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

2026/08/05 by Parvez Rasul
Mathematics · #math.AG #msc:14C17 #msc:14H60

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arxiv created 2026/08/05 · arxiv updated 2026/08/06

Abstract

Let C be a smooth projective curve over \mathbb C and let E be a vector bundle over C. Let 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 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 Quot_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 Quot_r,d(E). With the help of these constraints, we compute the (virtual) intersection numbers of f-classes on the Quot schemes.

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