Moduli spaces of semistable sheaves of dimension 1 on $\mathbb{P}^2$

TL;DR

This paper describes the structure of moduli spaces of semistable sheaves of dimension 1 on the projective plane, providing explicit descriptions, isomorphisms with projective bundles, and computing their classes in the Grothendieck group.

Contribution

It offers a new description of the moduli spaces using matrix classes and establishes isomorphisms with projective bundles over Hilbert schemes, along with class computations.

Findings

A description of $M(d, ext{1})$ as a projective bundle over a Hilbert scheme.

Explicit class computations for $M(4,1)$, $M(5,1)$, and $M(5,2)$.

Proof that $M(5,1)$ and $M(5,2)$ have the same class in the Grothendieck group.

Abstract

Let $M(d,\chi)$ be the moduli space of semistable sheaves of rank 0, Euler characteristic $\chi$ and first Chern class $dH (d>0)$, with $H$ the hyperplane class in $\mathbb{P}^2$. We give a description of $M(d,\chi)$, viewing each sheaf as a class of matrices with entries in $\bigoplus_{i\geq0}H^0(\mathcal{O}_{\mathbb{P}^2}(i))$. We show that there is a big open subset of $M(d,1)$ isomorphic to a projective bundle over an open subset of a Hilbert scheme of points on $\mathbb{P}^2.$ Finally we compute the classes of M(4,1), M(5,1) and M(5,2) in the Grothendieck group of varieties, especially we conclude that M(5,1) and M(5,2) are of the same class.