Automorphisms of relative Quot schemes

Abstract

Let $X\rightarrow S$ be a smooth family of projective curves over an algebraically closed field k of characteristic zero. Assume that both X and S are smooth projective varieties and let E be a vector bundle of rank r over X and $\mathbb {P}(E)$ be its projectivization. Fix $d\ge 1$. Let $\mathcal {Q}(E,d)$ be the relative quot scheme of torsion quotients of E of degree d. Then we show that if $r\ge 3$, then the identity component of the group of automorphisms of $\mathcal {Q}(E,d)$ over S is isomorphic to the identity component of the group of automorphisms of $\mathbb {P}(E)$ over S. We also show that under additional hypotheses, the same statement is true in the case when $r=2$. As a corollary, the identity component of the automorphism group of flag scheme of filtrations of torsion quotients of $\mathcal {O}^{r}_{C}$, where $r\ge 3$ and C a smooth projective curve of genus $\ge 2$ is computed.