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

Let Mα(d,χ) be the moduli space of αα-stable pairs (s,F)(s,F) on the projective plane P2P2 with Hilbert polynomial χ(F(m))=dm+χχ(F(m))=dm+χ. For sufficiently large αα (denoted by ∞∞), it is well known that the moduli space is isomorphic to the relative Hilbert scheme of points over the universal degree dd plane curve. For the general (d,χ)(d,χ), 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 αα-stable pairs space when (d,χ)=(5,2)(d,χ)=(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 M∞(5,2).