Automorphisms of relative Quot Schemes

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 $S$.

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