Moduli Spaces of $\alpha$-stable Pairs and Wall-Crossing on $\mathbb{P}^2$

TL;DR

This paper investigates how moduli spaces of $ ext{alpha}$-stable pairs on $ ext{P}^2$ change as the stability parameter varies, revealing geometric transformations and computing Poincaré polynomials for certain cases.

Contribution

It provides a detailed description of wall-crossing phenomena for $ ext{alpha}$-stable pairs on $ ext{P}^2$, including geometric descriptions of blow-ups and blow-downs, and computes Poincaré polynomials for stable sheaves.

Findings

Wall-crossing involves smooth blow-up and blow-down morphisms.

Explicit geometric descriptions of blow-up centers.

Poincaré polynomials of moduli spaces of stable sheaves.

Abstract

We study the wall-crossing of the moduli spaces $\mathbf{M}^\alpha (d,1)$ of $\alpha$-stable pairs with linear Hilbert polynomial $dm+1$ on the projective plane $\mathbb{P}^2$ as we alter the parameter $\alpha$. When $d$ is 4 and 5, at each wall, the moduli spaces are related by a smooth blow-up morphism followed by a smooth blow-down morphism, where one can describe the blow-up centers geometrically. As a byproduct, we obtain the Poincar\'e polynomials of the moduli space $\mathbf{M}(d,1)$ of stable sheaves. We also discuss the wall-crossing when the number of stable components in Jordan-H\"{o}lder filtrations is three.