Automorphisms of relative Quot Schemes

TL;DR

This paper computes the identity component of the automorphism group scheme of relative Quot schemes parameterizing torsion quotients of vector bundles over families of smooth projective curves, over an algebraically closed field of characteristic zero.

Contribution

It provides a detailed computation of the automorphism group scheme of relative Quot schemes in a geometric setting involving families of curves and vector bundles.

Findings

Determined the identity component of the automorphism group scheme.

Characterized automorphisms of relative Quot schemes over families of curves.

Extended understanding of symmetries in moduli spaces of sheaves.

Abstract

Let $k$ be an algebraically closed field of characteristic zero. Let $S$ be a smooth projective variety over $k$ and let $p_S:X\rightarrow S$ be a family of smooth projective curves over $S$. Let $E$ be a vector bundle over $X$. For $s\in S$ let $X_s$ be the fibre of $p_S$ over $s$ and let $E_s$ be the restriction of $E$ to $X_s$. Fix $d\geq 1$. Let $\mathcal Q(E,d)\to S$ be the relative Quot scheme parameterizing torsion quotients of $E_s$ over $X_s$ of degree $d$ for all $s\in S$. In this article we compute the identity component of relative automorphism group scheme which parameterizes automorphisms of $\mathcal Q(E,d)$ over $S$.

Equations

$$X{\times }_S\mathcal{Q}(E,d)$$

$${\pi }_{\mathcal{Q}}^{\ast }E\to \mathcal{B}\to 0.$$

$${\mathrm{Aut}}^o(\mathbb{P}(E)/S)\hookrightarrow {\mathrm{Aut}}^o(\mathcal{Q}(E,d)/S).$$

$$\mathbb{P}(E)\cong \mathbb{P}((g_1{)}^{\ast }E)$$

$${\pi }_{\mathcal{Q}}^{\ast }E\cong {\pi }_{\mathcal{Q}}^{\ast }{h}^{\ast }E\otimes {\pi }_{\mathcal{Q}}^{\ast }L\to (h\times id{)}^{\ast }\mathcal{B}\otimes {\pi }_{\mathcal{Q}}^{\ast }L$$

$${\text{Aut}}^o(\mathbb{P}(E)/S)\to {\text{Aut}}^o(\mathcal{Q}(E,d)/S).$$

$$[E_s\to B_d\to 0]\to [E_s\stackrel{{\Psi }_{g,s}}{\to }{h}^{\ast }(E_s\to B_d\to 0)\otimes L_s]$$

$$E_s\to {\mathcal{O}}_{X_s,x}/{\mathfrak{m}}_{X_s,x}^d$$

$$E_s\to {\mathcal{O}}_{X_s,h(x)}/{\mathfrak{m}}_{X_s,h(x)}^d$$

$$E_s\to E{|}_x\oplus \underset{i=1}{\overset{d-1}{⨁}}E{|}_{x_i}\to k_x\oplus \underset{i=1}{\overset{d-1}{⨁}}{k}_{x_i}.$$

$$H^0(\mathbb{P}(E),{\mathcal{T}}_{\mathbb{P}(E)/S})\hookrightarrow H^0(\mathcal{Q}(E,d),{\mathcal{T}}_{\mathcal{Q}(E,d)/S}).$$

$$\mathcal{U}:=\mathcal{Z}\setminus (\bigcup _{i,j}{\Delta }_{i,j}\bigcup \bigcup _{i,j,k}{\Delta }_{i,j,k,X}).$$

$$q:{\pi }_1^{\ast }E\to \underset{i=1}{\overset{d}{⨁}}{\pi }_1^{\ast }E{|}_{{\Delta }_i}\cong \underset{i=1}{\overset{d}{⨁}}{\pi }_{2,i}^{\ast }E{|}_{{\Delta }_i}\to \underset{i=1}{\overset{d}{⨁}}{\pi }_{2,i}^{\ast }\mathcal{O}(1){|}_{{\Delta }_i}.$$

$$E\to \underset{i=1}{\overset{d}{⨁}}E{|}_{p\circ p_i(u)}\to \underset{i=1}{\overset{d}{⨁}}{k}_{p\circ p_i(u)}$$

$$\Phi :\mathcal{U}\to \mathcal{Q}(E,d).$$

$$H^0(\mathcal{U},{\Phi }^{\ast }{\mathcal{T}}_{\mathcal{Q}(E,d)/S})=\underset{i=1}{\overset{d}{⨁}}{H}^0(\mathbb{P}(E),{\mathcal{T}}_{\mathbb{P}(E)/S}).$$

$${\Phi }^{\ast }{\mathcal{T}}_{\mathcal{Q}(E,d)/S}\cong \underset{i=1}{\overset{d}{⨁}}({\pi }_2{)}_{\ast }\mathcal{H}om(\mathcal{F}(E,d),{\pi }_{2,i}^{\ast }\mathcal{O}(1){|}_{{\Delta }_i}){|}_{\mathcal{U}}.$$

$$0\to \mathcal{A}(E,d)\to {\pi }_X^{\ast }E\to \mathcal{B}(E,d)\to 0.$$

$${\mathcal{T}}_{\mathcal{Q}(E,d)/S}=(p_{\mathcal{Q}}{)}_{\ast }\mathcal{H}om(\mathcal{A}(E,d),\mathcal{B}(E,d)).$$

$$X{\times }_S\mathcal{U}$$

$${\Phi }^{\ast }{\mathcal{T}}_{\mathcal{Q}(E,d)/S}=$$

$$\cong$$

$$(i{d}_X{\times }_S\Phi {)}^{\ast }\mathcal{H}om(\mathcal{A}(E,d),\mathcal{B}(E,d))$$

$$=\mathcal{H}om((i{d}_X{\times }_S\Phi {)}^{\ast }\mathcal{A}(E,d),(i{d}_X{\times }_S\Phi {)}^{\ast }\mathcal{B}(E,d)).$$

$$(i{d}_X{\times }_S\Phi {)}^{\ast }\mathcal{B}(E,d)\cong (\underset{i=1}{\overset{d}{⨁}}{\pi }_{2,i}^{\ast }\mathcal{O}(1){|}_{{\Delta }_i}){|}_{X{\times }_S\mathcal{U}}.$$

$${\Phi }^{\ast }\mathcal{A}(E,d)\cong \mathcal{F}(E,d){|}_{X{\times }_S\mathcal{U}}.$$

$$\mathcal{H}om({\pi }_{2,i}^{\ast }\mathcal{O}(1){|}_{{\Delta }_i},{\pi }_{2,j}^{\ast }\mathcal{O}(1){|}_{{\Delta }_j})=0.$$

$$\mathcal{H}om({\pi }_{2,i}^{\ast }\mathcal{O}(1){|}_{{\Delta }_i},{\pi }_{2,j}^{\ast }\mathcal{O}(1){|}_{{\Delta }_j})=\mathcal{H}om({\pi }_{2,i}^{\ast }\mathcal{O}(1){|}_{{\Delta }_i\cap {\Delta }_j},{\pi }_{2,j}^{\ast }\mathcal{O}(1){|}_{{\Delta }_j})$$

$$H^0(X{\times }_S\mathcal{U},\mathcal{H}om(({\pi }_1\times {\pi }_{2,j}{)}^{\ast }\mathcal{F}(E,1),{\pi }_{2,j}^{\ast }\mathcal{O}(1){|}_{{\Delta }_j}))=H^0(\mathbb{P}(E),{\mathcal{T}}_{\mathbb{P}(E)/S}).$$

$$H^0(X{\times }_S\mathcal{U},\mathcal{H}om(({\pi }_1\times {\pi }_{2,j}{)}^{\ast }\mathcal{F}(E,1),{\pi }_{2,j}^{\ast }\mathcal{O}(1){|}_{{\Delta }_j}))$$

$$=H^0(\mathcal{U},\mathcal{H}om(p_j^{\ast }\mathcal{F}(E,1){|}_{{\Delta }_1},p_j^{\ast }\mathcal{O}(1)))$$

$$=H^0(\mathcal{U},p_j^{\ast }(\mathcal{F}(E,1{)}^{\vee }{|}_{{\Delta }_1}\otimes \mathcal{O}(1))).$$

Full text

1. Introduction

Let $k$ be an algebraically closed field of characteristic zero. Let $Y\rightarrow S$ be a smooth morphism between two projective varieties over $k$. Associated to this morphism we have the automorphism group scheme ${\rm Aut}(Y/S)$ which parameterizes automorphisms of $Y$ over $S$. Let us denote identity component of ${\rm Aut}(Y/S)$ by ${\rm Aut}^{o}(Y/S)$. It is known that ${\rm Aut}^{o}(Y/S)$ is an algebraic group and if $\mathcal{T}_{Y/S}$ is the relative tangent bundle, then ${\rm Lie}({\rm Aut}^{o}(Y/S))=H^{0}(Y,\mathcal{T}_{Y/S})$ [MO67, Theorem 3.7], [Bri18, Theorem 2.3]. We refer to [Bri14], [Bri18] for other properties of this group scheme.

We refer to [HL10, Section 2] for definitions and properties of Quot Schemes in general. The Quot Scheme which we will study in this article can be defined in the following manner. Let $p_{S}:X\rightarrow S$ be a family of smooth projective curves over an algebraically closed field $k$ of characteristic zero. Assume $X$ and $S$ are smooth projective varieties. Let $E$ be a vector bundle over $X$ of rank $r$. For a closed point $s\in S$ let $X_{s}$ be the fibre of $p_{S}$ over $s$ and let $E_{s}$ be the restriction of $E$ to $X_{s}$. Fix $d\geq 1$. Then associated to the morphism $p_{S}$ and the vector bundle $E$ we have the relative Quot scheme $\pi_{S}:\mathcal{Q}(E,d)\rightarrow S$ whose closed points correspond to quotients $E_{s}\rightarrow B_{d}$, $\forall s\in S$ where $B_{d}$ is a torsion sheaf of degree $d$ over the smooth projective curve $X_{s}$ [HL10, Theorem 2.2.4]. It is known that $\mathcal{Q}(E,d)$ is a smooth projective variety [HL10, Proposition 2.2.8]. These schemes have been studied extensively. We refer the reader to [BGL94], [BDW96], [BDH15] for other properties of this scheme. In this article we compute the group scheme ${\rm Aut}^{o}(\mathcal{Q}(E,d)/S)$. We recall that in the case when $S$ is a point and $E$ the trivial bundle of rank $r$ this group scheme was computed in [BDH15]. In [BM16] this group scheme was computed in another special case. We refer to Corollary 3.2 and Corollary 3.5 where these results are stated explicitly.

Over $X$ we fix a certain ample line bundle $\mathcal{M}$ (this line bundle is defined just before Lemma 2.12). Then the main theorem of this article is

Theorem** (Theorem 2.15).**

Suppose either $r:={\rm rank}~{}E\geq 3$ or $r=2$, $E$ is semistable with respect to $\mathcal{M}$ and genus of $X_{s}$ is $\geq 2$ for $s\in S$. In both of these cases we have isomorphisms

1. (1)

${\rm Aut}^{o}(\mathcal{Q}(E,d))\cong{\rm Aut}^{o}(\mathbb{P}(E)/S)\,.$** 2. (2)

$H^{0}(\mathbb{P}(E),\mathcal{T}_{\mathbb{P}(E)/S})\cong H^{0}(\mathcal{Q}(E,d),\mathcal{T}_{\mathcal{Q}(E,d)/S})\,.$**

As consequences of Theorem 2.15 we deduce the results of [BDH15] and [BM16] as Corollary 3.2 and Corollary 3.5. We also compute the identity component of the automorphism group scheme of the flag scheme parameterizing chains of torsion quotients of trivial bundle over a smooth projective curve(Corollary 3.4). We refer to Section 3 for more details.

2. Main Theorem

Let us denote the projection $X\times_{S}\mathcal{Q}(E,d)\rightarrow X$ by $\pi_{X}$ and the projection $X\times_{S}\mathcal{Q}(E,d)\rightarrow\mathcal{Q}(E,d)$ by $p_{\mathcal{Q}}$ i.e. we have the following diagram:

[TABLE]

We denote the universal quotient on $X\times_{S}\mathcal{Q}(E,d)$ by

[TABLE]

Lemma 2.1**.**

We have a closed immersion of algebraic groups

[TABLE]

Proof.

By [MO67, Corollary 2.2] any automorphism $g\in\text{Aut}^{o}(\mathbb{P}(E)/S)$ descends to an automorphism $h\in\text{Aut}^{o}(X/S)$. Therefore we have the following diagram:

[TABLE]

Then $E\cong h^{*}E\otimes p^{*}L$ for some line bundle $L$ on $X$. Let us denote this isomorphism of bundles by $\Psi_{g}$. On $X\times_{S}\mathcal{Q}(E,d)$ consider the quotient

[TABLE]

where the first isomorphism is induced from $\Psi_{g}$ and the second morphism is the pullback of the universal quotient $\pi^{*}_{\mathcal{Q}}E\rightarrow\mathcal{B}\rightarrow 0$ under the map $(h\times{id})$ tensored with $\pi^{*}_{\mathcal{Q}}L$. This gives a quotient of $\pi^{*}_{\mathcal{Q}}E$ over $X\times_{S}\mathcal{Q}(E,d)$ and by the universal property of Quot schemes this induces an automorphism of $\mathcal{Q}(E,d)$. Hence, we have a homomorphism

[TABLE]

Next we show that this homomorphism is injective. At the level of closed points the above automorphism of $\mathcal{Q}(E,d)$ induced by $g\in{\rm Aut}^{o}(\mathbb{P}(E)/S)$ is given by

[TABLE]

Suppose $g\in\text{Aut}^{o}(\mathbb{P}(E)/S)$ induces the identity automorphism on $\mathcal{Q}(E,d)$. We will show that $g=\text{id}$. First we show that $h={\rm id}$. Fix $x\in X$ and let $p_{S}(x)=s\in S$. Consider any quotient

[TABLE]

where $\mathcal{O}_{X_{s},x}$ is the local ring of $X_{s}$ at $x$ and $\mathfrak{m}_{X_{s},x}$ is its maximal ideal. Then under the automorphism induced by $g$ the image of this quotient is of the form

[TABLE]

Hence if $g$ induces the identity automorphism of $\mathcal{Q}(E,d)$ then $h={\rm id}$. Next we show that $g=\text{id}$. Let $v\in\mathbb{P}(E)$, let $p(v)=x$ and $p_{S}(x)=s$. Then $v$ corresponds to a quotient of vector spaces $E|_{x}\xrightarrow{v}k$. Since, $h=\text{id}$, $p(g(v))=x$. is a quotient of the form $E|_{x}\rightarrow k$. Let us fix $d-1$ degree $1$ quotients $E\xrightarrow{v_{i}}k_{x_{i}}$ for $1\leq i\leq d-1$ such that all $x_{i},x$ are distinct and $p_{S}(x_{i})=p_{S}(x)=s$. Then the summation of these quotients gives us a point in $\mathcal{Q}(E,d)$

[TABLE]

Note that each of the quotients $E|_{x}\xrightarrow{h}k$ and $E|_{x_{i}}\rightarrow k$ can be recovered from the above degree $d$ quotient simply by restricting this quotient to the points $x$ and $x_{i}$ respectively. By assumption the automorphism induced by $g$ is identity. Therefore applying the automorphism induced by $g$ and restricting it to $x$, we get that $g(v)=v$. This completes the proof of injectivity. ∎

Corollary 2.2**.**

We have an inclusion of lie algebras

[TABLE]

Proof.

This follows from Lemma 2.1 and [MO67, Theorem 3.7]. ∎

Let $\mathcal{Z}$ be the fibered product of $d$ copies of $\mathbb{P}(E)$ over $S$. We will construct a rational map $\Phi:\mathcal{Z}\dashrightarrow\mathcal{Q}(E,d)$. Note that this map was already constructed in [Gan18, Section 2] in the special case when $S=\{{\rm pt}\}$ and $E=\mathcal{O}^{r}_{X}$. First we set some notations.

Notation 2.3**.**

1. (1)

Let $p:\mathbb{P}(E)\rightarrow X$ be the projection. 2. (2)

Let $p_{i}:\mathcal{Z}\to\mathbb{P}(E)$ be the $i$-th projection. 3. (3)

For, $i\neq j$, let $\Delta_{i,j}\hookrightarrow\mathcal{Z}$ be the closed subscheme given by the equation $p_{i}=p_{j}$. 4. (4)

For $i,j$ distinct let $\Delta_{i,j,X}\hookrightarrow\mathcal{Z}$ be the closed subscheme given by the equation $p\circ p_{i}=p\circ p_{j}$. 5. (5)

For $i,j,k$ all distinct, let $\Delta_{i,j,k,X}\hookrightarrow\mathcal{Z}$ be the closed subscheme given by the equation $p\circ p_{i}=p\circ p_{j}=p\circ p_{k}$. 6. (6)

Let $\pi_{1}:X\times_{S}\mathcal{Z}\to X$ and $\pi_{2}:X\times_{S}\mathcal{Z}\to\mathcal{Z}$ be the first and second projections respectively. 7. (7)

Let $p_{i}\circ\pi_{2}:X\times_{S}\mathcal{Z}\to\mathbb{P}(E)$ be denoted by $\pi_{2,i}$. 8. (8)

Let $\Delta_{i}\hookrightarrow X\times_{S}\mathcal{Z}$ be the closed subscheme given by the equation $\pi_{1}=p\circ\pi_{2,i}$.

We define an open set

[TABLE]

Consider the following compostion of morphisms over $X\times_{S}\mathcal{Z}$

[TABLE]

Let $u\in\mathcal{U}$. Then $q|_{X\times_{S}u}$ is the morphism

[TABLE]

where the map $E|_{p\circ p_{i}(u)}\to k_{p\circ p_{i}(u)}$ is given by $p_{i}(u)\in\mathbb{P}(E)$. Since $u\in\mathcal{U}$ for any $1\leq i\leq d$, there can exist atmost one pair $j\neq i$ such that $p\circ p_{i}(u)=p\circ p_{j}(u)$, and for such a pair $(i,j)$, $p_{i}(u)\neq p_{j}(u)$. Hence $q|_{X\times_{S}u}$ is a surjection. Therefore $q|_{X\times_{S}\mathcal{U}}$ is a surjection. By universal property of $\mathcal{Q}(E,d)$ the surjection $q|_{X\times_{S}\mathcal{U}}$ induces a map

[TABLE]

Then we prove the following proposition

Proposition** (Propositon 2.14).**

Suppose either $r\geq 3$ or $r=2$, $E$ is semistable with respect to $\mathcal{M}$ and genus of $C$ is $\geq 2$. In both of these cases we have an isomorphism

[TABLE]

To prove Proposition 2.14 we need a few lemmas. We define $\mathcal{F}(E,d):={\rm ker}~{}q$.

Lemma 2.4**.**

We have an isomorphism of vector bundles

[TABLE]

Proof.

Over $X\times_{S}\mathcal{Q}(E,d)$, we have the universal exact sequence:

[TABLE]

Then it is known that $\mathcal{A}(E,d)$ is a vector bundle of rank $r$ [Gan18, Lemma 2.2] and by [HL10, Proposition 2.2.7] we have

[TABLE]

Consider the following diagram:

[TABLE]

By Grauert’s theorem [Har77, Corollary 12.9], We get that

[TABLE]

Since $\mathcal{A}(E,d)$ is a vector bundle, we have

[TABLE]

By the definition of the map $\Phi$ we have

[TABLE]

Also

[TABLE]

since by [Gan18, Lemma 2.2] $\mathcal{F}(E,d)|_{X\times_{S}\mathcal{U}}$ is again a vector bundle of rank $r$ and there exists a surjection $\Phi^{*}\mathcal{A}(E,d)\twoheadrightarrow\mathcal{F}(E,d)|_{X\times_{S}\mathcal{U}}$. This completes the proof of the lemma. ∎

Lemma 2.5**.**

For $1\leq i,j\leq d$ and $i\neq j$ we have

[TABLE]

Proof.

By adjunction, we have

[TABLE]

Since $\Delta_{j}$ is an integral scheme and $\Delta_{i}\cap\Delta_{j}$ is a proper subset of $\Delta_{j}$, the later term in the above expression is zero. ∎

Lemma 2.6**.**

For any $1\leq j\leq d$ we have

[TABLE]

Proof.

The projection $\pi_{2}$ induces isomorphism $\Delta_{j}\xrightarrow{\sim}\mathcal{Z}$. Identifying $\Delta_{j}$ with $\mathcal{Z}$ we have

[TABLE]

Since $\mathcal{F}(E,1)$ is vector bundle over $\mathcal{Z}$ and codimension of $\mathcal{Z}\setminus\mathcal{U}\geq 2$ we have

[TABLE]

Using projection formula for the morphism $p_{j}$ we get that

[TABLE]

Now over $\mathbb{P}(E)$ we have

[TABLE]

This completes the proof of the lemma. ∎

Lemma 2.7**.**

For $1\leq i,j\leq d,~{}i\neq j$, we have an isomorphism of sheaves:

[TABLE]

Proof.

Consider the exact sequence:

[TABLE]

Applying $\mathscr{H}om(\text{ },\mathcal{O}_{\Delta_{j}})$ to the above exact sequence, we get:

[TABLE]

Therefore

[TABLE]

and the statement follows immediately from this. ∎

The following corollary follows immediately from Lemma 2.7.

Corollary 2.8**.**

We have an isomorphism of sheaves on $\mathcal{Z}$

[TABLE]

Lemma 2.9**.**

Fix $1\leq i,j\leq d$ with $i\neq j$. Then for $1\leq k,l,m\leq d$ with $k,l,m$ distinct we have

1. (1)

${\rm codim}(\Delta_{k,l}\cap\Delta_{i,j,X},\Delta_{i,j,X})\geq 2$* if $\{k,l\}\neq\{i,j\}$.* 2. (2)

${\rm codim}(\Delta_{i,j}\cap,\Delta_{i,j,X},\Delta_{i,j,X})=r$. 3. (3)

If $\{i,j\}\nsubseteq\{k,l,m\}$ then ${\rm codim}(\Delta_{k,l,m,X}\cap\Delta_{i,j,X},\Delta_{i,j,X})\geq 2$. 4. (4)

${\rm codim}(\Delta_{i,j,k,X},\Delta_{i,j,X})=1$.

Proof.

Without loss of generality we can assume $(i,j)=(1,2)$. Then

[TABLE]

1. (1)

Let $\{k,l\}\cap\{1,2\}=\emptyset$. Without loss of generality we can assume $(k,l)=(3,4)$. Then

[TABLE]

Let $\{k,l\}\cap\{1,2\}=\{2\}$. Without loss of generality we can assume $(k,l)=(2,3)$. Then

[TABLE]

Therefore in both these cases it have codimension $\geq 2$ in $\Delta_{1,2,X}$. 2. (2)

If $\{k,l\}=\{1,2\}$ then $\Delta_{1,2}\cong\mathbb{P}(E)\times_{S}\mathbb{P}(E)^{d-2}_{S}$. Hence it has codimension $r$ in $\Delta_{1,2,X}$. 3. (3)

Let $\{1,2\}\cap\{k,l,m\}=\emptyset$. Without loss of generality we can assume $(k,l,m)=(3,4,5)$. Then

[TABLE]

Let $\{i,j\}\cap\{k,l,m\}=\{i\}$. Without loss of generality we can assume $k=i=1$ and $(l,m)=(3,4)$. Then

[TABLE]

Hence in both of these two cases it has codimension $\geq 2$ in $\Delta_{1,2,X}$. 4. (4)

$\Delta_{1,2,k,X}\cong\mathbb{P}(E)\times_{X}\mathbb{P}(E)\times_{X}\mathbb{P}(E)\times_{S}\mathbb{P}(E)^{d-3}_{S}$. Hence it has codimension $1$ in $Y$.

∎

On $\Delta_{i,j,X}$ we define the line bundle

[TABLE]

By Lemma 2.9 we have that $\Delta_{i,j,k,X}\subset\Delta_{i,j,X}$ is a divisor on $\Delta_{i,j,X}$.

Lemma 2.10**.**

Fix $1\leq i,j\leq d$ with $i\neq j$. For any $n\geq 0$ we have

[TABLE]

Proof.

Let $f:\Delta_{i,j,X}\to\mathbb{P}(E)^{d-1}_{S}$ be the product of all the projections except the $i$-th projection. Then by projection formula

[TABLE]

for some line bundle $\mathcal{L}^{\prime}$ on $\mathbb{P}(E)^{d-1}_{S}$. Consider the following fibered diagram:

[TABLE]

Here $g_{i}$ is the composition of $i$-th projection from $\mathbb{P}(E)_{S}^{d-1}$ and the morphism $p:\mathbb{P}(E)\to X$. Since $g_{i}$ is flat we have

[TABLE]

Since $p_{*}\mathcal{O}(-1)=0$ we have that $f_{*}p^{*}_{i}\mathcal{O}(-1))=0$. This completes the proof of the lemma. ∎

Proposition 2.11**.**

Let $r={\rm rank}~{}E\geq 3$. For $1\leq i,j\leq d,~{}i\neq j$ we have

[TABLE]

Proof.

By Corollary 2.7 it is enough to show

[TABLE]

Since $r\geq 3$ by Lemma 2.9 we have

[TABLE]

Let $s\in H^{0}(V,\mathcal{L}|_{V})$. Then for some $n$ large enough, there exists a section $0\neq t\in H^{0}(\Delta_{i,j,X},\mathcal{O}(\sum\limits_{k\neq i,j}n\cdot\Delta_{1,2,k,X}))$ such that the section $st^{n}$ extends to a global section of $\mathcal{L}\otimes\mathcal{O}(\sum\limits_{k\neq i,j}n\cdot\Delta_{1,2,k,X})$. However by Lemma 2.10 there are no global sections of this line bundle and this completes the proof of the proposition. ∎

Since $p_{S}$ is a projective morphism, we have a $p_{S}$-ample line bundle $\mathcal{O}_{X}(1)$. Let $\mathcal{O}_{S}(1)$ be an ample line bundle on $S$. Then for $a\gg 0$ the line bundle $\mathcal{O}_{X}(1)\otimes\mathcal{O}_{S}(a)$ is an ample line bundle on $X$. We fix such an ample line bundle $\mathcal{M}$ on $X$.

Lemma 2.12**.**

Let $E$ be semistable with respect to $\mathcal{M}$, ${\rm rank}~{}E=2$ and genus of $X_{s}\geq 2$ for any $s\in S$. Fix $1\leq i\leq d$. Then for any $n\geq 0$ we have

[TABLE]

Proof.

Without loss of generality we can assume $i=1$. Let us denote the $j$-th projection from $(X)_{S}^{d}$ to $X$ by $p_{j,X}$. We define $X_{j}\subseteq(X)_{S}^{d}$ to be the closed set defined by the equation $p_{1,X}=p_{j,X}$. By projection formula we have

[TABLE]

Now we have the following exact sequence

[TABLE]

Note that $\mathcal{O}((n+1)X_{k})|_{X_{k}}=p^{*}_{1,X}\mathcal{T}^{n+1}_{X/S}$. Tensoring the above exact sequence with $p^{*}_{1,X}(S^{2n}(E)\otimes(\text{det}(E^{\vee}))^{n})\otimes p_{1.X}^{*}\mathcal{T}_{X/S}$ and applying $H^{0}$ we get that it is enough to show

[TABLE]

Applying projection formula for the morphism $p_{1,X}$ we get

[TABLE]

Hence it is enough to show that

[TABLE]

Now

[TABLE]

and ${\rm rank}~{}S^{2n}(E)=2n+1$. Therefore

[TABLE]

Since genus of each fibre of $X\rightarrow S$ is $\geq 2$, deg $\mathcal{T}_{X/S}<0$. Hence

[TABLE]

Since $E$ is semistable we have that the bundle $S^{2n}(E)\otimes(\text{det}(E^{\vee}))^{n}\otimes\mathcal{T}^{m}_{X/S}$ is also semistable with negative degree. Therefore it does not have any global section. ∎

Proposition 2.13**.**

Let $r={\rm rank}~{}E=2$, $E$ is semistable with respect to $\mathcal{M}$ and genus of $C$ is $\geq 2$. For $1\leq i,j\leq d,~{}i\neq j$ we have

[TABLE]

Proof.

By Corollary 2.7 it is enough to show

[TABLE]

Define the open set

[TABLE]

By Lemma 2.9 we have

[TABLE]

Therefore, to show that this space vanishes, it is enough to show that

[TABLE]

Now consider the following exact sequence:

[TABLE]

Tensoring the above exact sequence by $\mathcal{L}(n\cdot\sum\limits_{k\neq i,j}\Delta_{i,j,k,X})$ and applying $H^{0}$ we see that it is enough to show that

[TABLE]

Note that $\mathcal{O}(\Delta_{i,j})|_{\Delta_{i,j}}=p^{*}_{i}\mathcal{T}_{\mathbb{P}(E)/S}|_{\Delta_{i,j}}$. Then

[TABLE]

Identifying $\Delta_{i,j}$ with $\mathbb{P}(E)_{S}^{d-1}$ the statement follows from Lemma 2.12. ∎

Proposition 2.14**.**

Suppose either $r\geq 3$ or $r=2$, $E$ is semistable with respect to $\mathcal{M}$ and genus of $C$ is $\geq 2$. In both of these cases we have an isomorphism

[TABLE]

Proof.

By Lemma 2.4 we have

[TABLE]

Hence for a fixed $1\leq j\leq d$ it is enough to show

[TABLE]

Over $X\times_{S}\mathcal{U}$ we have the following commutative diagram:

[TABLE]

Using snake lemma for the above diagram we get the following exact sequence over $X\times_{S}\mathcal{U}$

[TABLE]

We apply $\mathscr{H}om($ $,\pi^{*}_{2,j}\mathcal{O}(1)|_{\Delta_{j}})$ and then the the functor $H^{0}$. Now the result follows from Lemma 2.5, Lemma 2.6 and Proposition 2.13. ∎

Theorem 2.15**.**

Suppose either $r:={\rm rank}~{}E\geq 3$ or $r=2$, $E$ is semistable with respect to $\mathcal{M}$ and genus of $X_{s}$ is $\geq 2$ for $s\in S$. In both of these cases we have isomorphisms

1. (1)

${\rm Aut}^{o}(\mathcal{Q}(E,d))\cong{\rm Aut}^{o}(\mathbb{P}(E)/S)$. 2. (2)

$H^{0}(\mathbb{P}(E),\mathcal{T}_{\mathbb{P}(E)/S})\cong H^{0}(\mathcal{Q}(E,d),\mathcal{T}_{\mathcal{Q}(E,d)/S})$.

Proof.

By Corollary 2.2 we have an inclusion of lie algebras:

[TABLE]

By Lemma 2.4 and Proposition 2.14 we have an inclusion

[TABLE]

Since $\mathcal{Z}\to\mathcal{Q}(E,d)$ is invariant under the action of the symmetric group $S_{d}$ we get that this inclusion factors through

[TABLE]

Comparing the dimensions we get that the (2.16) is an isomorphism. Hence the inclusion in Lemma 2.1 is an isomorphism. ∎

3. Applications

Corollary 3.1**.**

Suppose either $r\geq 3$ or $r=2$, $E$ is semistable with respect to $\mathcal{M}$ and genus of $C$ is $\geq 2$. Then we have the following left exact sequence of algebraic groups

[TABLE]

The corresponding sequence of lie algebras is given by

[TABLE]

Proof.

The left exactness of the above sequences follow from Theorem 2.15 and from the fact that $\text{Aut}^{o}(\mathbb{P}(E)/S)$ and its lie algebra fits into the above exact sequences. ∎

Corollary 3.2**.**

Let the genus of the fibres of $X\rightarrow S$ is $\geq 2$. Suppose either $r\geq 3$ or $r=2$ and $E$ is semistable with respect to $\mathcal{M}$. Then

1. (1)

${\rm Aut}^{o}(\mathcal{Q}(E,d)/S)=\text{GL}(E)/k^{*}$. 2. (2)

$H^{0}(\mathcal{Q}(E,d),\mathcal{T}_{\mathcal{Q}(E,d)/S})=H^{0}(X,\text{ad }E)$.

Proof.

If genus of each fibre is $\geq 2$ then $(p_{S})_{*}\mathcal{T}_{X/S}=0$. In particular $H^{0}(X,\mathcal{T}_{X/S})=0$. Hence $\text{Aut}^{o}(X/S)=0$. Now the corollary follows from Corollary 3.1. ∎

Taking $S$ to be a point and $E=\mathcal{O}^{r}_{C}$ in Corollary 3.2 we get [BDH15, Theorem 3.1] and [BDH15, Corollary 3.2].

Corollary 3.3**.**

Let $C$ be a smooth projective curve of genus $\geq 2$ over an algebraically closed field $k$ of characteristic zero. Then

1. (1)

${\rm Aut}^{o}(\mathcal{Q}(\mathcal{O}^{r}_{C},d)/S)=\text{PGL}(r)$. 2. (2)

$H^{0}(\mathcal{Q}(\mathcal{O}^{r}_{C},d),\mathcal{T}_{\mathcal{Q}(\mathcal{O}^{r}_{C},d)})=\mathfrak{s}l(r)$.

Let $C$ be a smooth projective curve of genus $\geq 2$ over an algebraically closed field $k$ of characteristic zero. Fix $\underline{d}=(d_{1},d_{2},\ldots,d_{k})\in\mathbb{N}^{k}$ with $d_{1}>d_{2}>\ldots>d_{k}$ and $r\geq 1$. Let $\mathcal{D}(r,\underline{d})$ be the flag scheme parametrizing chain of quotients of $\mathcal{O}^{r}_{C}\to B_{1}\to B_{2}\to\ldots\to B_{d}$ where $B_{i}$ is a torsion quotient of degree $d_{i}$ [HL10, 2.A.1]. It is known that $\mathcal{D}(r,\underline{d})$ is a smooth projective variety.

Corollary 3.4**.**

We have the following isomorphisms of algebraic groups and lie algebras

1. (1)

${\rm Aut}^{o}(\mathcal{D}(r,\underline{d}))\cong{\rm PGL}(r)$. 2. (2)

$H^{0}(\mathcal{D}(r,\underline{d}),\mathcal{T}_{D(r,\underline{d})})=\mathfrak{sl}(r)$.

Proof.

Let $\underline{d}^{\prime}:=(d_{2},d_{3},\ldots,d_{k})$. Over $C\times\mathcal{D}(r,\underline{d}^{\prime})$ we have the universal chain of filtrations:

[TABLE]

Then $D(r,\underline{d})$ is the relative quot scheme of torsion quotients of degree $d_{1}-d_{2}$ of the vector bundle $\mathcal{A}(r,d_{2})$ for the map

[TABLE]

By Corollary 3.2 we get that

[TABLE]

By [Gan18, Theorem 3.2.4, Theorem 5.1] the bundle $\mathcal{A}(r,d_{2})$ is stable with respect to certain polarisations on $C\times\mathcal{D}(r,\underline{d}^{\prime})$. Hence by Corollary 3.2 we have

[TABLE]

By induction on k we get that

[TABLE]

This completes the proof of the corollary. ∎

Let $C$ be a smooth projective curve over an algebraically closed field of characteristic zero. In [BM16] the authors computed the identity component of automorphism group scheme of a certain generalized quot scheme $\mathcal{Q}_{C}(r,d_{p},d_{z})$. We recall the definition of this scheme: Fix $r\geq 2,d_{p},d_{z}\geq 1$. Consider the quot scheme $\mathcal{Q}(\mathcal{O}^{r}_{C},d_{p})$ and the universal kernel bundle $\mathcal{A}(r,d_{p})$ over $C\times\mathcal{Q}(\mathcal{O}^{r}_{C},d_{p})$. Then $\mathcal{Q}(r,d_{p},d_{z})$ is defined as the relative Quot scheme associated to the projection $C\times\mathcal{Q}(\mathcal{O}^{r}_{C},d_{p})\to\mathcal{Q}(\mathcal{O}^{r}_{C},d_{p})$ and the bundle $\mathcal{A}(r,d_{p})^{\vee}$. By [Gan18, Theorem 3.2.4] $\mathcal{A}(r,d_{p})$ is stable with respect to certain polarisations. Hence $H^{0}(C\times\mathcal{Q}(r,d_{p}),{\rm ad}~{}\mathcal{A}(r,d_{p})^{\vee})=0$. Now from Theorem 2.15 we get the result proved in [BM16]:

Corollary 3.5**.**

[BM16, Theorem 2.1]** Let $C$ be a smooth projective curve of genus $\geq 2$ over an algebraically closed field $k$ of characteristic zero. We have the following isomorphisms of algebraic groups and lie algebras

1. (1)

${\rm Aut}^{o}(\mathcal{Q}(r,d_{p},d_{z}))\cong{\rm PGL}(r)$. 2. (2)

$H^{0}(\mathcal{Q}(r,d_{p},d_{z}),\mathcal{T}_{\mathcal{Q}(r,d_{p},d_{z})})=\mathfrak{sl}(r)$.

Corollary 3.6**.**

Let $C$ be a smooth projective curve over an algebraically closed field $k$. Let $E$ be a vector bundle of rank $\geq 3$ over $C$. Fix $d\geq 1$. Let $\mathcal{Q}(E,d)$ be the quot scheme of torsion quotients of $E$ of degree $d$. Then we have

1. (1)

If genus of $C=0$,i.e. $C\cong\mathbb{P}^{1}$, then

[TABLE] 2. (2)

If genus of $C=1$ and if $E$ is semistable then we have the following sequence of algebraic groups

[TABLE] 3. (3)

If $E$ is not semistable, then ${\rm Aut}^{o}(\mathcal{Q}(E,d))=\text{GL}(E)/k^{*}$.

Proof.

If $C\cong\mathbb{P}^{1}$ then any vector bundle $E$ admits a $\text{GL}(2)$ linearisation, in paricular we have a homomorphism GL$(2)\rightarrow\text{Aut}^{o}(\mathcal{Q}(E,d))$. This homomorphism factors through $\text{PGL}(2,k)$ and gives a section to the map $\text{Aut}^{o}(\mathcal{Q}(E,d))\rightarrow\text{PGL}(2,k)$. Therefore the left exact sequence in Corollary 3.1 is exact in this case and it splits.

From now on we assume that genus of $C$ is $1$ i.e. $C$ is an elliptic curve. Recall that a bundle $E$ is called semi-homogeneous if ${\rm Aut}^{o}(\mathbb{P}(E))\to{\rm Aut}^{o}(C)=C$ is surjective ([Muk78, Definition 5.2]). By [Muk78, Proposition 6.13] every semi-homogenous bundle is semistable. Hence (3) follows from Corollary 3.1. Let us assume $E$ is semistable. Then $E\cong\oplus E_{i}$, where $E_{i}$ are indecomposable of slope $\mu(E_{i})=\mu(E)$. By [Ati57, Theorem 10] any indecomposable bundle over $C$ is semi-homogenous and therefore by [Muk78, Proposition 6.9] we have that $E$ is semi-homogenous. Now (2) follows from Corollary 3.1. ∎