Virtual Poincar\'e polynomial of the space of stable pairs supported on quintic curves

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 $\mathrm{\infty }$), 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 Poincar'e polynomial of ${\mathbf{M}}^{\mathrm{\infty }}(5,2)$.