Virtual Poincaré polynomial of the space of stable pairs supported on quintic curves

Abstract

Let ${\mathbf{M}}^{\alpha }(d,\chi )$ be the moduli space of $\alpha$-stable pairs $(s,F)$ on the projective plane ${\mathbb{P}}^2$ with Hilbert polynomial $\chi (F(m))=dm+\chi$. For sufficiently large $\alpha$ (denoted by $\infty$), it is well known that the moduli space is isomorphic to the relative Hilbert scheme of points over the universal degree $d$ plane curves. For the general $(d,\chi )$, the relative Hilbert scheme does not have a bundle structure over the Hilbert scheme of points. In this paper, as the first non trivial such a case, we study the wall crossing of the $\alpha$-stable pairs space when $(d,\chi )=(5,2)$. As a direct corollary, by combining with Bridgeland wall crossing of the moduli space of stable sheaves, we compute the virtual Poincaré polynomial of ${\mathbf{M}}^{\infty }(5,2)$.

To appear in Journal of Geometry and Physics

References in corpus (5)

• The refined BPS index from stable pair invariants
• The Minimal Model Program for the Hilbert Scheme of Points on P^2 and Bridgeland Stability
• The birational geometry of moduli space of sheaves on the projective plane
• On the geometry of the moduli spaces of semistable sheaves supported on plane quartics
• Universal families of extensions of coherent systems