# 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.

## Key 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 $S$.