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

Abstract

We study the wall-crossing of the moduli spaces $\mathbf{M}^α(d,1)$ of $α$-stable pairs with linear Hilbert polynomial $dm+1$ on the projective plane $\mathbb{P}^2$ as we alter the parameter $α$. 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 Poincaré 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-Hölder filtrations is three.