English

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

Algebraic Geometry 2026-08-05 v1

Abstract

Let CC be a smooth projective curve over C\mathbb C and let EE be a vector bundle over CC. Let Quotr,d(E)\text{Quot}_{r,d}(E) denote the Quot scheme which parametrizes quotients of EE of rank rr and degree dd. Following Joyce's recipe [Joy21], we introduce Joyce's enumerative invariant for the Quot scheme Quotr,d(E)\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 Quotrank(E)1,d(E)\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 Quotr,d(E)\text{Quot}_{r,d}(E). With the help of these constraints, we compute the (virtual) intersection numbers of ff-classes on the Quot schemes.

Keywords

Cite

@article{arxiv.2608.04795,
  title  = {Joyce's invariant and Virasoro Constraints for Quot schemes on curves},
  author = {Parvez Rasul},
  journal= {arXiv preprint arXiv:2608.04795},
  year   = {2026}
}

Comments

Comments are welcome