in\mathbf{M}_{2}$, we have $\operatorname{H}^{0}({\mathcal{F}})\simeq\mathbb{C}^{2}$, whereas, for $[{\mathcal{F}}]$ in any of the other subvarieties, we have $\operatorname{H}^{0}({\mathcal{F}})\simeq\mathbb{C}$. Thus, $\mathbf{M}_{2}$ is disjoint from the other subvarieties. For $[{\mathcal{F}}]\in\mathbf{M}_{0}\cup\mathbf{M}_{2}^{\prime}$, $\operatorname{H}^{0}({\mathcal{F}})$ generates the structure sheaf of a curve $C$ of degree $(2,4)$, whereas, for $[{\mathcal{F}}]\in\mathbf{M}_{3}\cup\mathbf{M}_{4}$, $\operatorname{H}^{0}({\mathcal{F}})$ generates the structure sheaf of a curve $Q$ of degree $(2,3)$. Thus, $\mathbf{M}_{0}\cup\mathbf{M}_{2}^{\prime}$ is disjoint from $\mathbf{M}_{3}\cup\mathbf{M}_{4}$. For $[{\mathcal{F}}]\in\mathbf{M}_{0}$, the support of ${\mathcal{F}}/\mathcal{O}_{C}$ is not contained in a line of degree $(1,0)$, whereas, for $[{\mathcal{F}}]\in\mathbf{M}_{2}^{\prime}$, the support of ${\mathcal{F}}/\mathcal{O}_{C}$ is contained in a line of degree $(1,0)$. Thus, $\mathbf{M}_{0}$ is disjoint from $\mathbf{M}_{2}^{\prime}$. For $[{\mathcal{F}}]\in\mathbf{M}_{3}$, ${\mathcal{F}}/\mathcal{O}_{Q}$ has zero-dimensional torsion, whereas, for $[{\mathcal{F}}]\in\mathbf{M}_{4}$, ${\mathcal{F}}/\mathcal{O}_{Q}$ is pure. Thus, $\mathbf{M}_{3}$ is disjoint from $\mathbf{M}_{4}$. In conclusion, the subvarieties in question form a decomposition of $\mathbf{M}$. ∎

4. Variation of the moduli spaces of $\alpha$-semi-stable pairs

A coherent system $\Lambda=(\Gamma,{\mathcal{F}})$ on $\mathbb{P}^{1}\times\mathbb{P}^{1}$ consists of a coherent algebraic sheaf ${\mathcal{F}}$ on $\mathbb{P}^{1}\times\mathbb{P}^{1}$ and a vector subspace $\Gamma\subset\operatorname{H}^{0}({\mathcal{F}})$. Let $\alpha$ be a positive real number and let $P_{{\mathcal{F}}}(m,n)=rm+sn+t$ be the Hilbert polynomial of ${\mathcal{F}}$. We define the $\alpha$-slope of $\Lambda$ as the ratio

[TABLE]

We say that $\Lambda$ is $\alpha$-semi-stable, respectively, $\alpha$-stable, if ${\mathcal{F}}$ is pure and for any proper coherent subsystem $\Lambda^{\prime}\subset\Lambda$ we have $\operatorname{p}_{\alpha}(\Lambda^{\prime})\leq\operatorname{p}_{\alpha}(\Lambda)$, respectively, $\operatorname{p}_{\alpha}(\Lambda^{\prime})<\operatorname{p}_{\alpha}(\Lambda)$. According to [9] and [7], for fixed positive real number $\alpha$, non-negative integer $k$ and linear polynomial $P(m,n)$, there is a coarse moduli space, denoted $\operatorname{Syst}(\mathbb{P}^{1}\times\mathbb{P}^{1},\alpha,k,P)$, which is a projective scheme whose closed points are in a bijective correspondence with the set of S-equivalence classes of $\alpha$-semi-stable coherent systems $(\Gamma,{\mathcal{F}})$ on $\mathbb{P}^{1}\times\mathbb{P}^{1}$ for which $\dim\Gamma=k$ and $P_{{\mathcal{F}}}=P$. When $k=0$ this space is $\operatorname{M}(P)$. A coherent system for which $\dim\Gamma=1$ will be called a pair. Our main concern is with the moduli space of $\alpha$-semi-stable pairs $\operatorname{M}^{\alpha}(P)=\operatorname{Syst}(\mathbb{P}^{1}\times\mathbb{P}^{1},\alpha,1,P)$. It is known that there are finitely many positive rational numbers $\alpha_{1}<\ldots<\alpha_{n}$, called walls, such that the set of $\alpha$-semi-stable pairs with Hilbert polynomial $P$ remains unchanged as $\alpha$ varies in one of the intervals $(0,\alpha_{1})$, or $(\alpha_{i},\alpha_{i+1})$, or $(\alpha_{n},\infty)$. In fact, from the definition of $\alpha$-semi-stability, we can see that, if $\alpha$ is a wall, then there is a strictly $\alpha$-semi-stable pair, i.e. a pair $\Lambda$ for which there exists a subpair or quotient pair $\Lambda^{\prime}$, such that $\operatorname{p}_{\alpha}(\Lambda)=\operatorname{p}_{\alpha}(\Lambda^{\prime})$. This equation has only rational solutions in $\alpha$. For $\alpha\in(\alpha_{n},\infty)$ we write $\operatorname{M}^{\infty}(P)=\operatorname{M}^{\alpha}(P)$. For $\alpha\in(0,\alpha_{1})$ we write $\operatorname{M}^{0+}(P)=\operatorname{M}^{\alpha}(P)$. If $\gcd(r+s,t)=1$, then, from the definition of $\alpha$-semi-stability, we see that $(\Gamma,{\mathcal{F}})\in\operatorname{M}^{0+}(P)$ if and only if ${\mathcal{F}}$ is semi-stable. At the other extreme we have the following proposition due to Pandharipande and Thomas.

Proposition 4.1**.**

For $\alpha\gg 0$, a pair $\Lambda=(\Gamma,{\mathcal{F}})$ is $\alpha$-semi-stable if and only if ${\mathcal{F}}$ is pure and ${\mathcal{F}}/\mathcal{O}_{C}$ has dimension zero or is zero, where $\mathcal{O}_{C}$ is the subsheaf of ${\mathcal{F}}$ generated by $\Gamma$. In particular, $t\geq r+s-rs$. The scheme $\operatorname{M}^{\infty}(rm+sn+t)$ is isomorphic to the relative Hilbert scheme of zero-dimensional schemes of length $t-r-s+rs$ contained in curves of degree $(s,r)$.

Proof.

Assume that $(\Gamma,{\mathcal{F}})$ is $\alpha$-semi-stable for $\alpha\gg 0$. If $P_{\mathcal{O}_{C}}(m,n)=r^{\prime}m+s^{\prime}n+t^{\prime}$ with $r^{\prime}+s^{\prime}<r+s$, then

[TABLE]

which contradicts our hypothesis. Thus, $P_{\mathcal{O}_{C}}(m,n)=rm+sn+r+s-rs$. Conversely, assume that $\mathcal{O}_{C}$ has this Hilbert polynomial and that ${\mathcal{F}}$ is pure. Let $\Lambda^{\prime}=(\Gamma^{\prime},{\mathcal{F}}^{\prime})\subset\Lambda$ be a proper coherent subsystem with $P_{{\mathcal{F}}^{\prime}}(m,n)=r^{\prime}m+s^{\prime}n+t^{\prime}$. If $\Gamma^{\prime}=\{0\}$, then

[TABLE]

If $\Gamma^{\prime}=\Gamma$, then $\mathcal{O}_{C}\subset{\mathcal{F}}^{\prime}$, hence $r^{\prime}=r$, $s^{\prime}=s$, $t^{\prime}<t$, and we have

[TABLE]

The isomorphism between $\operatorname{M}^{\infty}(P)$ and the relative Hilbert scheme is a particular case of [13, Proposition B.8]. As a map, it is given by $(\Gamma,{\mathcal{F}})\mapsto(Z,C)$, where $Z\subset C$ is the subscheme introduced at Lemma 2.2. ∎

Corollary 4.2**.**

The scheme $\operatorname{M}^{\infty}(4m+2n+1)$ is isomorphic to a fiber bundle with fiber $\mathbb{P}^{11}$ and base the Hilbert scheme of three points in $\mathbb{P}^{1}\times\mathbb{P}^{1}$, so it is smooth.

Proof.

The relative Hilbert scheme of pairs $(Z,C)$, where $C\subset\mathbb{P}^{1}\times\mathbb{P}^{1}$ is a curve of degree $(2,4)$ and $Z\subset C$ is a subscheme of dimension zero and length $3$, has fiber $\mathbb{P}(\operatorname{H}^{0}({\mathcal{I}}_{Z}(2,4)))$ over $Z$. If $Z$ is not contained in a line of degree $(0,1)$ or $(1,0)$, then, from Lemma 2.1, we deduce that $\operatorname{H}^{0}({\mathcal{I}}_{Z}(2,4))\simeq\mathbb{C}^{12}$. If $Z$ is contained in such a line, then it is straightforward to check that $\operatorname{H}^{0}({\mathcal{I}}_{Z}(2,4))\simeq\mathbb{C}^{12}$. ∎

Lemma 4.3**.**

Assume that $\operatorname{M}^{\alpha}(rm+sn+t)\neq\emptyset$. Then $t\geq r+s-rs$. For $r$, $s$ non-negative integers, not both zero, and $\alpha\in(0,\infty)$, we have

[TABLE]

Proof.

We use induction on $r+s$. If $r+s=1$, or if there is no wall in $[\alpha,\infty)$, then $\operatorname{M}^{\alpha}(rm+sn+t)=\operatorname{M}^{\infty}(rm+sn+t)$ and the conclusion follows from Proposition 4.1. Assume that $r+s>1$ and that there is a wall $\alpha^{\prime}\in[\alpha,\infty)$. There is a pair $\Lambda\in\operatorname{M}^{\alpha^{\prime}}(rm+sn+t)$ and a subpair or quotient pair $\Lambda^{\prime}\in\operatorname{M}^{\alpha^{\prime}}(r^{\prime}m+s^{\prime}n+t^{\prime})$, such that $\operatorname{p}_{\alpha^{\prime}}(\Lambda)=\operatorname{p}_{\alpha^{\prime}}(\Lambda^{\prime})$. We have $0\leq r^{\prime}\leq r$, $0\leq s^{\prime}\leq s$, $1\leq r^{\prime}+s^{\prime}<r+s$,

[TABLE]

hence

[TABLE]

If $t=r+s-rs$, then there is no wall in $[\alpha,\infty)$, hence we have an isomorphism as in the lemma. ∎

Proposition 4.4**.**

With respect to $P(m,n)=4m+2n+1$ there are only two walls at $\alpha_{1}=5$ and $\alpha_{2}=11$.

Proof.

Assume that $\alpha$ is a wall. Then there are pairs $\Lambda\in\operatorname{M}^{\alpha}(4m+2n+1)$ and $\Lambda^{\prime}\in\operatorname{M}^{\alpha}(rm+sn+t)$ such that $\Lambda^{\prime}$ is a subpair or a quotient pair of $\Lambda$ and

[TABLE]

Here $0\leq r\leq 4$, $0\leq s\leq 2$, $1\leq r+s\leq 5$. By Lemma 4.3, we also have $t\geq r+s-rs$. Assume that $r=3$, $s=2$, $t\geq-1$. Equation (18) has solutions $\alpha_{1}=5$ for $t=0$ and $\alpha_{2}=11$ for $t=-1$. Assume that $r=2$, $s=2$, $t\geq 0$. Equation (18) has solution $\alpha=2$ for $t=0$. In this case either $\Lambda\in\operatorname{Ext}^{1}(\Lambda^{\prime},\Lambda^{\prime\prime})$ or $\Lambda\in\operatorname{Ext}^{1}(\Lambda^{\prime\prime},\Lambda^{\prime})$ for some $\Lambda^{\prime\prime}\in\operatorname{M}(2m+1)$. However, according to [1, Proposition 10], $\operatorname{M}(2m+1)=\emptyset$. Thus, there is no wall at $\alpha=2$. For all other choices of $r$ and $s$ equation (18) has no positive solution in $\alpha$. ∎

Denote $\mathbf{M}^{\alpha}=\operatorname{M}^{\alpha}(4m+2n+1)$. For $\alpha\in(11,\infty)$, write $\mathbf{M}^{\alpha}=\mathbf{M}^{\infty}$. For $\alpha\in(5,11)$, write $\mathbf{M}^{\alpha}=\mathbf{M}^{5+}=\mathbf{M}^{11-}$. For $\alpha\in(0,5)$, write $\mathbf{M}^{\alpha}=\mathbf{M}^{0+}$. The inclusions of sets of $\alpha$-semi-stable pairs induce the birational morphisms

[TABLE]

In view of Theorem 4.11, the above are flipping diagrams (consult [12, Remark 5] for details).

Remark 4.5**.**

From the proof of Proposition 4.4, we see that an S-equivalence class of strictly $\alpha$-semi-stable elements in $\mathbf{M}^{11}$ consists of (split or non-split) extensions of $(\Gamma_{1},{\mathcal{E}}_{1})$ by $(0,\mathcal{O}_{L}(1,0))$, together with the extensions of $(0,\mathcal{O}_{L}(1,0))$ by $(\Gamma_{1},{\mathcal{E}}_{1})$. Here $(\Gamma_{1},{\mathcal{E}}_{1})$ lies in $\operatorname{M}^{11}(3m+2n-1)$ and $L\subset\mathbb{P}^{1}\times\mathbb{P}^{1}$ is a line of degree $(0,1)$. We say, for short, that the strictly $\alpha$-semi-stable elements of $\mathbf{M}^{11}$ are of the form $(\Gamma_{1},{\mathcal{E}}_{1})\oplus(0,\mathcal{O}_{L}(1,0))$. According to Lemma 4.3 and Proposition 4.1, ${\mathcal{E}}_{1}\simeq\mathcal{O}_{Q}$ for a quintic curve $Q\subset\mathbb{P}^{1}\times\mathbb{P}^{1}$ of degree $(2,3)$. Thus, $\operatorname{M}^{11}(3m+2n-1)\simeq\mathbb{P}^{11}$.

Again from the proof of Proposition 4.4, we see that the strictly $\alpha$-semi-stable elements in $\mathbf{M}^{5}$ are of the form $(\Gamma,{\mathcal{E}})\oplus(0,\mathcal{O}_{L})$, where $(\Gamma,{\mathcal{E}})\in\operatorname{M}^{5}(3m+2n)$. We claim that $\operatorname{M}^{5}(3m+2n)\simeq\operatorname{M}^{\infty}(3m+2n)$. To see this, we will show that there are no walls relative to the Polynomial $P(m,n)=3m+2n$. As in the proof of Proposition 4.4, we attempt to solve the equation

[TABLE]

with $0\leq r\leq 3$, $0\leq s\leq 2$, $1\leq r+s\leq 4$, $t\geq r+s-rs$. For all choices of $r$ and $s$ we have $t\geq 0$, hence the above equation has no positive solutions in $\alpha$. From Proposition 4.1 we see that $\operatorname{M}^{5}(3m+2n)$ isomorphic to the universal quintic of degree $(2,3)$, so it is a $\mathbb{P}^{10}$-bundle over $\mathbb{P}^{1}\times\mathbb{P}^{1}$. More precisely, the elements in $\operatorname{M}^{5}(3m+2n)$ are of the form $(\operatorname{H}^{0}(\mathcal{O}_{Q}(p)),\mathcal{O}_{Q}(p))$, where $\mathcal{O}_{Q}(p)$ is a non-split extension of $\mathbb{C}_{p}$ by $\mathcal{O}_{Q}$.

Proposition 4.6**.**

Let $Q\subset\mathbb{P}^{1}\times\mathbb{P}^{1}$ be a quintic curve of degree $(2,3)$, let $p\in Q$ be a point, let $\mathcal{O}_{Q}(p)$ be a non-split extension of $\mathbb{C}_{p}$ by $\mathcal{O}_{Q}$, and let $L\subset\mathbb{P}^{1}\times\mathbb{P}^{1}$ be a line of degree $(0,1)$. Then any non-split extension sheaf ${\mathcal{F}}$ as in (13) is semi-stable. The set of such sheaves is the closure of $\mathbf{M}_{3}$ in $\mathbf{M}$. The boundary $\overline{\mathbf{M}}_{3}\setminus\mathbf{M}_{3}$ is contained in $\mathbf{M}_{2}$, more precisely, it consists of extensions as in (9) in which $C=Q\cup L$ and $p\in Q$.

Proof.

The case when $\operatorname{H}^{0}({\mathcal{F}})\simeq\mathbb{C}$ was examined at Proposition 3.2, so we need only consider the case when $\operatorname{H}^{0}({\mathcal{F}})\simeq\mathbb{C}^{2}$. In this case the canonical morphism $\mathcal{O}\to\mathcal{O}_{L}$ lifts to a morphism $\mathcal{O}\to{\mathcal{F}}$, hence we can combine resolution (14) with the standard resolution of $\mathcal{O}_{L}$ to obtain the resolution

[TABLE]

[TABLE]

where $\varphi_{11}\neq 0$, $\varphi_{12}\neq 0$, and $\varphi_{23}$ and $\varphi_{33}$ are linearly independent. Note that $p$ is given by the equations $\varphi_{11}=0$, $\varphi_{12}=0$. From the snake lemma, we obtain an extension

[TABLE]

where ${\mathcal{F}}^{\prime}$ is given by the resolution

[TABLE]

[TABLE]

We claim that ${\mathcal{F}}^{\prime}\simeq\mathcal{O}_{C}(0,1)$, where $C=Q\cup L$. In view of Proposition 3.1, the claim implies that ${\mathcal{F}}$ is semi-stable, in fact $[{\mathcal{F}}]\in\mathbf{M}_{2}$. It remains to prove the claim. Let ${\mathcal{K}}$ be the kernel of the canonical morphism $\mathcal{O}_{C}\to\mathcal{O}_{Q}$. Since ${\mathcal{K}}$ has no zero-dimensional torsion and $P_{\mathcal{K}}=m-1$, ${\mathcal{K}}\simeq\mathcal{O}_{L}(-2,0)$. Applying ${\mathcal{H}om}(-,\omega)$ to the exact sequence

[TABLE]

yields the exact sequence

[TABLE]

which is the same as the exact sequence

[TABLE]

Since $\operatorname{H}^{0}(\mathcal{O}_{C}(0,1))\simeq\mathbb{C}^{2}$, the canonical morphism $\mathcal{O}\to\mathcal{O}_{L}$ lifts to a morphism $\mathcal{O}\to\mathcal{O}_{C}(0,1)$, hence the canonical resolutions of $\mathcal{O}_{Q}$ and $\mathcal{O}_{L}$ can be combined into a resolution of the form

[TABLE]

[TABLE]

Since $\mathcal{O}_{C}(0,1)$ is a non-split extension of $\mathcal{O}_{L}$ by $\mathcal{O}_{Q}$, $\psi_{12}$ and $\varphi_{33}$ are linearly independent. It is clear now that the matrices representing $\varphi^{\prime}$ and $\psi$ are equivalent under elementary row and column operations. We conclude that ${\mathcal{F}}^{\prime}\simeq\mathcal{O}_{C}(0,1)$. ∎

The preimages of the sets of strictly semi-stable elements are the flipping loci:

[TABLE]

Proposition 4.7**.**

Consider $\Lambda_{1}\in\operatorname{M}^{11}(3m+2n-1)$, $\Lambda_{2}\in\operatorname{M}(m+2)$, $\Lambda_{3}\in\operatorname{M}^{5}(3m+2n)$, and $\Lambda_{4}\in\operatorname{M}(m+1)$.

1. (i)

Over a point $(\Lambda_{1},\Lambda_{2})$, $F^{\infty}$ has fiber $\mathbb{P}(\operatorname{Ext}^{1}(\Lambda_{1},\Lambda_{2}))$. 2. (ii)

Over a point $(\Lambda_{1},\Lambda_{2})$, $F^{11}$ has fiber $\mathbb{P}(\operatorname{Ext}^{1}(\Lambda_{2},\Lambda_{1}))$. 3. (iii)

Over a point $(\Lambda_{3},\Lambda_{4})$, $F^{5}$ has fiber $\mathbb{P}(\operatorname{Ext}^{1}(\Lambda_{3},\Lambda_{4}))$. 4. (iv)

Over a point $(\Lambda_{3},\Lambda_{4})$, $F^{0}$ has fiber $\mathbb{P}(\operatorname{Ext}^{1}(\Lambda_{4},\Lambda_{3}))$.

Proof.

(i) We refer to the argument at [12, Remark 2].

(ii) Assume that $\Lambda=(\Gamma,{\mathcal{F}})\in F^{11}$ lies over $(\Lambda_{1},\Lambda_{2})$. Then $\Lambda$ is a non-split extension of $\Lambda_{1}$ by $\Lambda_{2}$, or, vice versa, of $\Lambda_{2}$ by $\Lambda_{1}$. If $\Lambda_{2}\subset\Lambda$, then

[TABLE]

which violates the semi-stability of $\Lambda$. Thus $\Lambda\in\mathbb{P}(\operatorname{Ext}^{1}(\Lambda_{2},\Lambda_{1}))$. Conversely, given such $\Lambda$, we need to show that $\Lambda\in\mathbf{M}^{\alpha}$ for $\alpha\in(5,11)$. Write $\Lambda_{1}=(\Gamma_{1},\mathcal{O}_{Q})$, $\Lambda_{2}=(0,\mathcal{O}_{L}(1,0))$. We have a non-split extension of sheaves

[TABLE]

Let $\Lambda^{\prime}=(\Gamma^{\prime},{\mathcal{F}}^{\prime})$ be a proper coherent subsystem of $\Lambda$. Let ${\mathcal{G}}$ be the image of ${\mathcal{F}}^{\prime}$ in $\mathcal{O}_{L}(1,0)$. If ${\mathcal{F}}^{\prime}\cap\mathcal{O}_{Q}=\{0\}$, then ${\mathcal{G}}\neq\mathcal{O}_{L}(1,0)$, forcing $\chi({\mathcal{F}}^{\prime})=\chi({\mathcal{G}})\leq 1$. If ${\mathcal{F}}^{\prime}\cap\mathcal{O}_{Q}\neq\{0\}$, then $\chi({\mathcal{F}}^{\prime}\cap\mathcal{O}_{Q})\leq-1$ because, by virtue of [1, Lemma 9], $\mathcal{O}_{Q}$ is semi-stable. We have in this case $\chi({\mathcal{F}}^{\prime})=\chi({\mathcal{F}}^{\prime}\cap\mathcal{O}_{Q})+\chi({\mathcal{G}})\leq-1+2=1$. If $\Gamma^{\prime}=\{0\}$, then

[TABLE]

Assume now that $\Gamma^{\prime}\neq\{0\}$. Then $\Gamma^{\prime}=\Gamma=\operatorname{H}^{0}(\mathcal{O}_{Q})$, hence $\mathcal{O}_{Q}\subset{\mathcal{F}}^{\prime}$. If $\mathcal{O}_{Q}={\mathcal{F}}^{\prime}$, then

[TABLE]

If $\mathcal{O}_{Q}\subsetneqq{\mathcal{F}}^{\prime}$, then $r({\mathcal{F}}^{\prime})+s({\mathcal{F}}^{\prime})=6$, hence $\chi({\mathcal{F}}^{\prime})\leq 0$, and hence

[TABLE]

In all cases we have the inequality $\operatorname{p}_{\alpha}(\Lambda^{\prime})<\operatorname{p}_{\alpha}(\Lambda)$, hence $\Lambda\in\mathbf{M}^{\alpha}$, for $\alpha\in(5,11)$.

(iii) We will show that every $\Lambda=(\Gamma,{\mathcal{F}})\in\mathbb{P}(\operatorname{Ext}^{1}(\Lambda_{3},\Lambda_{4}))$ gives a point in $\mathbf{M}^{\alpha}$ for $\alpha\in(5,11)$. Write $\Lambda_{3}=(\Gamma_{3},\mathcal{O}_{Q}(p))$, $\Lambda_{4}=(0,\mathcal{O}_{L})$. We have a, possibly split, extension of sheaves

[TABLE]

Let $\Lambda^{\prime}=(\Gamma^{\prime},{\mathcal{F}}^{\prime})$ be a proper coherent subsystem of $\Lambda$. Let ${\mathcal{G}}$ be the image of ${\mathcal{F}}^{\prime}$ in $\mathcal{O}_{Q}(p)$. Using the fact that $\mathcal{O}_{Q}$ is semi-stable, it is easy to see that $\mathcal{O}_{Q}(p)$ is semi-stable, as well. Thus, $\chi({\mathcal{G}})\leq 0$, hence $\chi({\mathcal{F}}^{\prime})=\chi({\mathcal{F}}^{\prime}\cap\mathcal{O}_{L})+\chi({\mathcal{G}})\leq 1+0=1$. If $\Gamma^{\prime}=\{0\}$, then

[TABLE]

Assume now that $\Gamma^{\prime}\neq\{0\}$, i.e. $\Gamma^{\prime}=\Gamma$. Then $\mathcal{O}_{Q}\subset{\mathcal{G}}$. If ${\mathcal{F}}^{\prime}\cap\mathcal{O}_{L}=\{0\}$, then ${\mathcal{F}}^{\prime}\ncong\mathcal{O}_{Q}(p)$, otherwise $\Lambda\simeq\Lambda_{3}\oplus\Lambda_{4}$. In this case ${\mathcal{F}}^{\prime}\simeq\mathcal{O}_{Q}$, hence

[TABLE]

Assume now that ${\mathcal{F}}^{\prime}\cap\mathcal{O}_{L}\neq\{0\}$. Then $r({\mathcal{F}}^{\prime})+s({\mathcal{F}}^{\prime})=6$, hence $\chi({\mathcal{F}}^{\prime})\leq 0$, and hence $\operatorname{p}_{\alpha}(\Lambda^{\prime})<\operatorname{p}_{\alpha}(\Lambda)$.

(iv) If $(\Gamma,{\mathcal{F}})\in\mathbb{P}(\operatorname{Ext}^{1}(\Lambda_{4},\Lambda_{3}))$, then we have the non-split extension (13), hence, by Proposition 4.6, ${\mathcal{F}}$ is semi-stable. Thus $(\Gamma,{\mathcal{F}})\in\mathbf{M}^{0+}$, i.e. $(\Gamma,{\mathcal{F}})\in F^{0}$. ∎

Proposition 4.8**.**

([7, Corollaire 1.6])* Let $\Lambda=(\Gamma,{\mathcal{F}})$ and $\Lambda^{\prime}=(\Gamma^{\prime},{\mathcal{F}}^{\prime})$ be two coherent systems on a separated scheme of finite type over $\mathbb{C}$. Then there is a long exact sequence*

[TABLE]

Proposition 4.9**.**

The flipping loci $F^{\infty}$, $F^{11}$, $F^{5}$, $F^{0}$ are smooth bundles with fibers $\mathbb{P}^{3}$, $\mathbb{P}^{1}$, $\mathbb{P}^{2}$, respectively, $\mathbb{P}^{1}$.

Proof.

We need to determine the extension spaces of pairs occurring at Proposition 4.7.

(i) Choose $\Lambda_{1}=(\Gamma_{1},\mathcal{O}_{Q})$ and $\Lambda_{2}=(0,\mathcal{O}_{L}(1,0))$. From Proposition 4.8 we have the long exact sequence

[TABLE]

For $\alpha\gg 0$, $\Lambda_{1}$ and $\Lambda_{2}$ are $\alpha$-stable coherent systems of different slopes, hence $\operatorname{Hom}(\Lambda_{1},\Lambda_{2})=\{0\}$. From the short exact sequence

[TABLE]

we obtain the long exact sequence

[TABLE]

Combining the last two long exact sequences, we obtain the isomorphism $\operatorname{Ext}^{1}(\Lambda_{1},\Lambda_{2})\simeq\mathbb{C}^{4}$.

(ii) From Proposition 4.8, we have the exact sequence

[TABLE]

From resolution (19), we obtain the exact sequence

[TABLE]

Combining the last two exact sequences, we obtain the isomorphism $\operatorname{Ext}^{1}(\Lambda_{2},\Lambda_{1})\simeq\mathbb{C}^{2}$.

(iii) Choose $\Lambda_{3}=(\Gamma,\mathcal{O}_{Q}(p))$ and $\Lambda_{4}=(0,\mathcal{O}_{L})$. From Proposition 4.8, we have the long exact sequence

[TABLE]

From resolution (14) we obtain the exact sequence

[TABLE]

Combining the last two exact sequences, it follows that $\operatorname{Ext}^{1}(\Lambda_{3},\Lambda_{4})\simeq\mathbb{C}^{3}$.

(iv) From Proposition 4.8, we obtain the exact sequence

[TABLE]

From resolution (14) we obtain the exact sequence

[TABLE]

Combining the last two exact sequences, it follows that $\operatorname{Ext}^{1}(\Lambda_{4},\Lambda_{3})\simeq\mathbb{C}^{2}$. ∎

Lemma 4.10**.**

(i)* For $\Lambda\in F^{11}$ we have $\operatorname{Ext}^{2}(\Lambda,\Lambda)=\{0\}$.

(ii) For $\Lambda\in F^{0}$ we have $\operatorname{Ext}^{2}(\Lambda,\Lambda)=\{0\}$.*

Proof.

(i) In view of the exact sequence

[TABLE]

it is enough to show that $\operatorname{Ext}^{2}(\Lambda_{i},\Lambda_{j})=\{0\}$ for $i,j=1,2$. From Proposition 4.8, we have the exact sequence

[TABLE]

The group on the right vanishes because $\operatorname{H}^{0}(\mathcal{O}_{Q}(-3,-2))=\{0\}$. Thus, $\operatorname{Ext}^{2}(\Lambda_{1},\Lambda_{2})$ $=\{0\}$. From the exact sequence

[TABLE]

we obtain the vanishing of $\operatorname{Ext}^{2}(\Lambda_{2},\Lambda_{1})$. From the exact sequence

[TABLE]

we obtain the vanishing of $\operatorname{Ext}^{2}(\Lambda_{2},\Lambda_{2})$. From Proposition 4.8, we have the exact sequence

[TABLE]

According to [7, Théorème 3.12], $\operatorname{Ext}^{1}(\Lambda_{1},\Lambda_{1})$ is isomorphic to the tangent space of $\operatorname{M}^{11}(3m+2n-1)\simeq\mathbb{P}^{11}$ (see Remark 4.5) at $\Lambda_{1}$, so it is isomorphic to $\mathbb{C}^{11}$. From resolution (19), we obtain the exact sequence

[TABLE]

Combining the last two exact sequences we obtain the vanishing of $\operatorname{Ext}^{2}(\Lambda_{1},\Lambda_{1})$.

(ii) As above, we need to prove that $\operatorname{Ext}^{2}(\Lambda_{i},\Lambda_{j})=\{0\}$ for $i,j=3,4$. From Proposition 4.8, we have the exact sequence

[TABLE]

Thus, $\operatorname{Ext}^{2}(\Lambda_{3},\Lambda_{4})=\{0\}$. From the exact sequence

[TABLE]

we obtain the vanishing of $\operatorname{Ext}^{2}(\Lambda_{4},\Lambda_{3})$. From the exact sequence

[TABLE]

we obtain the vanishing of $\operatorname{Ext}^{2}(\Lambda_{4},\Lambda_{4})$. From Proposition 4.8, we have the exact sequence

[TABLE]

From resolution (14), we obtain the exact sequence

[TABLE]

Since $\operatorname{Hom}(\mathcal{O}_{Q}(p),\mathcal{O}_{Q}(p))\simeq\mathbb{C}$, it follows that

[TABLE]

According to [7, Théorème 3.12], $\operatorname{Ext}^{1}(\Lambda_{3},\Lambda_{3})$ is isomorphic to the tangent space at $\Lambda_{3}$ of $\operatorname{M}^{5}(3m+2n)$, which, according to Remark 4.5, is smooth of dimension $12$. We obtain the vanishing of $\operatorname{Ext}^{2}(\Lambda_{3},\Lambda_{3})$. ∎

Theorem 4.11**.**

Let $\mathbf{M}^{\alpha}$ be the moduli space of $\alpha$-semi-stable pairs on $\mathbb{P}^{1}\times\mathbb{P}^{1}$ with Hilbert polynomial $P(m,n)=4m+2n+1$. We have the following blowing up diagrams

[TABLE]

Here $\beta_{\infty}$ is the blow-up along $F^{\infty}$ and $\beta_{11}$ is the contraction of the exceptional divisor $\widetilde{F}^{\infty}$ in the direction of $\mathbb{P}^{3}$, where we view $\widetilde{F}^{\infty}$ as a $\mathbb{P}^{3}\times\mathbb{P}^{1}$-bundle with base $\operatorname{M}^{11}(3m+2n-1)\times\operatorname{M}(m+2)$. Likewise, $\beta_{5}$ is the blow-up along $F^{5}$ and $\beta_{0}$ is the contraction of the exceptional divisor $\widetilde{F}^{5}$ in the direction of $\mathbb{P}^{2}$, where we view $\widetilde{F}^{5}$ as a $\mathbb{P}^{2}\times\mathbb{P}^{1}$-bundle over $\operatorname{M}^{5}(3m+2n)\times\operatorname{M}(m+1)$.

Proof.

A birational morphism $\beta_{11}\colon\widetilde{\mathbf{M}}^{\infty}\to\mathbf{M}^{11-}$ can be constructed as at [3, Theorem 3.3] such that $\beta_{11}$ contracts $\widetilde{F}^{\infty}$ in the direction of $\mathbb{P}^{3}$, $\beta_{11}$ is an isomorphism outside $F^{11}$, and $\beta_{11}^{-1}(x)\simeq\mathbb{P}^{3}$ for any $x\in F^{11}$. We now apply the Universal Property of the blow-up [6, p. 604] to deduce that $\beta_{11}$ is a blow-up with center $F^{11}$. For this we need to know that $\mathbf{M}^{11-}$ and $F^{11}$ are smooth. By Corollary 4.2, $\mathbf{M}^{\infty}$ is smooth, by Proposition 4.9, the blowing up center $F^{\infty}$ is smooth, hence $\widetilde{\mathbf{M}}^{\infty}$ is smooth, too. Since $\beta_{11}$ is an isomorphism outside $F^{11}$, $\mathbf{M}^{11-}\setminus F^{11}$ is smooth. Since all points of $\mathbf{M}^{11-}$ are $\alpha$-stable, we can apply the Smoothness Criterion [7, Théorème 3.12], which states that $\Lambda\in\mathbf{M}^{11-}$ is a smooth point if $\operatorname{Ext}^{2}(\Lambda,\Lambda)=\{0\}$. Thus, in view of Lemma 4.10(i), $\mathbf{M}^{11-}$ is smooth at every point of $F^{11}$. The smoothness of $F^{11}$ was proved at Proposition 4.9.

For the second blow-up diagram we reason analogously, using the facts that $F^{5}$ and $F^{0}$ are smooth, and using Lemma 4.10(ii). ∎

According to [7, Théorème 4.3], there is a universal family $(\widetilde{\Gamma},\widetilde{{\mathcal{F}}})$ of coherent systems on $\mathbf{M}^{0+}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$. In particular, $\widetilde{{\mathcal{F}}}$ is a family of semi-stable sheaves on $\mathbb{P}^{1}\times\mathbb{P}^{1}$ with Hilbert polynomial $4m+2n+1$, which is flat over $\mathbf{M}^{0+}$. It induces the so called forgetful morphism $\phi\colon\mathbf{M}^{0+}\to\mathbf{M}$. We have $\phi(\Gamma,{\mathcal{F}})=[{\mathcal{F}}]$.

Proposition 4.12**.**

The forgetful morphism $\phi\colon\mathbf{M}^{0+}\to\mathbf{M}$ is a blow-up with center the Brill-Noether locus $\mathbf{M}_{2}$.

Proof.

According to Proposition 2.3(ii), for $[{\mathcal{F}}]\in\mathbf{M}\setminus\mathbf{M}_{2}$ we have $\operatorname{H}^{0}({\mathcal{F}})\simeq\mathbb{C}$, hence $\phi^{-1}([{\mathcal{F}}])=(\operatorname{H}^{0}({\mathcal{F}}),{\mathcal{F}})$ is a single point. Thus, $\phi$ is an isomorphism away from $\mathbf{M}_{2}$. According to Proposition 3.1, for $[{\mathcal{F}}]\in\mathbf{M}_{2}$ we have $\operatorname{H}^{0}({\mathcal{F}})\simeq\mathbb{C}^{2}$, hence $\phi^{-1}([{\mathcal{F}}])\simeq\mathbb{P}^{1}$. Taking into account that $\mathbf{M}$ and $\mathbf{M}_{2}$ are smooth, we can apply the Universal Property of the blow-up [6, p. 604] to conclude that $\phi$ is a blow-up with center $\mathbf{M}_{2}$. ∎

Proof of Theorem 1.2. By virtue of Proposition 4.12, we have the relation

[TABLE]

According to Proposition 3.1, we have the relation

[TABLE]

By virtue of Theorem 4.11, we have the relation

[TABLE]

In view of Corollary 4.2 and Remark 4.5, we have the relation

[TABLE]

According to [5, Theorem 0.1], we have the equation

[TABLE]

The final result reads

[TABLE]

Acknowledgement. The author would like to thank Jean-Marc Drézet for several helpful discussions.