# Rank 2 wall-crossing and the Serre correspondence

Research paper by **A. Gholampour, M. Kool**

Indexed on: **10 Feb '16**Published on: **10 Feb '16**Published in: **Mathematics - Algebraic Geometry**

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#### Abstract

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.