Wall-crossing for punctual Quot-schemes

TL;DR

This paper investigates the virtual geometry of punctual Quot-schemes on various algebraic varieties, introducing new methods based on wall-crossing to describe their fundamental classes and invariants, including a novel 12-fold correspondence and a Nekrasov genus formula.

Contribution

It provides a comprehensive description of virtual fundamental classes for punctual Quot-schemes across different dimensions using wall-crossing techniques, extending previous results and establishing new invariants and correspondences.

Findings

Complete description of virtual fundamental classes on curves, surfaces, and fourfolds.

New 12-fold correspondence relating Segre and Verlinde invariants.

A closed formula for the Nekrasov genus, supporting Nekrasov's conjecture.

Abstract

We study punctual quot-schemes of torsion-free sheaves $E_Y$ on smooth projective curves, surfaces and Calabi--Yau fourfolds via their virtual geometry. Our goal is to give a complete description of the virtual fundamental classes and their tautological integrals. In the fourfold case, we first construct these classes under additional conditions. We use novel methods relying on the wall-crossing of Joyce. Our results include -the dependence of the cobordism classes on the torsion-free sheaf $E_Y$ where $Y$ is a surface, -relations to the previous results in the literature, which addressed the case of a trivial $E_Y$, -a new 12-fold correspondence relating Segre and Verlinde invariants for curves, surfaces and Calabi-Yau fourfolds based on the one observed by Arbesfeld-Johnson-Lim-Oprea-Pandharipande in dimensions one and two, -a closed formula for the Nekrasov genus, which gives a compact analogue of Nekrasov's conjecture. As our techniques are orthogonal to the original literature, we make our work independent by proving a new combinatorial identity in arXiv:2111.09868