Birational geometry of the moduli space of pure sheaves on quadric surface

TL;DR

This paper investigates the birational geometry of a specific moduli space of stable sheaves on a quadric surface, describing its structure through a sequence of blow-ups and blow-downs, and relating it to a projective bundle over a Grassmannian.

Contribution

It provides a detailed birational description of the moduli space of stable sheaves with given invariants on a quadric surface, including explicit geometric transformations.

Findings

Explicit birational map constructed between the moduli space and a projective bundle

Description of the moduli space via smooth blow-ups and blow-downs

Connection established with a Grassmannian-based projective bundle

Abstract

We study birational geometry of the moduli space of stable sheaves on a quadric surface with Hilbert polynomial $5m + 1$ and $c_{1} = (2, 3)$. We describe a birational map between the moduli space and a projective bundle over a Grassmannian as a composition of smooth blow-ups/downs.

Equations

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

$${\mathrm{elm}}_Z(\mathcal{F}):=\ker (\mathcal{F}\twoheadrightarrow \mathcal{F}{|}_Z\twoheadrightarrow \mathcal{Q}).$$

$${\mathcal{O}}_Q(0,1)⟶{\mathcal{O}}_Q(1,2{)}^{\oplus 2}$$

$$0\to \mathcal{F}\otimes {\mathrm{H}}^0({\mathcal{O}}_Q(0,1))\to \mathcal{E}\otimes {\mathrm{H}}^0({\mathcal{O}}_Q(1,2))\to \mathcal{U}\to 0$$

$$\frac{P(F^{\prime })(m)+\delta \cdot \alpha }{r(F^{\prime })}\le (<)\frac{P(F)(m))+\alpha }{r(F)}$$

$$0\to (s^{\ast })\otimes {\mathcal{O}}_Q\to G\to F^D(2,2)\to 0.$$

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

$$\mathrm{Ker}(({\mathcal{L}}^{\ast },\mathcal{E})\twoheadrightarrow ({\mathcal{L}}^{\ast }{|}_{{\mathbf{M}}_2^{+}},\mathcal{E}{|}_{{\mathbf{M}}_2^{+}\times Q})\twoheadrightarrow ({\mathcal{L}}^{{}^{\prime \prime }},{\mathcal{O}}_{\mathcal{Z}})).$$

$$0\to {\mathrm{Ext}}^0((0,{\mathcal{O}}_Q(-2,-2)),(s,F))\to {\mathrm{Ext}}^1((s,F),(s,F))\to {\mathrm{Ext}}^1((s,{\mathcal{O}}_Q(0,1)),(s,F))\to \cdots .$$

$$0\to \mathrm{Hom}(s,{\mathrm{H}}^0(F)/⟨s⟩)\to {\mathrm{Ext}}^1((s,{\mathcal{O}}_Q(0,1)),(s,F))\to {\mathrm{Ext}}^1({\mathcal{O}}_Q(0,1),F)\stackrel{\varphi }{\to }\mathrm{Hom}(s,{\mathrm{H}}^1(F)).$$

$$({\mathbf{M}}^{+}\setminus {\mathbf{M}}_1^{+})\times Q\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$P(\mathbf{M})=q^{13}+3q^{12}+8q^{11}+10q^{10}+11q^9+11q^8+11q^7+11q^6+11q^5+11q^4+10q^3+8q^2+3q+1.$$

$$P(\mathbf{M})=P({\mathbb{P}}^{11})-P({\mathbb{P}}^1)+P({\mathbf{M}}^{-})=P({\mathbb{P}}^{11})-P({\mathbb{P}}^1)+P({\mathbb{P}}^9)(P(\mathbf{G})+(P({\mathbb{P}}^2)-1)P({\mathbb{P}}^1)).$$

Full text

1. Introduction

The geometry of the moduli space of sheaves on a del Pezzo surface has been studied in various viewpoints, for instance curve counting, the strange duality conjecture, and birational geometry via Bridgeland stability. For a detailed description of the motivation, see [CM15] and references therein. In this paper we continue the study of birational geometry of the moduli space of torsion sheaves on a del Pezzo surface, which was initiated in [CM15]. More precisely, here we construct a flip between the moduli space of sheaves and a projective bundle, and show that their common blown-up space is the moduli space of stable pairs ([LP93]), in the case of a quadric surface.

Let $Q\cong\mathbb{P}^{1}\times\mathbb{P}^{1}$ be a smooth quadric surface in $\mathbb{P}^{3}$ with a very ample polarization $L:=\mathcal{O}_{Q}(1,1)$. For the convenience of the reader, we start with a list of relevant moduli spaces.

Definition 1.1**.**

1. (1)

Let $\mathbf{M}:=\mathbf{M}_{L}(Q,(2,3),5m+1)$ be the moduli space of stable sheaves $F$ on $Q$ with $c_{1}(F)=c_{1}(\mathcal{O}_{Q}(2,3))$ and $\chi(F(m))=5m+1$. 2. (2)

Let $\mathbf{M}^{\alpha}:=\mathbf{M}^{\alpha}_{L}(Q,(2,3),5m+1)$ be the moduli space of $\alpha$-stable pairs $(s,F)$ with $c_{1}(F)=c_{1}(\mathcal{O}_{Q}(2,3))$ and $\chi(F(m))=5m+1$ ([LP93] and [He98, Theorem 2.6]). 3. (3)

Let $\mathbf{G}=\mathrm{Gr}(2,4)$ and let $\mathbf{G}_{1}$ be the blow-up of $\mathbf{G}$ along $\mathbb{P}^{1}$ (Section 2.1). 4. (4)

Let $\mathbf{P}:=\mathbb{P}(\mathcal{U})$ and $\mathbf{P}^{-}:=\mathbb{P}(\mathcal{U}^{-})$, where $\mathcal{U}$ (resp. $\mathcal{U}^{-}$) is a rank $10$ vector bundle over $\mathbf{G}$ (resp. $\mathbf{G}_{1}$) defined in (2) in Section 2.1 (resp. Section 3.3).

The aim of this paper is to explain and justify the following commutative diagram between moduli spaces.

[TABLE]

We have to explain two flips (dashed arrows) on the diagram.

One of key ingredients is the elementary modification of vector bundles ([Mar73]), sheaves ([HL10, Section 2.B]), and pairs ([CC16, Section 2.2]). It has been widely used in the study of sheaves on a smooth projective variety. Let $\mathcal{F}$ be a vector bundle on a smooth projective variety $X$ and $\mathcal{Q}$ be a vector bundle on a smooth divisor $Z\subset X$ with a surjective map $\mathcal{F}|_{Z}\twoheadrightarrow\mathcal{Q}$. The elementary modification of $\mathcal{F}$ along $Z$ is the kernel of the composition

[TABLE]

A similar definition is valid for sheaves and pairs, too.

On $\mathbf{G}_{1}$, let $\mathcal{U}^{-}:=\mathrm{elm}_{Y_{10}}(u^{*}\mathcal{U})$ be the elementary transformation of $u^{*}\mathcal{U}$ along a smooth divisor $Y_{10}$ (Section 2.1).

Proposition 1.2**.**

Let $\mathbf{P}^{-}=\mathbb{P}(\mathcal{U}^{-})$. The flip $\mathbf{P}^{-}\dashrightarrow\mathbb{P}(u^{*}\mathcal{U})=\mathbf{G}_{1}\times_{\mathbf{G}}\mathbb{P}(\mathcal{U})$ is a composition of a blow-up and a blow-down. The blow-up center in $\mathbf{P}^{-}$ (resp. $\mathbb{P}(u^{*}\mathcal{U})$) is a $\mathbb{P}^{1}$ (resp. $\mathbb{P}^{7}$)-bundle over $Y_{10}$.

Theorem 1.3**.**

There is a flip between $\mathbf{M}$ and $\mathbf{P}^{-}$ which is a blow-up followed by a blow-down, and the master space is $\mathbf{M}^{+}$, the moduli space of $+$-stable pairs.

As applications, we compute the Poincaré polynomial of $\mathbf{M}$ and show the rationality of $\mathbf{M}$ (Corollary 3.8) which were obtained by Maican by different methods ([Mai16]). Since each step of the birational transform is described in terms of blow-ups/downs along explicit subvarieties, in principle the cohomology ring and the Chow ring of $\mathbf{M}$ can be obtained from that of $\mathbf{G}$. Also one may aim for the completion of Mori’s program for $\mathbf{M}$. We will carry on these projects in forthcoming papers.

2. Relevant moduli spaces

In this section we give definitions and basic properties of some relevant moduli spaces.

2.1. Grassmannian as a moduli space of Kronecker quiver representations

The moduli space of representations of a Kronecker quiver parametrizes the isomorphism classes of stable sheaf homomorphisms

[TABLE]

up to the natural action of the automorphism group $\mathbb{C}^{*}\times\mathrm{GL}_{2}/\mathbb{C}^{*}\cong\mathrm{GL}_{2}$. For two vector spaces $E$ and $F$ of dimension $1$ and $2$ respectively and $V^{*}:=\mathrm{H}^{0}(Q,L)$, the moduli space is constructed as $\mathbf{G}:=\mathrm{Hom}(F,V^{*}\otimes E)/\!/\mathrm{GL}_{2}\cong V^{*}\otimes E\otimes F^{*}/\!/\mathrm{GL}_{2}$ with an appropriate linearization ([Kin94]). Note that the $\mathrm{GL}_{2}$ acts as a row operation on the space of $2\times 4$ matrices, $\mathbf{G}\cong\mathrm{Gr}(2,4)$.

Let $\mathbf{H}(n):=\mathrm{Hilb}^{n}(Q)$, the Hilbert scheme of $n$ points on $Q$. $\mathbf{H}(2)$ is birational to $\mathbf{G}$ because a general $Z\in\mathbf{H}(2)$, $I_{Z}(2,3)$ has a resolution of the form (1). For any $Z\in\mathbf{H}(2)$, let $\ell_{Z}$ be the unique line in $\mathbb{P}^{3}\supset Q$ containing $Z$. Then either $\ell_{Z}\cap Q=Z$ or $\ell_{Z}\subset Q$. In the second case, the class of $\ell_{Z}$ is of the type $(1,0)$ or $(0,1)$. Let $Y_{10}$ (resp. $Y_{01}$) be the locus of subschemes such that $\ell_{Z}$ is a line of the type $(1,0)$ (resp. $(0,1)$). Then $Y_{10}$ and $Y_{01}$ are two disjoint subvarieties which are isomorphic to a $\mathbb{P}^{2}$-bundle over $\mathbb{P}^{1}$.

Proposition 2.1** ([BC13, Example 6.1]).**

There exists a morphism $t:\mathbf{H}(2)\longrightarrow\mathbf{G}_{1}\stackrel{{\scriptstyle u}}{{\longrightarrow}}\mathbf{G}$. The first (resp. the second) map contracts the divisor $Y_{01}$ (resp. $Y_{10}$) to $\mathbb{P}^{1}$. If $\ell_{Z}\cap Q=Z$, then $t(Z)=I_{Z}(2,3)$. If $Z\in Y_{10}$, then $t(Z)=E_{10}\in\mathbb{P}(\mathrm{Ext}^{1}(\mathcal{O}_{Q}(1,3),\mathcal{O}_{\ell_{Z}}(1)))=\{\mathrm{pt}\}$. If $Z\in Y_{01}$, then $t(Z)=E_{01}\in\mathbb{P}(\mathrm{Ext}^{1}(\mathcal{O}_{Q}(2,2),\mathcal{O}_{\ell_{Z}}))=\{\mathrm{pt}\}$.

There is a universal morphism $\phi:p_{1}^{*}\mathcal{F}\otimes p_{2}^{*}\mathcal{O}_{Q}(0,1)\to p_{1}^{*}\mathcal{E}\otimes p_{2}^{*}\mathcal{O}_{Q}(1,2)$ where $p_{1}:\mathbf{G}\times Q\to\mathbf{G}$ and $p_{2}:\mathbf{G}\times Q\to Q$ are two projections ([Kin94]). Let $\mathcal{U}$ be the cokernel of $p_{1*}\phi$. On the stable locus, $p_{1*}\phi$ is injective. Thus we have an exact sequence

[TABLE]

and $\mathcal{U}$ is a rank $10$ vector bundle. Let $\mathbf{P}:=\mathbb{P}(\mathcal{U})$.

2.2. Moduli space $\mathbf{M}$ of stable sheaves

Recall that $\mathbf{M}:=\mathbf{M}_{L}(Q,(2,3),5m+1)$ is the moduli space of stable sheaves $F$ on $Q$ with $c_{1}(F)=c_{1}(\mathcal{O}_{Q}(2,3))$ and $\chi(F(m))=5m+1$. There are four types of points in $\mathbf{M}$ ([Mai16, Theorem 1.1]). Let $C\in|\mathcal{O}_{Q}(2,3)|$.

1. (0)

$F=\mathcal{O}_{C}(p+q)$, where the line $\langle p,q\rangle$ is not contained in $Q$; 2. (1)

$F=\mathcal{O}_{C}(p+q)$, where the line $\langle p,q\rangle$ in $Q$ is of type $(1,0)$; 3. (2)

$F=\mathcal{O}_{C}(0,1)$; 4. (3)

$F$ fits into a non-split extension $0\to\mathcal{O}_{E}\to F\to\mathcal{O}_{\ell}\to 0$ where $E$ is a $(2,2)$-curve and $\ell$ is a $(0,1)$-line.

Let $\mathbf{M}_{i}$ be the locus of sheaves of the form ($i$). Then $\mathbf{M}_{i}$ is a subvariety of codimension $i$. $\mathbf{M}_{1}$ is a $\mathbb{P}^{9}$-bundle over $\mathbb{P}^{2}\times\mathbb{P}^{1}$. $\mathbf{M}_{2}$ is isomorphic to $|\mathcal{O}_{Q}(2,3)|$. Finally, $\mathbf{M}_{3}$ is a $\mathbb{P}^{1}$-bundle over $|\mathcal{O}_{Q}(2,2)|\times|\mathcal{O}_{Q}(0,1)|$. $\mathbf{M}_{1}\cap\mathbf{M}_{2}=\mathbf{M}_{1}\cap\mathbf{M}_{3}=\emptyset$, but $\mathbf{M}_{23}:=\mathbf{M}_{2}\cap\mathbf{M}_{3}\cong|\mathcal{O}_{Q}(2,2)|\times|\mathcal{O}_{Q}(0,1)|$ ([Mai16, Theorem 1.1]). Note that $\dim\mathrm{H}^{0}(F)=1$ in general, but $\mathbf{M}_{2}$ parametrizes sheaves that $\dim\mathrm{H}^{0}(F)=2$.

2.3. Moduli spaces of stable pairs

A pair $(s,F)$ consists of $F\in\textsf{Coh}(Q)$ and a section $\mathcal{O}_{Q}\stackrel{{\scriptstyle s}}{{\to}}F$. Fix $\alpha\in\mathbb{Q}_{>0}$. A pair $(s,F)$ is called $\alpha$-semistable (resp. $\alpha$-stable) if $F$ is pure and for any proper subsheaf $F^{\prime}\subset F$, the inequality

[TABLE]

holds for $m\gg 0$. Here $\delta=1$ if the section $s$ factors through $F^{\prime}$ and $\delta=0$ otherwise. Let $\mathbf{M}^{\alpha}:=\mathbf{M}_{L}^{\alpha}(Q,(2,3),5m+1)$ be the moduli space of $S$-equivalence classes of $\alpha$-semistable pairs whose support have a class $c_{1}(\mathcal{O}_{Q}(2,3))$ ([LP93, Theorem 4.12] and [He98, Theorem 2.6]). The extremal case that $\alpha$ is sufficiently large (resp. small) is denoted by $\alpha=\infty$ (resp. $\alpha=+$). The deformation theory of pairs is studied in [He98, Corollary 1.6 and Corollary 3.6].

Proposition 2.2**.**

1. (1)

There exists a natural forgetful map $r:\mathbf{M}^{+}\longrightarrow\mathbf{M}$ which maps $(s,F)$ to $F$. 2. (2)

(**[He98, Section 4.4]**) The moduli space $\mathbf{M}^{\infty}$ of $\infty$-stable pairs is isomorphic to the relative Hilbert scheme of two points on the complete linear system $|\mathcal{O}_{Q}(2,3)|$.

The birational map $\mathbf{M}^{\infty}\dashrightarrow\mathbf{M}^{+}$ is analyzed in [Mai16, Theorem 5.7]. It turns out that this is a single flip over $\mathbf{M}^{4}$ and is a composition of a smooth blow-up and a smooth blow-down. The blow-up center $\mathbf{M}_{3}^{\infty}$ is isomorphic to a $\mathbb{P}^{2}$-bundle over $|\mathcal{O}_{Q}(2,2)|\times|\mathcal{O}_{Q}(0,1)|$ where a fiber $\mathbb{P}^{2}$ parameterizes two points lying on a $(0,1)$-line. After the flip, the flipped locus on $\mathbf{M}^{+}$ is $\mathbf{M}_{3}^{+}$.

For the forgetful map $r:\mathbf{M}^{+}\to\mathbf{M}$, we define $\mathbf{M}_{i}^{+}:=r^{-1}(\mathbf{M}_{i})$ if $i\neq 3$ and $\mathbf{M}_{3}^{+}$ is the proper transform of $\mathbf{M}_{3}$. It contracts $\mathbf{M}_{2}^{+}$, which is a $\mathbb{P}^{1}$-bundle over $\mathbf{M}_{2}$ and $\mathbf{M}^{+}\setminus\mathbf{M}_{2}^{+}\cong\mathbf{M}\setminus\mathbf{M}_{2}$. Maican proved that $r$ is a smooth blow-up along the Brill-Noether locus $\mathbf{M}_{2}$ ([Mai16, Proposition 5.8]).

3. Decomposition of the birational map between $\mathbf{M}$ and $\mathbf{P}$

In this section we prove Proposition 1.2 and Theorem 1.3 by describing the birational map between $\mathbf{M}$ and $\mathbf{P}$.

3.1. Construction of a birational map $\mathbf{M}^{+}\dashrightarrow\mathbf{P}$

Lemma 3.1**.**

There exists a surjective morphism $w:\mathbf{M}^{+}\longrightarrow\mathbf{G}$ which maps $(s,\mathcal{O}_{C}(p+q))\in\mathbf{M}_{0}^{+}$ to $I_{\{p,q\}}(2,3)$, maps $(s,\mathcal{O}_{C}(p+q))\in\mathbf{M}_{1}^{+}$ to the line $\langle p,q\rangle$ of the type $(1,0)$, maps $(s,F)\in\mathbf{M}_{2}^{+}$ to a $(0,1)$-line determined by a section, and maps $(s,F)\in\mathbf{M}_{3}^{+}$ to $\ell$ (see Section 2.2 for the notation), a $(0,1)$-line.

Proof.

By Proposition 2.2, $\mathbf{M}^{\infty}$ is the relative Hilbert scheme of 2 points on the universal $(2,3)$-curves, which is a $\mathbb{P}^{9}$-bundle over $\mathbf{H}(2)$ ([CC16, Lemma 2.3]). By composing with $t:\mathbf{H}(2)\rightarrow\mathbf{G}$ in Proposition 2.1, we have a morphism $\mathbf{M}^{\infty}\to\mathbf{G}$. On the other hand, since the flip $\mathbf{M}^{\infty}\to\mathbf{M}^{+}$ is the composition of a single blow-up/down, the blown-up space $\widetilde{\mathbf{M}}^{\infty}$ admits two morphisms to $\mathbf{M}^{\infty}$ and $\mathbf{M}^{+}$, and the flipped locus is $\mathbf{M}_{3}^{+}$. Note that each point in $\mathbf{M}_{3}^{+}$ can be regarded as a collection of data $(E,\ell,e)$ where $E$ is a $(2,2)$-curve, $\ell$ is a $(0,1)$-line, and $e\in\mathbb{P}\mathrm{Ext}^{1}(\mathcal{O}_{\ell},\mathcal{O}_{E})$. The fiber $\widetilde{\mathbf{M}}^{\infty}\to\mathbf{M}^{+}$ over the point in the blow-up center $\mathbf{M}_{3}^{+}$ is a $\mathbb{P}^{2}$ which parameterizes two points on $\ell$. The composition map $\widetilde{\mathbf{M}}^{\infty}\to\mathbf{M}^{\infty}\to\mathbf{G}$ is constant along the $\mathbb{P}^{2}$, because $\mathbf{G}$ does not remember points on the line $\ell\subset Q$. By the rigidity lemma, $\widetilde{\mathbf{M}}^{\infty}\rightarrow\mathbf{G}$ factors through $\mathbf{M}^{+}$ and we obtain a map $w:\mathbf{M}^{+}\rightarrow\mathbf{G}$. ∎

Note that $\mathbf{M}_{1}^{+}\cong\mathbf{M}_{1}$ is a $\mathbb{P}^{9}$-bundle over $\mathbb{P}^{2}\times\mathbb{P}^{1}$ and $\mathbf{M}_{2}^{+}$ is a $\mathbb{P}^{1}$-bundle over $|\mathcal{O}_{Q}(2,3)|\cong\mathbb{P}^{11}$. They are disjoint divisors on $\mathbf{M}^{+}$.

Proposition 3.2**.**

There is a birational morphism $q:\mathbf{M}^{+}\setminus\mathbf{M}_{1}^{+}\to\mathbf{P}=\mathbb{P}(\mathcal{U})$ such that $p\circ q:\mathbf{M}^{+}\setminus\mathbf{M}_{1}^{+}\to\mathbf{P}\to\mathbf{G}$ coincides with $w|_{\mathbf{M}^{+}\setminus\mathbf{M}_{1}^{+}}$ in Lemma 3.1. Furthermore, $q$ is the smooth blow-down along $\mathbf{M}_{2}^{+}$.

The proof consists of several steps. Since $\mathbf{P}=\mathbb{P}(\mathcal{U})$ is a projective bundle over $\mathbf{G}$, it is sufficient to construct a surjective homomorphism $w^{*}\mathcal{U}^{*}\to\mathcal{L}\to 0$ over $\mathbf{M}^{+}\setminus\mathbf{M}_{1}^{+}$ for some $\mathcal{L}\in\mathrm{Pic}(\mathbf{M}^{+}\setminus\mathbf{M}_{1}^{+})$, or equivalently, a bundle morphism $0\to\mathcal{L}^{*}\to w^{*}\mathcal{U}$.

Recall that a family $(\mathcal{L},\mathcal{F})$ of pairs on a scheme $S$ is a collection of data $\mathcal{L}\in\mathrm{Pic}(S)$, $\mathcal{F}\in\mathsf{Coh}(S\times Q)$, which is a flat family of pure sheaves, and a surjective morphism $\mathcal{E}xt^{2}_{\pi}(\mathcal{F},\omega_{\pi})\twoheadrightarrow\mathcal{L}$ where $\pi:S\times Q\to S$ is the projection and $\omega_{\pi}$ is the relatively dualizing sheaf (See [LP93, Section 4.3] for the explanation why we take the dual.). Now let $(\mathcal{L},\mathcal{F})$ be the universal pair ([He98, Theorem 4.8]) on $\mathbf{M}^{+}\times Q$. By applying $\mathcal{H}om(-,\mathcal{O})$ to $\mathcal{E}xt^{2}_{\pi}(\mathcal{F},\omega_{\pi})\twoheadrightarrow\mathcal{L}$, we obtain $0\to\mathcal{L}^{*}\to\mathcal{H}om(\mathcal{E}xt^{2}_{\pi}(\mathcal{F},\omega_{\pi}),\mathcal{O})$. It can be shown that $\mathcal{H}om(\mathcal{E}xt^{2}_{\pi}(\mathcal{F},\omega_{\pi}),\mathcal{O})\cong\mathcal{E}xt^{1}_{\pi}(\mathcal{E}xt^{1}(\mathcal{F},\mathcal{O}),\mathcal{O})$ (see [CM15, Section 3.2]). So we have a non-zero element $e\in\mathrm{Hom}(\mathcal{L}^{*},\mathcal{E}xt_{\pi}^{1}(\mathcal{E}xt^{1}(\mathcal{F},\mathcal{O}),\mathcal{O}))\cong\mathrm{Ext}^{1}(\mathcal{E}xt^{1}(\mathcal{F},\mathcal{O}),\pi^{*}\mathcal{L})$ ([CM15, Section 3.2]), which provides $0\rightarrow{{\pi}^{*}\mathcal{L}}\rightarrow{\mathcal{E}}\rightarrow{\mathcal{E}xt^{1}(\mathcal{F},\mathcal{O})}\rightarrow 0$ on $\mathbf{M}^{+}\times Q$. By taking $\mathcal{H}om_{\pi}(-,\omega_{\pi})$, we have $\mathcal{E}xt_{\pi}^{2}(\mathcal{E},\omega_{\pi})\rightarrow\mathcal{E}xt_{\pi}^{2}(\pi^{*}\mathcal{L},\omega_{\pi})\cong\mathcal{L}^{*}\to 0$ because $\mathcal{L}$ is a line bundle. This implies the existence of a flat family of pairs $(\mathcal{L}^{*},\mathcal{E})$ on $\mathbf{M}^{+}\times Q$. We may explicitly describe this construction fiberwisely in the following way. Let $(s,F)\in\mathbf{M}^{+}$. Let $F^{D}:=\mathcal{E}xt^{1}(F,\omega_{Q})$. For a non-zero section $s\in\mathrm{H}^{0}(F)\cong\mathrm{H}^{1}(F^{D})^{*}\cong\mathrm{Ext}^{1}(F^{D}(2,2),(s^{*})\otimes\mathcal{O}_{Q})$, we have a pair $(s^{*},G)$ given by

[TABLE]

Lemma 3.3**.**

The map $(s,F)\mapsto(s^{*},G)$ defines a dominant rational map $\mathbf{M}^{+}\dashrightarrow\mathbf{P}=\mathbb{P}(\mathcal{U})$, which is regular on $\mathbf{M}^{+}\setminus(\mathbf{M}_{1}^{+}\sqcup\mathbf{M}_{2}^{+})$.

Proof.

Since we have a relative construction of pairs, it suffices to describe the extension $(s^{*},G)$ set theoretically. If $(s,F)\in\mathbf{M}_{0}^{+}\sqcup\mathbf{M}_{1}^{+}$, then $F\cong\mathcal{O}_{C}(p+q)\cong I_{Z,C}^{D}(0,-1)$ for some curve $C$ and $Z=\{p,q\}\in\mathbf{H}(2)$ such that the line $\ell_{Z}$ containing $Z$ is not in $Q$ ([He98, Section 4.4]). Then $F^{D}(2,2)\cong I_{Z,C}(2,3)$. Since $\mathrm{Ext}^{1}(F^{D}(2,2),\mathcal{O}_{Q})\cong\mathrm{H}^{1}(F^{D})^{*}\cong\mathrm{H}^{0}(F)\cong\mathbb{C}$, from $0\rightarrow{\mathcal{O}_{Q}(-2,-3)\cong I_{C,Q}}\rightarrow{I_{Z,Q}}\rightarrow{I_{Z,C}}\rightarrow 0$, we obtain $G=I_{Z,Q}(2,3)$. If $(s,F)\in\mathbf{M}_{0}^{+}$, then we have an element $(s^{*},G)\in\mathbf{P}$ because $G$ has a resolution of the form $\mathcal{O}_{Q}(0,1)\to\mathcal{O}_{Q}(1,2)^{\oplus 2}$. However, if $(s,F)\in\mathbf{M}_{1}^{+}$, then we have $0\rightarrow{I_{\ell_{Z},Q}(2,3)}\rightarrow{G=I_{Z,Q}(2,3)}\rightarrow{I_{Z,\ell_{Z}}(2,3)}\rightarrow 0$ and $I_{\ell_{Z},Q}(2,3)=\mathcal{O}_{Q}(1,3)$, $I_{Z,\ell_{Z}}(2,3)=\mathcal{O}_{\ell_{Z}}(1)$. In particular, $\mathrm{Hom}(\mathcal{O}_{Q}(1,3),G)\neq 0$ and $G$ does not admit a resolution $\mathcal{O}_{Q}(0,1)\to\mathcal{O}_{Q}(1,2)^{\oplus 2}$. So $G\notin\mathbf{G}$.

Suppose that $(s,F)\in\mathbf{M}_{3}^{+}\setminus\mathbf{M}_{2}^{+}$. Then $F$ fits into a non-split extension $0\rightarrow{\mathcal{O}_{E}}\rightarrow{F}\rightarrow{\mathcal{O}_{\ell}}\rightarrow 0$. Apply $\mathcal{H}om(-,\omega_{Q})$, then we have $0\rightarrow{\mathcal{O}_{\ell}(0,1)}\rightarrow{F^{D}(2,2)}\rightarrow{\mathcal{O}_{E}(2,2)}\rightarrow 0$. Since $\mathrm{Ext}^{1}(\mathcal{O}_{E}(2,2),\mathcal{O}_{Q})\cong\mathrm{Ext}^{1}(F^{D}(2,2),\mathcal{O}_{Q})\cong\mathbb{C}$, the sheaf $G$ is given by the pull-back:

[TABLE]

By applying the snake lemma to (4), we conclude that the unique non-split extension $G$ lies on $0\rightarrow{\mathcal{O}_{\ell}(0,1)}\rightarrow{G}\rightarrow{\mathcal{O}_{Q}(2,2)}\rightarrow 0$. Hence $G\in\mathbf{G}$ (Proposition 2.1) and we have an element $(s^{*},G)\in\mathbf{P}$.

Now suppose that $(s,F)\in\mathbf{M}_{2}^{+}$, so $F=\mathcal{O}_{C}(0,1)$. Then $F^{D}(2,2)=\mathcal{O}_{C}(2,2)$. So we have $0\rightarrow{(s^{*})\otimes\mathcal{O}_{Q}}\rightarrow{G}\rightarrow{\mathcal{O}_{C}(2,2)}\rightarrow 0$. By the snake lemma (Consult the proof of [CM15, Lemma 3.7].), $G$ fits into $0\rightarrow{\mathcal{O}_{Q}(2,2)}\rightarrow{G}\rightarrow{\mathcal{O}_{\ell}}\rightarrow 0$ where $\ell$ is the line of type $(0,1)$ determined by the section $s$. So $\mathrm{Hom}(\mathcal{O}_{Q}(2,2),G)\neq 0$ and this implies $G$ does not admit a resolution $\mathcal{O}_{Q}(0,1)\to\mathcal{O}_{Q}(1,2)^{\oplus 2}$. Thus the correspondence is not well-defined on $\mathbf{M}_{2}^{+}$. ∎

3.2. The first elementary modification and the extension of the domain

We can extend the morphism in Lemma 3.3 by applying an elementary modification of pairs ([CC16, Section 2.2]) on $\mathbf{M}_{2}^{+}$.

Lemma 3.4**.**

There exists an exact sequence of pairs $0\rightarrow{(0,K)}\rightarrow{(\mathcal{L}^{*}|_{\mathbf{M}_{2}^{+}},\mathcal{E}|_{\mathbf{M}_{2}^{+}\times Q})}\rightarrow{(\mathcal{L}^{{}^{\prime\prime}},\mathcal{O}_{\mathcal{Z}})}\rightarrow 0$ where $\mathcal{Z}$ is the pull-back of the universal family of $(0,1)$-lines to $\mathbf{M}_{2}^{+}\times Q$ and $K_{\{m\}\times Q}\cong\mathcal{O}_{Q}(2,2)$ for $m=[(s,F)]\in\mathbf{M}_{2}^{+}$.

Proof.

The last part of the proof of Lemma 3.3 tells us that there is an exact sequence of sheaves $0\rightarrow{K}\rightarrow{\mathcal{E}|_{\mathbf{M}_{2}^{+}\times Q}}\rightarrow{\mathcal{O}_{\mathcal{Z}}}\rightarrow 0$. Now it is sufficient to show that for each fiber $G=\mathcal{E}|_{\{(s,F)\}\times Q}$, the section $s^{*}$ of $G$ does not come from $\mathrm{H}^{0}(\mathcal{O}_{Q}(2,2))$. If it is, we have an injection $\mathcal{O}_{Q}\subset\mathcal{O}_{Q}(2,2)$ whose cokernel is $\mathcal{O}_{E}(2,2)$ for some elliptic curve $E$. By the snake lemma once again, we obtain $0\rightarrow{\mathcal{O}_{E}(2,2)}\rightarrow{F^{D}(2,2)=\mathcal{O}_{C}(2,2)}\rightarrow{\mathcal{O}_{\ell}}\rightarrow 0$. It violates the stability of $F^{D}(2,2)$. ∎

Let $(\mathcal{L}^{\prime},\mathcal{E}^{\prime})$ be the elementary modification of $(\mathcal{L}^{*},\mathcal{E})$ along $\mathbf{M}_{2}^{+}$, that is,

[TABLE]

Lemma 3.5**.**

For a point $m=[(s,F=O_{C}(0,1))]\in\mathbf{M}_{2}^{+}$, the modified pair $(\mathcal{L}^{\prime},\mathcal{E}^{\prime})|_{\{m\}\times Q}$ fits into a non-split exact sequence $0\rightarrow{(s^{\prime},\mathcal{O}_{\ell})}\rightarrow{(s^{\prime},\mathcal{E}^{\prime}|_{\{m\}\times Q})}\rightarrow{(0,\mathcal{O}_{Q}(2,2))}\rightarrow 0$ where $\ell$ is a $(0,1)$-line.

Proof.

An elementary modification of pairs interchanges the sub pair with the quotient pair ([He98, Lemma 4.24]). Thus we obtain the sequence. It remains to show that the sequence is non-split. We will show that the normal bundle $\mathcal{N}_{\mathbf{M}_{2}^{+}/\mathbf{M}^{+}}$ at $m$ is canonically isomorphic to $\mathrm{H}^{0}(\mathcal{O}_{\ell})^{*}$. Then the element $m$ corresponds to the projective equivalent class of nonzero elements in $\mathrm{H}^{0}(\mathcal{O}_{\ell})^{*}\cong\mathrm{Ext}^{1}((0,\mathcal{O}_{Q}(2,2)),(s^{\prime},\mathcal{O}_{\ell}))$, so it is non-split.

The pair $(s,F)$ fits into $0\rightarrow{(0,\mathcal{O}_{Q}(-2,-2))}\rightarrow{(s,\mathcal{O}_{Q}(0,1))}\rightarrow{(s,F)}\rightarrow 0$. Thus we have

[TABLE]

The first term $\mathrm{Ext}^{0}((0,\mathcal{O}_{Q}(-2,-2)),(s,F))\cong\mathrm{H}^{0}(\mathcal{O}_{C}(2,3))\cong\mathbb{C}^{11}$ is the deformation space of curves $C$ on $Q$. The second term $\mathrm{Ext}^{1}((s,F),(s,F))$ is $\mathcal{T}_{m}\mathbf{M}^{+}$ ([He98, Theorem 3.12]). For the third term, by [He98, Theorem 3.12], we have

[TABLE]

The first term $\mathrm{Hom}(s,\mathrm{H}^{0}(F)/\langle s\rangle)=\mathbb{C}$ is the deformation space of the line $\ell$ in $Q$ determined by the section $s$. By Serre duality, $\phi:\mathrm{H}^{0}(\mathcal{O}_{Q}(0,1))^{*}\to\mathrm{H}^{0}(\mathcal{O}_{Q})^{*}$ and the kernel is $\mathrm{H}^{0}(\mathcal{O}_{\ell}(0,1))^{*}\cong\mathrm{H}^{0}(\mathcal{O}_{\ell})^{*}$. This proves our assertion. ∎

Recall that the modified pair $(\mathcal{L}^{\prime},\mathcal{E}^{\prime})$ provides a natural surjection $\mathcal{E}xt_{\pi}^{2}(\mathcal{E}^{\prime},\omega_{\pi})\twoheadrightarrow\mathcal{L}^{\prime}$ on $\mathbf{M}^{+}\times Q$. It is straightforward to check that $\mathcal{E}xt_{\pi}^{2}(\mathcal{E}^{\prime},\omega_{\pi})$ has rank $10$ at each fiber, thus it is locally free.

Proof of Proposition 3.2.

We claim that there exists a surjection $w^{*}\mathcal{U}^{*}\rightarrow\mathcal{L}^{\prime}\rightarrow 0$ up to a twisting by a line bundle on $\mathbf{M}^{+}\setminus\mathbf{M}_{1}^{+}$. Then there is a morphism $\mathbf{M}^{+}\setminus\mathbf{M}_{1}^{+}\rightarrow\mathbf{P}$.

Consider the following commutative diagram

[TABLE]

Note that $\mathcal{U}=\pi_{*}(\mathcal{W})$ where $\mathcal{W}=\mathrm{coker}(\phi)$ is the universal quotient on $\mathbf{G}\times Q$ (Section 2.1). One can check that $\mathcal{W}$ is flat over $\mathbf{G}$. By its construction of $w$, $\mathcal{E}^{\prime}|_{\{m\}\times Q}\cong{w^{\prime}}^{*}\mathcal{W}|_{\{m\}\times Q}$ restricted to each point $m\in\mathbf{M}^{+}\setminus\mathbf{M}_{1}^{+}$. The universal property of $\mathbf{G}$ (as a quiver representation space [Kin94, Proposition 5.6]) tells us that ${w^{\prime}}^{*}\mathcal{W}\cong\mathcal{E}^{\prime}$ up to a twisting by a line bundle on $\mathbf{M}^{+}\setminus\mathbf{M}_{1}^{+}$. The base change property implies that there exists a natural isomorphism (up to a twisting by a line bundle) $w^{*}\mathcal{U}=w^{*}(\pi_{*}\mathcal{W})\cong\pi_{*}({w^{\prime}}^{*}\mathcal{W})=\pi_{*}\mathcal{E}^{\prime}\cong\mathcal{E}xt_{\pi}^{2}(\mathcal{E}^{\prime},\omega_{\pi})^{*}$ by [LP93, Corollary 8.19]. Hence we have $w^{*}\mathcal{U}^{*}\cong(w^{*}\mathcal{U})^{*}\cong(\pi_{*}(\mathcal{E}^{\prime}))^{*}\cong\mathcal{E}xt_{\pi}^{2}(\mathcal{E}^{\prime},\omega_{\pi})\twoheadrightarrow\mathcal{L}^{\prime}$. Therefore we obtain a morphism $q:\mathbf{M}^{+}\setminus\mathbf{M}_{1}^{+}\to\mathbf{P}$.

By the proof of Lemma 3.5, the modified pair does not depend on the choice of a $(2,3)$-curve, so $q:\mathbf{M}^{+}\setminus\mathbf{M}_{1}^{+}\to\mathbf{P}\setminus p^{-1}(t(Y_{10}))$ is indeed a contraction of $\mathbf{M}_{2}^{+}$ and the image of $\mathbf{M}_{2}^{+}$ is $Y_{01}$. Recall that the exceptional divisor $\mathbf{M}_{2}^{+}$ is $|\mathcal{O}_{Q}(2,3)|\times|\mathcal{O}_{Q}(0,1)|\cong\mathbb{P}^{11}\times\mathbb{P}^{1}$. Note that the sheaf $F$ in the pair $(s,F)\in\mathbf{M}_{2}^{+}$ is parametrized by $\mathbb{P}^{11}=|\mathcal{O}_{Q}(2,3)|=\mathbb{P}\mathrm{Ext}^{1}(\mathcal{O}_{Q}(-2,-2)[1],\mathcal{O}_{Q}(0,1))$. It follows also from the fact that each $F$ fits into a triangle $0\rightarrow{\mathcal{O}_{Q}(0,1)}\rightarrow{F}\rightarrow{\mathcal{O}_{Q}(-2,-2)[1]}\rightarrow 0$. By analyzing $T_{F}\mathbf{M}=\mathrm{Ext}^{1}(F,F)$ (which is similar to [CC16, Lemma 3.4]), one can see that $\mathcal{N}_{\mathbf{M}_{2}/\mathbf{M}}|_{\mathbb{P}^{11}}\cong\mathrm{Ext}^{1}(\mathcal{O}_{Q}(0,1),\mathcal{O}_{Q}(-2,-2)[1])\otimes\mathcal{O}_{\mathbb{P}^{11}}(-1)\cong\mathrm{H}^{0}(\mathcal{O}_{Q}(0,1))^{*}\otimes\mathcal{O}_{\mathbb{P}^{11}}(-1)$. Thus $\mathcal{N}_{\mathbf{M}_{2}^{+}/\mathbf{M}^{+}}\cong\mathcal{O}_{\mathbb{P}^{11}\times\mathbb{P}^{1}}(-1,-1)$ and $q$ is a smooth blow-down by Fujiki-Nakano criterion. ∎

Thus we have two different contractions of $\mathbf{M}^{+}$, one is $\mathbf{M}$ obtained by contracting all $\mathbb{P}^{1}$-fibers on $\mathbf{M}_{2}^{+}$, and the other is:

Definition 3.6**.**

Let $\mathbf{M}^{-}$ be the contraction of $\mathbf{M}^{+}$ which is obtained by contracting all $\mathbb{P}^{11}$-fibers on $\mathbf{M}_{2}^{+}$. We define $\mathbf{M}_{i}^{-}$ as the image of $\mathbf{M}_{i}^{+}$ for the contraction $\mathbf{M}^{+}\to\mathbf{M}^{-}$.

3.3. The second elementary modification and $\mathbf{M}^{-}$

Recall that $u:\mathbf{G}_{1}\rightarrow\mathbf{G}$ is the blow-up of $\mathbf{G}$ along the $\mathbb{P}^{1}$ parameterizing $(1,0)$-lines in $Q$, and $Y_{10}$ is the exceptional divisor. Let $\mathcal{W}$ be the cokernel of the universal morphism $\phi$ on $\mathbf{G}\times Q$ in Section 2.1. Let $\mathcal{V}:=(u\times\mathrm{id})^{*}\mathcal{W}$ be the pull-back of $\mathcal{W}$ along the map $u\times\mathrm{id}:\mathbf{G}_{1}\times Q\rightarrow\mathbf{G}\times Q$. Then for $([\ell],t)\in Y_{10}$, $\mathcal{V}|_{([\ell],t)\times Q}$ fits into a non-split exact sequence $0\rightarrow{\mathcal{O}_{\ell}(1)}\rightarrow{\mathcal{V}|_{([\ell],t)\times Q}}\rightarrow{\mathcal{O}_{Q}(1,3)}\rightarrow 0$. By relativizing it over $Y_{10}\times Q$, we obtain $0\rightarrow{\mathcal{S}}\rightarrow{\mathcal{V}|_{Y_{10}\times Q}}\rightarrow{\mathcal{Q}}\rightarrow 0$. Let $\mathcal{V}^{-}$ be the elementary modification $\mathrm{elm}_{Y_{10}\times Q}(\mathcal{V},\mathcal{Q}):=\mathrm{ker}(\mathcal{V}\twoheadrightarrow\mathcal{V}|_{Y_{10}\times Q}\twoheadrightarrow\mathcal{Q})$ along $Y_{10}\times Q$. Note that over $([\ell],t)\in\mathbf{G}_{1}$, $\mathcal{V}^{-}|_{([\ell],t)\times Q}$ fits into a non-split exact sequence $0\rightarrow{\mathcal{O}_{Q}(1,3)}\rightarrow{\mathcal{V}^{-}|_{([\ell],t)\times Q}}\rightarrow{\mathcal{O}_{\ell}(1)}\rightarrow 0$ because the elementary modification interchanges the sub/quotient sheaves. Let $\pi_{1}:\mathbf{G}_{1}\times Q\rightarrow\mathbf{G}_{1}$ be the projection into the first factor. Then $\mathcal{U}^{-}:=\pi_{1*}\mathcal{V}^{-}$ is a rank $10$ bundle over $\mathbf{G}_{1}$. Let $\mathbf{P}^{-}:=\mathbb{P}(\mathcal{U}^{-})$.

The following proposition completes the proof of Theorem 1.3.

Proposition 3.7**.**

The projective bundle $\mathbf{P}^{-}$ is isomorphic to $\mathbf{M}^{-}$ in Definition 3.6.

Proof.

Since the elementary modification has been done locally around $Y_{10}\times Q$, $\mathbb{P}(u^{*}\mathcal{U})$ and $\mathbf{P}^{-}$ are isomorphic over $\mathbf{G}_{1}\setminus Y_{10}$. On the other hand, set theoretically, it is straightforward to see that the image of $q$ is $\mathbf{P}\setminus p^{-1}(t(Y_{10}))$, where $p:\mathbf{P}\to\mathbf{G}$ is the structure morphism. So we have a birational morphism $\mathbf{M}^{+}\setminus\mathbf{M}_{1}^{+}\to\mathbf{P}\setminus p^{-1}(t(Y_{10}))\cong\mathbb{P}(u^{*}\mathcal{U})\setminus p^{-1}(Y_{10})\cong\mathbf{P}^{-}\setminus p^{-1}(Y_{10})$ (here we used the same notation $p$ for the projections $\mathbb{P}(u^{*}\mathcal{U})\to\mathbf{G}_{1}$ and $\mathbf{P}^{-}\to\mathbf{G}_{1}$). By Proposition 3.2, this map is a blow-down along $\mathbf{M}_{2}^{+}$, thus we have an isomorphism $\tau:\mathbf{P}^{-}\setminus p^{-1}(Y_{10})\to\mathbf{M}^{-}\setminus\mathbf{M}_{1}^{-}$. So we have a birational map $\tau:\mathbf{P}^{-}\dashrightarrow\mathbf{M}^{-}$, where its undefined locus is $p^{-1}(Y_{10})$.

On the other hand, since the flipped locus for $\mathbf{M}^{\infty}\dashrightarrow\mathbf{M}^{+}$ is $\mathbf{M}_{3}^{+}$, we have an isomorphism $\mathbf{M}^{-}\setminus(\mathbf{M}_{2}^{-}\cup\mathbf{M}_{3}^{-})\cong\mathbf{M}^{+}\setminus(\mathbf{M}_{2}^{+}\cup\mathbf{M}_{3}^{+})\cong\mathbf{M}^{\infty}\setminus(\mathbf{M}_{2}^{\infty}\cup\mathbf{M}_{3}^{\infty})$ (Here $\mathbf{M}_{i}^{\infty}$ is defined in an obvious way.). Also $\tau^{-1}(\mathbf{M}_{2}^{-}\cup\mathbf{M}_{3}^{-})=p^{-1}(Y_{01})$. Hence if we restrict the domain of $\tau$, then we have $\sigma:\mathbf{P}^{-}\setminus p^{-1}(Y_{01})\dashrightarrow\mathbf{M}^{-}\setminus(\mathbf{M}_{2}^{-}\cup\mathbf{M}_{3}^{-})\cong\mathbf{M}^{\infty}\setminus(\mathbf{M}_{2}^{\infty}\cup\mathbf{M}_{3}^{\infty})$ whose undefined locus is $p^{-1}(Y_{10})$. Therefore $\sigma$ can be regarded as a map into a relative Hilbert scheme. Note that $\mathbf{M}_{2}^{\infty}\cup\mathbf{M}_{3}^{\infty}$ is the locus of $(2,3)$-curves passing through two points lying on a $(0,1)$-line.

We claim that $\sigma$ is extended to a morphism $\tilde{\sigma}:\mathbf{P}^{-}\setminus p^{-1}(Y_{01})\to\mathbf{M}^{-}$ such that $\tilde{\sigma}(p^{-1}(Y_{10}))=\mathbf{M}_{1}^{-}\cong\mathbf{M}_{1}^{\infty}$. To show this, it is enough to check that $\mathcal{V}^{-}$ over $Y_{10}$ provides a flat family of the twisted ideal sheaf of Hilbert scheme of two points lying on $(1,0)$-type lines. Note that $\mathcal{V}^{-}$ fits into a non-split extension $0\rightarrow{\mathcal{O}_{Q}(1,3)}\rightarrow{\mathcal{V}^{-}|_{([\ell],t)\times Q}}\rightarrow{\mathcal{O}_{\ell}(1)}\rightarrow 0$. By a diagram chasing similar to the second paragraph of the proof of Lemma 3.3, one can check that $\mathcal{V}^{-}|_{([\ell],t)\times Q}\cong I_{Z,Q}(2,3)$ where $Z\subset\ell$ and $\ell$ is a $(1,0)$-line.

Now two maps $\tau$ and $\tilde{\sigma}$ coincide over the intersection $\mathbf{P}^{-}\setminus p^{-1}(Y_{10}\cup Y_{01})$ of domains, so we have a birational morphism $\mathbf{P}^{-}\to\mathbf{M}^{-}$. Since $\rho(\mathbf{P}^{-})=3=\rho(\mathbf{M}^{-})$ and both of them are smooth, this map is an isomorphism. ∎

The modification on $\mathbf{G}_{1}\times Q$ descends to $\mathbf{G}_{1}$. Then Proposition 1.2 follows from a general result of Maruyama ([Mar73]).

Proof of Proposition 1.2.

Let $\pi_{1}:\mathbf{G}_{1}\times Q\to\mathbf{G}_{1}$ be the projection. We claim that $\mathcal{U}^{-}=\mathrm{elm}_{Y_{10}}(u^{*}\mathcal{U},\pi_{1*}\mathcal{Q})\cong\pi_{1*}\mathrm{elm}_{Y_{10}\times Q}(\mathcal{V},\mathcal{Q})$. Indeed, from $0\rightarrow{\mathcal{V}^{-}}\rightarrow{\mathcal{V}}\rightarrow{\mathcal{Q}}\rightarrow 0$, we have $0\to\pi_{1*}\mathcal{V}^{-}\to\pi_{1*}\mathcal{V}=u^{*}\mathcal{U}\to\pi_{1*}\mathcal{Q}\to R^{1}\pi_{1*}\mathcal{V}^{-}\to R^{1}\pi_{1*}\mathcal{V}$. It is sufficient to show that $R^{1}\pi_{1*}\mathcal{V}^{-}=0$. By using the resolution of $\mathcal{V}$ given by the universal morphism $\phi$, we have $R^{1}\pi_{1*}\mathcal{V}=0$. Over $\mathbf{G}_{1}\setminus Y_{10}$, the last two terms are isomorphic. Over $Y_{10}$, from $\mathrm{H}^{1}(\mathcal{O}_{Q}(1,3))=\mathrm{H}^{1}(\mathcal{O}_{\ell}(1))=0$ and the description of $\mathcal{V}^{-}|_{([\ell],t)}$, we obtain $R^{1}\pi_{1*}\mathcal{V}^{-}=0$.

Note that $u^{*}\mathcal{U}|_{Y_{10}}$ fits into a vector bundle sequence $0\rightarrow{\pi_{1*}\mathcal{S}}\rightarrow{u^{*}\mathcal{U}|_{Y_{10}}}\rightarrow{\pi_{1*}\mathcal{Q}}\rightarrow 0$ and $\mathrm{rank}\;\pi_{1*}\mathcal{S}=2$ and $\mathrm{rank}\;\pi_{1*}\mathcal{Q}=8$. The result follows from [Mar73, Theorem 1.3]. ∎

As a direct application of Theorem 1.3, we compute the Poincaré polynomial of $\mathbf{M}$ which matches with the result in [Mai16, Theorem 1.2].

Corollary 3.8**.**

1. (1)

The moduli space $\mathbf{M}$ is rational; 2. (2)

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

[TABLE]

Proof.

Now $\mathbf{M}$ is birational to a $\mathbb{P}^{9}$-bundle over $\mathbf{G}$, so we obtain Item (1). Item (2) is a straightforward calculation using

[TABLE]

∎