Rank 2 wall-crossing and the Serre correspondence

TL;DR

This paper investigates the generating functions of Quot schemes of rank 2 reflexive sheaves on 3-folds, revealing they are related to MacMahon functions and deriving explicit formulas in specific cases using Serre correspondence.

Contribution

It establishes a new connection between Quot schemes of rank 2 sheaves and MacMahon functions, extending Hall algebra techniques via Serre correspondence.

Findings

Generating functions are powers of the MacMahon function times a polynomial.

Polynomial generating functions relate to Quot schemes supported where the sheaf is not locally free.

Explicit product formulas are derived for equivariant cases on \\mathbb{C}^3.

Abstract

We study Quot schemes of 0-dimensional quotients of sheaves on 3-folds $X$. When the sheaf $\mathcal{R}$ is rank 2 and reflexive, we prove that the generating function of Euler characteristics of these Quot schemes is a power of the MacMahon function times a polynomial. This polynomial is itself the generating function of Euler characteristics of Quot schemes of a certain 0-dimensional sheaf, which is supported on the locus where $\mathcal{R}$ is not locally free. In the case $X = \mathbb{C}^3$ and $\mathcal{R}$ is equivariant, we use our result to prove an explicit product formula for the generating function. This formula was first found using localization techniques in previous joint work with B. Young. Our results follow from R. Hartshorne's Serre correspondence and a rank 2 version of a Hall algebra calculation by J. Stoppa and R.P. Thomas.