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

Let $\mathbf{M}^α(d,χ)$ be the moduli space of $α$-stable pairs $(s,F)$ on the projective plane $\mathbb{P}^2$ with Hilbert polynomial $χ(F(m))=dm+χ$. For sufficiently large $α$ (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 $(d,χ)$, 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 $α$-stable pairs space when $(d,χ)=(5,2)$. As a direct corollary, by combining with Bridgeland wall crossing of the moduli space of stable sheaves, we compute the virtual Poincaré polynomial of $\mathbf{M}^{\infty}(5,2)$.