The geometry of the moduli space of one-dimensional sheaves

TL;DR

This paper studies the geometric structure of the moduli space of stable sheaves on the projective plane, determining its effective and nef cones, and analyzing its stable base locus decomposition, with implications for Betti numbers and physics predictions.

Contribution

It introduces a method using wall-crossing and intersection theory to explicitly compute the cones and base locus decomposition of the moduli space.

Findings

Effective and nef cones of 1 space 1 are determined.

Stable base locus decomposition of 1 space for d=6 is presented.

Betti numbers of the moduli spaces are computed, confirming physics predictions.

Abstract

Let $\mathbf{M}_d$ be the moduli space of stable sheaves on $\mathbb{P}^2$ with Hilbert polynomial $dm+1$. In this paper, we determine the effective and the nef cone of the space $\mathbf{M}_d$ by natural geometric divisors. Main idea is to use the wall-crossing on the space of Bridgeland stability conditions and to compute the intersection numbers of divisors with curves by using the Grothendieck-Riemann-Roch theorem. We also present the stable base locus decomposition of the space $\mathbf{M}_6$. As a byproduct, we obtain the Betti numbers of the moduli spaces, which confirm the prediction in physics.