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