Automorphisms of the generalized quot schemes

TL;DR

This paper determines the automorphism group of a generalized quot scheme over a Riemann surface and shows that isomorphic schemes imply isomorphic surfaces, revealing deep geometric connections.

Contribution

It computes the automorphism group of the generalized quot scheme and establishes a criterion for surface isomorphism based on scheme isomorphism.

Findings

Connected component of automorphism group is PGL(r,C)

Automorphism group characterization for the scheme

Isomorphism of schemes implies isomorphism of Riemann surfaces

Abstract

Given a compact connected Riemann surface $X$ of genus $g \geq 2$, and integers $r\geq 2$, $d_p > 0$ and $d_z > 0$, in \cite{BDHW}, a generalized quot scheme ${\mathcal Q}_X(r,d_p,d_z)$ was introduced. Our aim here is to compute the holomorphic automorphism group of ${\mathcal Q}_X(r,d_p,d_z)$. It is shown that the connected component of $\text{Aut}( {\mathcal Q}_X(r,d_p,d_z))$ containing the identity automorphism is $\text{PGL}(r,{\mathbb C})$. As an application of it, we prove that if the generalized quot schemes of two Riemann surfaces are holomorphically isomorphic, then the two Riemann surfaces themselves are isomorphic.