Rank 2 wall-crossing and the Serre correspondence

M. Kool, A. Gholampour

Published 2016-02-09Version 1

Let $\mathcal{R}$ be a rank 2 reflexive sheaf on a smooth projective 3-fold $X$. We are interested in the Euler characteristics of the Quot schemes $\mathrm{Quot}(\mathcal{R},n)$ of 0-dimensional quotients of $\mathcal{R}$ of length $n$. Provided $\mathcal{R}$ admits a cosection cutting out a 1-dimensional subscheme, we prove that the generating function of these Euler characteristics is equal to $M(q)^{2e(X)}$ times a polynomial of degree $c_3(\mathcal{R})$. This polynomial is the generating function of Euler characteristics of $\mathrm{Quot}(\mathcal{E}{\it{xt}}^1(\mathcal{R},\mathcal{O}_X),n)$. Here $\mathcal{E}{\it{xt}}^1(\mathcal{R},\mathcal{O}_X)$ is a 0-dimensional sheaf supported at the points where $\mathcal{R}$ is not locally free. Since $\mathcal{R}$ is reflexive, it admits a 2-term resolution by vector bundles. In the case the vector bundles are of rank 1 and 3, $\mathcal{E}{\it{xt}}^1(\mathcal{R},\mathcal{O}_X)$ is a structure sheaf. This observation is used to prove a closed product formula for the Euler characteristics of $\mathrm{Quot}(\mathcal{R},n)$ in the case $X = \mathbb{C}^3$ and $\mathcal{R}$ is $T$-equivariant. This formula was first found using localization techniques and the double dimer model by the authors and B. Young. Our result follows from R. Hartshorne's Serre correspondence and a rank 2 version of the Hall algebra calculation of J. Stoppa and R. P. Thomas.