Joyce's invariant and Virasoro Constraints for Quot schemes on curves

Abstract

Let $C$ be a smooth projective curve over $\mathbb C$ and let $E$ be a vector bundle over $C$. Let $\text{Quot}_{r,d}(E)$ denote the Quot scheme which parametrizes quotients of $E$ of rank $r$ and degree $d$. Following Joyce's recipe [Joy21], we introduce Joyce's enumerative invariant for the Quot scheme $\text{Quot}_{r,d}(E)$. The invariant can be viewed as a generalization of the virtual fundamental cycle of the Quot scheme. We evaluate intersection pairings on the Quot scheme $\text{Quot}_{\text{rank}(E)-1,d}(E)$ by computing its invariant explicitly. Following the reformulation of sheaf-theoretic Virasoro constraints in terms of Joyce's vertex algebra framework in [BLM24], we give a proof of the Virasoro constraints for the Quot scheme $\text{Quot}_{r,d}(E)$. With the help of these constraints, we compute the (virtual) intersection numbers of $f$-classes on the Quot schemes.