On the geometry of the moduli space of sheaves supported on curves of genus four contained in a quadric surface

TL;DR

This paper investigates the structure of the moduli space of stable sheaves supported on genus four curves within a quadric surface, demonstrating its rationality and computing its Betti numbers.

Contribution

It establishes the rationality of the moduli space and calculates its Betti numbers using the variation of alpha-semi-stable pairs.

Findings

The moduli space is rational.

Betti numbers are explicitly computed.

Variation of alpha-stability is used in the analysis.

Abstract

We study the moduli space of stable sheaves of Euler characteristic 1 supported on curves of bidegree (3, 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.

Tables

Bundle	$F_1^{\infty }$	$F_1^{11}$	$F_2^{\infty }$	$F_2^{11}$	$F_1^{5+}$	$F_1^{5-}$	$F_2^{5+}$	$F_2^{5-}$	$F^2$	$F^0$
Fiber	${\mathbb{P}}^4$	${\mathbb{P}}^2$	${\mathbb{P}}^4$	${\mathbb{P}}^2$	${\mathbb{P}}^3$	${\mathbb{P}}^2$	${\mathbb{P}}^3$	${\mathbb{P}}^2$	${\mathbb{P}}^4$	${\mathbb{P}}^3$

Equations

$$p(\mathcal{E})=\frac{t}{r+s}\le \frac{1}{6}=p(\mathcal{F}).$$

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

$$\begin{gathered} {\xi }^{19}+3{\xi }^{18}+10{\xi }^{17}+22{\xi }^{16}+41{\xi }^{15}+53{\xi }^{14}+60{\xi }^{13}+62{\xi }^{12}+63{\xi }^{11}+63{\xi }^{10} \\ +63{\xi }^9+63{\xi }^8+62{\xi }^7+60{\xi }^6+53{\xi }^5+41{\xi }^4+22{\xi }^3+10{\xi }^2+3\xi +1. \end{gathered}$$

$$p_{\alpha }(\Gamma ,\mathcal{F})=\frac{\alpha +t}{r+s}.$$

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

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

$$0⟶\mathcal{O}(-1,-4)⟶\mathcal{O}(-1,0)\oplus \mathcal{O}(0,-4)⟶{\mathcal{I}}_Z⟶0.$$

$$0⟶\mathcal{O}(2,-1)⟶\mathcal{O}(2,3)\oplus \mathcal{O}(3,-1)⟶{\mathcal{I}}_Z(3,3)⟶0$$

$$H^2({\mathcal{I}}_Z(1,1))$$

$$H^2({\mathcal{I}}_Z(1,0))$$

$${\dim }_{\mathbb{C}}{H}^1({\mathcal{I}}_Z)$$

$${\dim }_{\mathbb{C}}{H}^1({\mathcal{I}}_Z(1,0))$$

$${\dim }_{\mathbb{C}}{H}^1({\mathcal{I}}_Z(0,1))$$

$$0$$

$$0⟶\mathcal{K}er({\phi }_1)\stackrel{{\phi }_3}{⟶}d\mathcal{O}⟶{\mathcal{I}}_Z(1,1)⟶\mathcal{K}er({\phi }_2)/\mathcal{I}m({\phi }_1)⟶0.$$

$$P_{E_1^{-1,1}}$$

$$=P_{\mathcal{I}m({\phi }_1)}+P_{\mathcal{K}er({\phi }_2)/\mathcal{I}m({\phi }_1)}+P_{d\mathcal{O}}$$

$$=P_{3\mathcal{O}(-1,-1)}-P_{\mathcal{K}er({\phi }_1)}+P_{{\mathcal{I}}_Z(1,1)}-P_{d\mathcal{O}}+P_{\mathcal{K}er({\phi }_1)}+P_{d\mathcal{O}}$$

$$=P_{3\mathcal{O}(-1,-1)}+P_{{\mathcal{I}}_Z(1,1)}$$

$$=P_{3\mathcal{O}(-1,-1)}+P_{\mathcal{O}(1,1)}-P_{{\mathcal{O}}_Z}$$

$$=3mn+(m+2)(n+2)-4$$

$$=4mn+2m+2n.$$

$$2\mathcal{O}(0,-1)⟶E_1^{-1,1}⟶2\mathcal{O}(-1,0).$$

$$0⟶3\mathcal{O}(-1,-1)⟶2\mathcal{O}(0,-1)\oplus 2\mathcal{O}(-1,0)⟶{\mathcal{I}}_Z(1,1)⟶0.$$

$$\left[\begin{array}[]{cccc}1\otimes z&0&x\otimes 1&y\otimes 1\end{array}\right],$$

$$\left[\begin{array}[]{cccc}1\otimes z&1\otimes w&x\otimes 1&0\end{array}\right],$$

$$\left[\begin{array}[]{cccc}1\otimes z&1\otimes w&x\otimes 1&y\otimes 1\end{array}\right].$$

$$0⟶\mathcal{K}er({\phi }_2)(2,2)⟶2\mathcal{O}(2,1)\oplus 2\mathcal{O}(1,2)\stackrel{{\phi }_2(2,2)}{\to }\mathcal{O}(2,2)⟶0$$

$$0⟶\mathcal{O}⟶{\mathcal{I}}_Z(1,1)⟶\mathcal{K}er({\phi }_2)/3\mathcal{O}(-1,-1)⟶0.$$

$$0⟶{\mathcal{I}}_Z(3,3)⟶\mathcal{O}(3,3)⟶{\mathcal{O}}_Z⟶0$$

$$\frac{\alpha +t}{r+s}=\frac{\alpha +1}{6}$$

$${\mathbf{M}}^{\infty }\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$M^{0+}(3m+2n-1)\times M(n+2)\text{and}{M}^{0+}(2m+3n-1)\times M(m+2).$$

$$M^{0+}(3m+2n)\times M(n+1)\text{and}{M}^{0+}(2m+3n)\times M(m+1).$$

$$0⟶\mathcal{O}(-2,-2)\oplus \mathcal{O}(-1,-3)\stackrel{\phi }{⟶}\mathcal{O}(-1,-2)\oplus \mathcal{O}⟶{\mathcal{E}}_3⟶0,$$

Full text

1. Introduction

Given a polynomial $P(m,n)$ in two variables with rational coefficients we denote by $\operatorname{M}(P)$ the moduli space of sheaves, on the product of two complex projective lines $\mathbb{P}^{1}\times\mathbb{P}^{1}$, that have Hilbert polynomial $P$ and are semi-stable with respect to the polarization $\mathcal{O}(1,1)$. This paper will be focused on the geometry of $\mathbf{M}=\operatorname{M}(3m+3n+1)$. A sheaf ${\mathcal{F}}\in\mathbf{M}$ satisfies $\chi({\mathcal{F}})=1$ and is supported on a curve of bidegree $(3,3)$. The semi-stability of ${\mathcal{F}}$ is given by two conditions: ${\mathcal{F}}$ is pure, meaning that there are no subsheaves with support of dimension zero, and for any subsheaf ${\mathcal{E}}\subset{\mathcal{F}}$ with Hilbert polynomial $P_{{\mathcal{E}}}(m,n)=rm+sn+t$ we have the inequality of slopes

[TABLE]

From the general construction [13] of moduli spaces of semi-stable sheaves with a fixed Hilbert polynomial we know that $\mathbf{M}$ is projective and from [9] we know that $\mathbf{M}$ is smooth, irreducible, of dimension $19$. We refer to the introductory section of [10] for more details about moduli spaces of sheaves on $\mathbb{P}^{1}\times\mathbb{P}^{1}$ with linear Hilbert polynomial.

For a projective variety $X$ we define the Poincaré polynomial

[TABLE]

The varieties with which we will be concerned in this paper will have no odd homology, so the above will truly be a polynomial expression.

Theorem 1.1**.**

The Poincaré polynomial of $\mathbf{M}$ is

[TABLE]

The Euler characteristic of $\mathbf{M}$ is $756$.

The Poincaré polynomial of $\operatorname{M}(3m+2n+1)$ was computed in [4], [10] and [5] by different methods. The Poincaré polynomial of $\operatorname{M}(4m+2n+1)$ was computed in [11]. To prove Theorem 1.1 we will use the same method we used in [10] and [11], namely variation of moduli spaces of $\alpha$-semi-stable pairs. A pair $(\Gamma,{\mathcal{F}})$ consists of a coherent sheaf ${\mathcal{F}}$ on $\mathbb{P}^{1}\times\mathbb{P}^{1}$ and a $1$-dimensional vector subspace $\Gamma\subset\operatorname{H}^{0}({\mathcal{F}})$. Fix a positive rational number $\alpha$. We restrict our attention to the case when ${\mathcal{F}}$ has linear Hilbert polynomial $P_{{\mathcal{F}}}(m,n)=rm+sn+t$. Then the $\alpha$-slope of $(\Gamma,{\mathcal{F}})$ is

[TABLE]

The $\alpha$-semi-stability of $(\Gamma,{\mathcal{F}})$ is given by three conditions: ${\mathcal{F}}$ is pure, for any subsheaf ${\mathcal{E}}\subset{\mathcal{F}}$ we have $\operatorname{p}({\mathcal{E}})\leq\operatorname{p}_{\alpha}({\mathcal{F}})$ and for any subpair $(\Gamma,{\mathcal{E}})\subset(\Gamma,{\mathcal{F}})$ we have $\operatorname{p}_{\alpha}(\Gamma,{\mathcal{E}})\leq\operatorname{p}_{\alpha}(\Gamma,{\mathcal{F}})$. Moduli spaces $\operatorname{M}^{\alpha}(P)$ of $\alpha$-semi-stable pairs with fixed Hilbert polynomial $P$ have been constructed in [8] and [7]. The positive real axis can be divided into finitely many intervals $(\alpha_{0},\alpha_{1})$, $(\alpha_{1},\alpha_{2})$, $\ldots$, $(\alpha_{k},\alpha_{k+1})$, where $\alpha_{0}=0$, $\alpha_{k+1}=\infty$, $\alpha_{i}$ are rational numbers, such that $\operatorname{M}^{\alpha}(P)$ remains unchanged when $\alpha$ varies in an interval, and $\operatorname{M}^{\alpha}(P)$ changes when $\alpha$ crosses the boundaries between intervals (which are called walls). We write $\operatorname{M}^{0+}(P)=\operatorname{M}^{\alpha}(P)$ for $\alpha\in(0,\alpha_{1})$ and $\operatorname{M}^{\infty}(P)=\operatorname{M}^{\alpha}(P)$ for $\alpha\in(\alpha_{k},\infty)$.

The Poincaré polynomial of $\mathbf{M}$ is related to the Poincaré polynomials of $\operatorname{M}^{0+}(3m+3n+1)$ and $\operatorname{M}^{0+}(3m+3n-1)$ by the formula from Proposition 4.1. These in turn can be related to the Poincaré polynomials of $\operatorname{M}^{\infty}(3m+3n+1)$ and $\operatorname{M}^{\infty}(3m+3n-1)$, which are certain flag Hilbert schemes. In Section 3 we explain the relationship by a series of blow-ups and blow-downs between $\operatorname{M}^{\infty}(3m+3n+1)$ and $\operatorname{M}^{0+}(3m+3n+1)$. In Section 4 we explain the relationship between $\operatorname{M}^{\infty}(3m+3n-1)$ and $\operatorname{M}^{0+}(3m+3n-1)$. In Section 2 we prove that $\operatorname{M}^{\infty}(3m+3n+1)$ is smooth.

2. Preliminaries

The purpose of this section is to prove Proposition 2.2, which will be needed later in the analysis of $\operatorname{M}^{0+}(3m+3n+1)$.

For any coherent sheaf ${\mathcal{F}}$ on $\mathbb{P}^{1}\times\mathbb{P}^{1}$, [2, Lemma 1] provides a spectral sequence converging to ${\mathcal{F}}$, with first level $E_{1}$ given by the tableau

[TABLE]

All the remaining $E_{1}^{ij}$ are zero. The sheaves $E_{1}^{-1,j}$ fit into the exact sequences

[TABLE]

Lemma 2.1**.**

Let $Z\subset\mathbb{P}^{1}\times\mathbb{P}^{1}$ be a zero-dimensional subscheme of length $4$. Then $\operatorname{H}^{1}({\mathcal{I}}_{Z}(3,3))=0$.

Proof.

Assume first that $Z$ is contained in a line of bidegree $(1,0)$. Then we have a resolution

[TABLE]

From the long exact sequence in cohomology associated to the short exact sequence

[TABLE]

we get $\operatorname{H}^{1}({\mathcal{I}}_{Z}(3,3))=0$. The same argument applies if $Z$ is contained in a line of bidegree $(0,1)$. For the remaining part of this proof we will assume that $Z$ is not contained in a line. 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 sequence (2) with $j=0$, we get $E_{1}^{-1,0}=0$. Also, $\operatorname{H}^{0}({\mathcal{I}}_{Z})=0$, hence $E_{1}^{-2,0}=0$. By Serre duality

[TABLE]

hence $E_{1}^{0,2}=0$, $E_{1}^{-2,2}=0$ and, from sequence (2) with $j=2$ we get $E_{1}^{-1,2}=0$. We have

[TABLE]

Denote $d=\dim_{\mathbb{C}}\operatorname{H}^{1}({\mathcal{I}}_{Z}(1,1))=\dim_{\mathbb{C}}\operatorname{H}^{0}({\mathcal{I}}_{Z}(1,1))$. Since $Z$ is not contained in a line, $Z$ is not contained in the intersection of two conics, hence $d\leq 1$. Tableau (1) now takes the simplified form

[TABLE]

The convergence of the spectral sequence means that $\varphi_{2}$ is surjective and that we have an exact sequence

[TABLE]

From this we can compute the Hilbert polynomial of $E_{1}^{-1,1}$:

[TABLE]

On the other hand we have the exact sequence (2) with $j=1$:

[TABLE]

Since $P_{E_{1}^{-1,1}}=P_{2\mathcal{O}(0,-1)}+P_{2\mathcal{O}(-1,0)}$, this sequence is also exact on the left and on the right, hence $E_{1}^{-1,1}\simeq 2\mathcal{O}(0,-1)\oplus 2\mathcal{O}(-1,0)$.

Assume that $d=0$. Then the exact sequence (3) yields the resolution

[TABLE]

From the long exact cohomology sequence we get $\operatorname{H}^{1}({\mathcal{I}}_{Z}(3,3))=0$.

Assume that $d=1$. Then ${\mathcal{K}er}(\varphi_{1})=0$, otherwise ${\mathcal{C}oker}(\varphi_{3})$ would be a non-zero torsion subsheaf of ${\mathcal{I}}_{Z}(1,1)$, which is absurd. Let $V_{1}$ and $V_{2}$ be vector spaces over $\mathbb{C}$ of dimension $2$ and make the identification $\mathbb{P}^{1}\times\mathbb{P}^{1}=\mathbb{P}(V_{1})\times\mathbb{P}(V_{2})$. We choose a basis $\{x,y\}$ of $V_{1}^{*}$ and a basis $\{z,w\}$ of $V_{2}^{*}$. Since $\varphi_{2}$ is surjective and $\varphi_{1}$ is injective, it is easy to see that $\varphi_{2}$ has one of the following canonical forms:

[TABLE]

[TABLE]

[TABLE]

In each case it is clear that $\varphi_{2}(2,2)$ is surjective on global sections. From the long exact cohomology sequence associated to the short exact sequence

[TABLE]

we get $\operatorname{H}^{1}({\mathcal{K}er}(\varphi_{2})(2,2))=0$. The exact sequence (3) becomes

[TABLE]

From long exact cohomology sequences we deduce that $\operatorname{H}^{1}({\mathcal{I}}_{Z}(3,3))=0$. ∎

Proposition 2.2**.**

Let $\operatorname{H}$ be the flag Hilbert scheme of zero-dimensional subschemes of length $4$ contained in curves of bidegree $(3,3)$ in $\mathbb{P}^{1}\times\mathbb{P}^{1}$. Then $\operatorname{H}$ is a $\mathbb{P}^{11}$-bundle over $\operatorname{Hilb}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(4)$, so it is smooth.

Proof.

The fiber of the canonical morphism $\operatorname{H}\to\operatorname{Hilb}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(4)$ over a zero-dimensional subscheme $Z$ is $\mathbb{P}(\operatorname{H}^{0}({\mathcal{I}}_{Z}(3,3)))$. By virtue of Lemma 2.1, the exact sequence

[TABLE]

is also exact on global sections. From this we get $\dim_{\mathbb{C}}\operatorname{H}^{0}({\mathcal{I}}_{Z}(3,3))=12$. ∎

3. Variation of $\operatorname{M}^{\alpha}(3m+3n+1)$

Proposition 3.1**.**

With respect to the polynomial $P(m,n)=3m+3n+1$ there are exactly three walls at $\alpha_{1}=2$, $\alpha_{2}=5$ and $\alpha_{3}=11$.

Proof.

As at [10, Proposition 5.2], we need to solve the equation

[TABLE]

with integers $0\leq r\leq 3$, $0\leq s\leq 3$, $t\geq r+s-rs$, and $\alpha>0$ a rational number, the case when $(r,s)=(3,3)$ being excluded. Assume that $(r,s)=(3,2)$ or $(2,3)$ and $t\geq-1$. Equation (4) becomes $\alpha=5-6t$, which has solutions $\alpha_{3}=11$ for $t=-1$ and $\alpha_{2}=5$ for $t=0$. Assume that $(r,s)=(2,2)$, $t\geq 0$. Equation (4) becomes $\alpha=2-3t$, which has solution $\alpha_{1}=2$ for $t=0$. For all other choices of $r$ and $s$ equation (4) has no positive solution in $\alpha$. ∎

Denote $\mathbf{M}^{\alpha}=\operatorname{M}^{\alpha}(3m+3n+1)$. For $\alpha\in(11,\infty)$ we write $\mathbf{M}^{\alpha}=\mathbf{M}^{\infty}$. For $\alpha\in(5,11)$ we write $\mathbf{M}^{\alpha}=\mathbf{M}^{5+}=\mathbf{M}^{11-}$. For $\alpha\in(2,5)$ we write $\mathbf{M}^{\alpha}=\mathbf{M}^{2+}=\mathbf{M}^{5-}$. For $\alpha\in(0,2)$ we write $\mathbf{M}^{\alpha}=\mathbf{M}^{0+}$. The inclusions of sets of $\alpha$-semi-stable pairs induce the flipping diagrams

[TABLE]

in which all maps are birational.

The following proposition is a particular case of [12, Proposition B8]:

Proposition 3.2**.**

The variety $\mathbf{M}^{\infty}$ is isomorphic to the flag Hilbert scheme of zero-dimensional subschemes of length $4$ contained in curves of bidegree $(3,3)$ in $\mathbb{P}^{1}\times\mathbb{P}^{1}$.

Corollary 3.3**.**

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

Proof.

By Propositions 2.2 and 3.2, $\mathbf{M}^{\infty}$ is rational. Since $\mathbf{M}^{0+}$ is birational to $\mathbf{M}^{\infty}$, also $\mathbf{M}^{0+}$ is rational. The forgetful morphism $\mathbf{M}^{0+}\to\mathbf{M}$, $(\Gamma,{\mathcal{F}})\mapsto[{\mathcal{F}}]$ is birational because, for a generic sheaf ${\mathcal{F}}\in\mathbf{M}$, we have $\operatorname{H}^{0}({\mathcal{F}})\simeq\mathbb{C}$. Thus, $\mathbf{M}$ is also rational. ∎

Remark 3.4**.**

From the proof of Proposition 3.1 we see that the strictly $\alpha$-semi-stable locus in $\mathbf{M}^{11}$ has two connected components:

[TABLE]

According to [10, Theorem 2.2], a point in $\operatorname{M}^{0+}(3m+2n-1)$ is of the form $(\Gamma_{1},{\mathcal{E}}_{1})$, where ${\mathcal{E}}_{1}$ is the structure sheaf of a curve of bidegree $(2,3)$ and $\Gamma_{1}=\operatorname{H}^{0}({\mathcal{E}}_{1})$. Thus, $\operatorname{M}^{0+}(3m+2n-1)\simeq\mathbb{P}^{11}$. A point in $\operatorname{M}(n+2)$ is of the form $\mathcal{O}_{L}(0,1)$, where $L\subset\mathbb{P}^{1}\times\mathbb{P}^{1}$ is a line of bidegree $(1,0)$. Thus, $\operatorname{M}(n+2)\simeq\mathbb{P}^{1}$.

The strictly $\alpha$-semi-stable locus in $\mathbf{M}^{5}$ has two connected components:

[TABLE]

According to [11, Proposition 3.3], the points in $\operatorname{M}^{0+}(3m+2n)$ are of the form $(\Gamma_{3},{\mathcal{E}}_{3})$, where $\Gamma_{3}=\operatorname{H}^{0}({\mathcal{E}}_{3})$ and ${\mathcal{E}}_{3}$ has resolution

[TABLE]

with $\varphi_{11}\neq 0$, $\varphi_{12}\neq 0$. Moreover, $\operatorname{M}^{0+}(3m+2n)$ is isomorphic to the universal quintic of bidegree $(2,3)$, so it is a $\mathbb{P}^{10}$-bundle over $\mathbb{P}^{1}\times\mathbb{P}^{1}$. A point in $\operatorname{M}(n+1)$ is of the form $\mathcal{O}_{L}$, hence $\operatorname{M}(n+1)\simeq\mathbb{P}^{1}$.

The strictly $\alpha$-semi-stable locus in $\mathbf{M}^{2}$ is of the form $\operatorname{M}^{0+}(2m+2n)\times\operatorname{M}(m+n+1)$. By [10, Remark 5.4], a point in $\operatorname{M}^{0+}(2m+2n)$ is of the form $(\Gamma_{5},{\mathcal{E}}_{5})$, where ${\mathcal{E}}_{5}$ is the structure sheaf of a curve of bidegree $(2,2)$ and $\Gamma_{5}=\operatorname{H}^{0}({\mathcal{E}}_{5})$. Thus, $\operatorname{M}^{0+}(2m+2n)\simeq\mathbb{P}^{8}$. According to [1, Proposition 11], a sheaf ${\mathcal{E}}_{6}\in\operatorname{M}(m+n+1)$ is the structure sheaf of a curve of bidegree $(1,1)$. Thus, $\operatorname{M}(m+n+1)\simeq\mathbb{P}^{3}$.

Consider the flipping loci

[TABLE]

Over a point $(\Lambda_{1},\Lambda_{2})\in\operatorname{M}^{0+}(3m+2n-1)\times\operatorname{M}(n+2)$, $F^{\infty}_{1}$ has fiber $\mathbb{P}(\operatorname{Ext}^{1}(\Lambda_{1},\Lambda_{2}))$ and $F^{11}_{1}$ has fiber $\mathbb{P}(\operatorname{Ext}^{1}(\Lambda_{2},\Lambda_{1}))$. Over a point $(\Lambda_{1}^{\prime},\Lambda_{2}^{\prime})\in\operatorname{M}^{0+}(2m+3n-1)\times\operatorname{M}(m+2)$, $F^{\infty}_{2}$ has fiber $\mathbb{P}(\operatorname{Ext}^{1}(\Lambda_{1}^{\prime},\Lambda_{2}^{\prime}))$ and $F^{11}_{2}$ has fiber $\mathbb{P}(\operatorname{Ext}^{1}(\Lambda_{2}^{\prime},\Lambda_{1}^{\prime}))$. Over a point $(\Lambda_{3},\Lambda_{4})\in\operatorname{M}^{0+}(3m+2n)\times\operatorname{M}(n+1)$, $F^{5+}_{1}$ has fiber $\mathbb{P}(\operatorname{Ext}^{1}(\Lambda_{3},\Lambda_{4}))$ and $F^{5-}_{1}$ has fiber $\mathbb{P}(\operatorname{Ext}^{1}(\Lambda_{4},\Lambda_{3}))$. Over a point $(\Lambda_{3}^{\prime},\Lambda_{4}^{\prime})\in\operatorname{M}^{0+}(2m+3n)\times\operatorname{M}(m+1)$, $F^{5+}_{2}$ has fiber $\mathbb{P}(\operatorname{Ext}^{1}(\Lambda_{3}^{\prime},\Lambda_{4}^{\prime}))$ and $F^{5-}_{2}$ has fiber $\mathbb{P}(\operatorname{Ext}^{1}(\Lambda_{4}^{\prime},\Lambda_{3}^{\prime}))$. Over a point $(\Lambda_{5},\Lambda_{6})\in\operatorname{M}^{0+}(2m+2n)\times\operatorname{M}(m+n+1)$, $F^{2}$ has fiber $\mathbb{P}(\operatorname{Ext}^{1}(\Lambda_{5},\Lambda_{6}))$ and $F^{0}$ has fiber $\mathbb{P}(\operatorname{Ext}^{1}(\Lambda_{6},\Lambda_{5}))$.

Proposition 3.5**.**

The flipping loci are smooth bundles with fibers indicated in Table 1 below.

Proof.

Choose $\Lambda_{1}=(\Gamma_{1},{\mathcal{E}}_{1})$ and $\Lambda_{2}=(0,\mathcal{O}_{L}(0,1))$. From [7, Corollaire 1.6] we have the exact sequence

[TABLE]

From the short exact sequence

[TABLE]

we get the long exact sequence

[TABLE]

Combining these exact sequences we obtain $\operatorname{Ext}^{1}(\Lambda_{1},\Lambda_{2})\simeq\mathbb{C}^{5}$.

From [7, Corollaire 1.6] we have the exact sequence

[TABLE]

From the short exact sequence

[TABLE]

we get the long exact sequence

[TABLE]

From these exact sequences we see that $\operatorname{Ext}^{1}(\Lambda_{2},\Lambda_{1})\simeq\mathbb{C}^{3}$.

Choose $\Lambda_{3}=(\Gamma_{3},{\mathcal{E}}_{3})$ and $\Lambda_{4}=(0,\mathcal{O}_{L})$. From [7, Corollaire 1.6] we have the exact sequence

[TABLE]

From resolution (5) we get the long exact sequence

[TABLE]

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

From [7, Corollaire 1.6] we have the exact sequence

[TABLE]

From resolution (5) we get the long exact sequence

[TABLE]

From these exact sequences we get $\operatorname{Ext}^{1}(\Lambda_{4},\Lambda_{3})\simeq\mathbb{C}^{3}$.

Choose $\Lambda_{5}=(\Gamma_{5},{\mathcal{E}}_{5})$ and $\Lambda_{6}=(\Gamma_{6},{\mathcal{E}}_{6})$. From [7, Corollaire 1.6] we have the exact sequence

[TABLE]

From the short exact sequence

[TABLE]

we get the long exact sequence

[TABLE]

From these exact sequences we get $\operatorname{Ext}^{1}(\Lambda_{5},\Lambda_{6})\simeq\mathbb{C}^{5}$.

From [7, Corollaire 1.6] we have the exact sequence

[TABLE]

From the short exact sequence

[TABLE]

we get the long exact sequence

[TABLE]

From these exact sequences we get $\operatorname{Ext}^{1}(\Lambda_{6},\Lambda_{5})\simeq\mathbb{C}^{4}$. ∎

Lemma 3.6**.**

1. (i)

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

For $\Lambda\in F^{5-}$ we have $\operatorname{Ext}^{2}(\Lambda,\Lambda)=0$. 3. (iii)

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

Proof.

(i) It is enough to consider the case when $\Lambda\in F^{11}_{1}$, the case when $\Lambda\in F^{11}_{2}$ being obtained by symmetry. In view of the exact sequence

[TABLE]

it suffices to show that $\operatorname{Ext}^{2}(\Lambda_{i},\Lambda_{j})=0$ for $i,j=1,2$. From [7, Corollaire 1.6] we have the exact sequence

[TABLE]

The group on the right vanishes because $\mathcal{O}_{L}$ is stable, by [10, Proposition 3.2], ${\mathcal{E}}_{1}\otimes\omega$ is stable, and $\operatorname{p}(\mathcal{O}_{L}(0,1))>\operatorname{p}({\mathcal{E}}_{1}\otimes\omega)$. Thus, $\operatorname{Ext}^{2}(\Lambda_{1},\Lambda_{2})=0$. From the exact sequence

[TABLE]

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

[TABLE]

we get the vanishing of $\operatorname{Ext}^{2}(\Lambda_{2},\Lambda_{2})$. From [7, Corollaire 1.6] we have the exact sequence

[TABLE]

The space $\operatorname{Ext}^{1}(\Lambda_{1},\Lambda_{1})$ is isomorphic to the tangent space of $\operatorname{M}^{0+}(3m+2n-1)$ at $\Lambda_{1}$, so it is isomorphic to $\mathbb{C}^{11}$. From the short exact sequence (6) we get the long exact sequence

[TABLE]

Thus, $\operatorname{Ext}^{1}({\mathcal{E}}_{1},{\mathcal{E}}_{1})\simeq\mathbb{C}^{13}$. The vanishing of $\operatorname{Ext}^{2}(\Lambda_{1},\Lambda_{1})$ follows from this.

(ii) From [7, Corollaire 1.6] we have the exact sequence

[TABLE]

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

[TABLE]

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

[TABLE]

we get the vanishing of $\operatorname{Ext}^{2}(\Lambda_{4},\Lambda_{4})$. From [7, Corollaire 1.6] we have the exact sequence

[TABLE]

The space $\operatorname{Ext}^{1}(\Lambda_{3},\Lambda_{3})$ is isomorphic to the tangent space of $\operatorname{M}^{0+}(3m+2n)$ at $\Lambda_{3}$, so it is isomorphic to $\mathbb{C}^{12}$. From resolution (5) we have the long exact sequence

[TABLE]

Thus, $\operatorname{Ext}^{1}({\mathcal{E}}_{3},{\mathcal{E}}_{3})\simeq\mathbb{C}^{13}$. The vanishing of $\operatorname{Ext}^{2}(\Lambda_{3},\Lambda_{3})$ follows from this.

(iii) From [7, Corollaire 1.6] we have the exact sequence

[TABLE]

Thus, $\operatorname{Ext}^{2}(\Lambda_{5},\Lambda_{6})=0$. From the exact sequence

[TABLE]

we get the vanishing of $\operatorname{Ext}^{2}(\Lambda_{6},\Lambda_{5})$. From the exact sequence

[TABLE]

we get the vanishing of $\operatorname{Ext}^{2}(\Lambda_{6},\Lambda_{6})$. From [7, Corollaire 1.6] we have the exact sequence

[TABLE]

The space $\operatorname{Ext}^{1}(\Lambda_{5},\Lambda_{5})$ is isomorphic to the tangent space of $\operatorname{M}^{0+}(2m+2n)$ at $\Lambda_{5}$, so it is isomorphic to $\mathbb{C}^{8}$. From the short exact sequence (7) we get the long exact sequence

[TABLE]

Thus, $\operatorname{Ext}^{1}({\mathcal{E}}_{5},{\mathcal{E}}_{5})\simeq\mathbb{C}^{9}$. From this we get the vanishing of $\operatorname{Ext}^{2}(\Lambda_{5},\Lambda_{5})$. ∎

Theorem 3.7**.**

Let $\mathbf{M}^{\alpha}$ be the moduli space of $\alpha$-semi-stable pairs on $\mathbb{P}^{1}\times\mathbb{P}^{1}$ having Hilbert polynomial $P(m,n)=3m+3n+1$. We have the following commutative diagrams expressing the variation of $\mathbf{M}^{\alpha}$ as $\alpha$ crosses the walls:

[TABLE]

Here each $\beta_{i}$ is the blow-up with center $F^{i}$. Moreover, $\beta_{11}$ contracts the component $\widetilde{F}^{\infty}_{1}$ of the exceptional divisor $\widetilde{F}^{\infty}$ in the direction of $\mathbb{P}^{4}$, where we view $\widetilde{F}^{\infty}_{1}$ as a $\mathbb{P}^{4}\times\mathbb{P}^{2}$-bundle over $\operatorname{M}^{0+}(3m+2n-1)\times\operatorname{M}(n+2)$; $\beta_{11}$ contracts the component $\widetilde{F}^{\infty}_{2}$ of $\widetilde{F}^{\infty}$ in the direction of $\mathbb{P}^{4}$, where we view $\widetilde{F}^{\infty}_{2}$ as a $\mathbb{P}^{4}\times\mathbb{P}^{2}$-bundle over $\operatorname{M}^{0+}(2m+3n-1)\times\operatorname{M}(m+2)$; $\beta_{5-}$ contracts the component $\widetilde{F}^{5+}_{1}$ of the exceptional divisor $\widetilde{F}^{5+}$ in the direction of $\mathbb{P}^{3}$, where we view $\widetilde{F}^{5+}_{1}$ as a $\mathbb{P}^{3}\times\mathbb{P}^{2}$-bundle over $\operatorname{M}^{0+}(3m+2n)\times\operatorname{M}(n+1)$; $\beta_{5-}$ contracts the component $\widetilde{F}^{5+}_{2}$ of $\widetilde{F}^{5+}$ in the direction of $\mathbb{P}^{3}$, where we view $\widetilde{F}^{5+}_{2}$ as a $\mathbb{P}^{3}\times\mathbb{P}^{2}$-bundle over $\operatorname{M}^{0+}(2m+3n)\times\operatorname{M}(m+1)$; $\beta_{0}$ contracts the exceptional divisor $\widetilde{F}^{2}$ in the direction of $\mathbb{P}^{4}$, where we view $\widetilde{F}^{2}$ as a $\mathbb{P}^{4}\times\mathbb{P}^{3}$-bundle over $\operatorname{M}^{0+}(2m+2n)\times\operatorname{M}(m+n+1)$. The spaces $\mathbf{M}^{11-}$, $\mathbf{M}^{5-}$ and $\mathbf{M}^{0+}$ are smooth.

Proof.

The proof of this theorem is analogous to the proof of [10, Theorem 5.7]. By Propositions 3.2 and 2.2, $\mathbf{M}^{\infty}$ is smooth. We consider the blow-up $\beta_{\infty}$ along the subvariety $F^{\infty}$, which, according to Proposition 3.5, is smooth. We construct $\beta_{11}$, which contracts $\widetilde{F}^{\infty}_{1}$ into $F^{11}_{1}$ and $\widetilde{F}^{\infty}_{2}$ into $F^{11}_{2}$. To prove that $\beta_{11}$ is a blow-up with center $F^{11}$ we use the Universal Property of the blow-up, which requires two ingredients: that $F^{11}$ be smooth and that $\mathbf{M}^{11-}$ be smooth. We proved the first property at Proposition 3.5 and the second property at Lemma 3.6(i). For the other squares in the diagram we proceed analogously. ∎

4. Variation of $\operatorname{M}^{\alpha}(3m+3n-1)$

Proposition 4.1**.**

We have the following equation of Poincaré polynomials:

[TABLE]

The proof of this proposition is analogous to the proof of [3, Lemma 5.1]. The Poincaré polynomial of $\operatorname{M}^{0+}(3m+3n+1)$ can be computed by relating it to the Poincaré polynomial of $\operatorname{M}^{\infty}(3m+3n+1)$ via Theorem 3.7. Likewise, in order to compute $\operatorname{P}(\operatorname{M}^{0+}(3m+3n-1))$, we will find a relation between $\operatorname{M}^{0+}(3m+3n-1)$ and $\operatorname{M}^{\infty}(3m+3n-1)$.

Proposition 4.2**.**

With respect to the polynomial $P(m,n)=3m+3n-1$ there is exactly one wall at $\alpha=1$.

Proof.

We need to solve the equation

[TABLE]

with integers $0\leq r\leq 3$, $0\leq s\leq 3$, $t\geq r+s-rs$, $(r,s)\neq(3,3)$, and $\alpha>0$ a rational number. Assume that $(r,s)=(3,2)$ or $(2,3)$ and $t\geq-1$. Equation (8) becomes $\alpha=-5-6t$, which has solution $\alpha=1$ for $t=-1$. For all other choices of $r$ and $s$ equation (8) has no positive solution in $\alpha$. ∎

For $\alpha\in(1,\infty)$ we write $\operatorname{M}^{\alpha}(3m+3n-1)=\operatorname{M}^{\infty}(3m+3n-1)$. For $\alpha\in(0,1)$ we write $\operatorname{M}^{\alpha}(3m+3n-1)=\operatorname{M}^{0+}(3m+3n-1)$. The inclusions of sets of $\alpha$-semi-stable pairs induce the flipping diagram

[TABLE]

in which both maps are birational.

Proposition 4.3**.**

The variety $\operatorname{M}^{\infty}(3m+3n-1)$ is isomorphic to the flag Hilbert scheme of zero-dimensional subschemes of length $2$ contained in curves of bidegree $(3,3)$ in $\mathbb{P}^{1}\times\mathbb{P}^{1}$.

In particular, $\operatorname{M}^{\infty}(3m+3n-1)$ is a bundle with base $\operatorname{Hilb}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(2)$ and fiber $\mathbb{P}^{13}$.

Remark 4.4**.**

From the proof of Proposition 4.2 we see that the strictly $\alpha$-semi-stable locus in $\operatorname{M}^{1}(3m+3n-1)$ has two connected components:

[TABLE]

A point in $\operatorname{M}(n)$ is of the form $\mathcal{O}_{L}(0,-1)$, where $L\subset\mathbb{P}^{1}\times\mathbb{P}^{1}$ is a line of bidegree $(1,0)$.

Consider the flipping loci

[TABLE]

Over a point $(\Lambda_{1},\Lambda_{7})\in\operatorname{M}^{0+}(3m+2n-1)\times\operatorname{M}(n)$, $F^{\infty}_{1}$ has fiber $\mathbb{P}(\operatorname{Ext}^{1}(\Lambda_{1},\Lambda_{7}))$ and $F^{0}_{1}$ has fiber $\mathbb{P}(\operatorname{Ext}^{1}(\Lambda_{7},\Lambda_{1}))$. Over a point $(\Lambda_{1}^{\prime},\Lambda_{7}^{\prime})\in\operatorname{M}^{0+}(2m+3n-1)\times\operatorname{M}(m)$, $F^{\infty}_{2}$ has fiber $\mathbb{P}(\operatorname{Ext}^{1}(\Lambda_{1}^{\prime},\Lambda_{7}^{\prime}))$ and $F^{0}_{2}$ has fiber $\mathbb{P}(\operatorname{Ext}^{1}(\Lambda_{7}^{\prime},\Lambda_{1}^{\prime}))$.

Proposition 4.5**.**

The flipping loci $F^{\infty}_{1}$, $F^{\infty}_{2}$, $F^{0}_{1}$, $F^{0}_{2}$ are smooth bundles with fiber $\mathbb{P}^{2}$.

Proof.

Choose $\Lambda_{1}=(\Gamma_{1},{\mathcal{E}}_{1})$ and $\Lambda_{7}=(0,\mathcal{O}_{L}(0,-1))$. From [7, Corollaire 1.6] we have the exact sequence

[TABLE]

From resolution (6) we get the long exact sequence

[TABLE]

Thus, $\operatorname{Ext}^{1}(\Lambda_{1},\Lambda_{7})\simeq\mathbb{C}^{3}$. From [7, Corollaire 1.6] we have the exact sequence

[TABLE]

From the short exact sequence

[TABLE]

we get the long exact sequence

[TABLE]

From these exact sequences we obtain $\operatorname{Ext}^{1}(\Lambda_{7},\Lambda_{1})\simeq\mathbb{C}^{3}$. ∎

Lemma 4.6**.**

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

Proof.

It is enough to consider the case when $\Lambda\in F^{0}_{1}$, the case when $\Lambda\in F^{2}_{0}$ being obtained by symmetry. In view of the exact sequence

[TABLE]

it is enough to show that $\operatorname{Ext}^{2}(\Lambda_{i},\Lambda_{j})=0$ for $i,j=1,7$. To obtain these vanishings we proceed exactly as in the proof of Lemma 3.6(i) with $\Lambda_{2}$ replaced by $\Lambda_{7}$. ∎

Theorem 4.7**.**

The space $\operatorname{M}^{0+}(3m+3n-1)$ is smooth. We have the following commutative diagram expressing the variation of $\operatorname{M}^{\alpha}(3m+3n-1)$ as $\alpha$ crosses the wall:

[TABLE]

Here $\beta_{\infty}$ is the blow-up with center $F^{\infty}$ and $\beta_{0}$ is the blow-up with center $F^{0}$. The exceptional divisor $\widetilde{F}^{\infty}$ has two connected components: $\widetilde{F}^{\infty}_{1}$ and $\widetilde{F}^{\infty}_{2}$. We view $\widetilde{F}^{\infty}_{1}$ as a $\mathbb{P}^{2}\times\mathbb{P}^{2}$-bundle over $\operatorname{M}^{0+}(3m+2n-1)\times\operatorname{M}(n)$ and $\widetilde{F}^{\infty}_{2}$ as a $\mathbb{P}^{2}\times\mathbb{P}^{2}$-bundle over $\operatorname{M}^{0+}(2m+3n-1)\times\operatorname{M}(m)$. Thus, $\beta_{0}$ contracts $\widetilde{F}^{\infty}_{1}$ in the direction of the first $\mathbb{P}^{2}$ into $F^{0}_{1}$ and it contracts $\widetilde{F}^{\infty}_{2}$ in the direction of the first $\mathbb{P}^{2}$ into $F^{0}_{2}$.

Proof.

The proof of this theorem is analogous to the proof of Theorem 3.7 and makes use of Proposition 4.5 and Lemma 4.6 in a similar fashion. ∎

Proof of Theorem 1.1. By Theorem 3.7, we have

[TABLE]

By Proposition 3.2, Proposition 2.2 and Remark 3.4, we have further

[TABLE]

By Theorem 4.7, we have

[TABLE]

By Proposition 4.3, we have further

[TABLE]

According to [6, Theorem 0.1],

[TABLE]

From Proposition 4.1 we finally obtain

[TABLE]