Torelli theorems for Quot schemes of vector bundles on curves

Abstract

For a vector bundle $E$ on a smooth projective curve $C$, one defines the Quot scheme $Q_d(E,C)$ parametrizing subsheaves of $E$ having length $d$ torsion quotients. If $E_i$ is a vector bundle of rank $r\ge 2$ on a smooth projective curve $C_i$ for $i=1,2$, isomorphisms between ${\mathbb{P}}_{C_i}(E_i)$ preserving the projective bundle structures induce isomorphisms of $Q_d(E_i,C_i)$, called natural isomorphisms. We show that for $d\ge 2$ all isomorphisms between $Q_d(E_i,C_i)$ are natural, except for a non-natural involution of $Q_2({\mathcal{O}}_C^{\oplus r},C)$. This gives a complete description of the automorphism group of $Q_d(E,C)$, generalizing previous works of Biswas-Dhillon-Hurtubise and Gangopadhyay. This also shows that one can reconstruct the curve from the Quot scheme, a Torelli-type theorem. As a key step in our proof, we show that any isomorphism between symmetric powers of smooth projective curves preserving big diagonals is natural, which is interesting in its own right.

arXiv comment20 pages. Comments are welcome