Virtual invariants of Quot schemes of points on threefolds

TL;DR

This paper develops a method to define and compute virtual invariants of Quot schemes of points on threefolds, confirming a conjecture through reduction to toric cases and localization techniques.

Contribution

It constructs an almost perfect obstruction theory for these Quot schemes and computes their invariants, verifying Ricolfi's conjecture.

Findings

Established a virtual class in degree zero for Quot schemes on threefolds.

Reduced the computation to toric cases using cobordism and degeneration arguments.

Solved the toric case via localization and equivariant formulas.

Abstract

We construct an almost perfect obstruction theory of virtual dimension zero on the Quot scheme parametrizing zero-dimensional quotients of a locally free sheaf on a smooth projective $3$-fold. This gives a virtual class in degree zero and therefore allows one to define virtual invariants of the Quot scheme. We compute these invariants proving a conjecture by Ricolfi. The computation is done by reducing to the toric case via cobordism theory and a degeneration argument. The toric case is solved by reducing to the computation on the Quot scheme of points on $\mathbb{A}^3$ via torus localization, the torus-equivariant Siebert's formula for almost perfect obstruction theories and the torus-equivariant Jouanolou trick.