Moduli spaces of α-stable pairs and wall-crossing on ℙ²

Abstract

We study the wall-crossing of the moduli spaces M^α(d,1) of α-stable pairs with linear Hilbert polynomial dm + 1 on the projective plane ℙ² as we alter the parameter α. When d is 4 or 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 spaces M(d,1) of stable sheaves. We also discuss the wall-crossing when the number of stable components in Jordan–Hölder filtrations is three.