Moduli Spaces of $\alpha$-stable Pairs and Wall-Crossing on ${\mathbb{P}}^2$

We study the wall-crossing of the moduli spaces ${\mathbf{M}}^{\alpha }(d,1)$ of $\alpha$-stable pairs with linear Hilbert polynomial $dm+1$ on the projective plane ${\mathbb{P}}^2$ as we alter the parameter $\alpha$. When $d$ is 4 and 5, at each wall, the moduli spaces are related by a smooth blow-up morphism followed by a smooth blow-down morphism, where one can describe the blow-up centers geometrically. As a byproduct, we obtain the Poincar'e polynomials of the moduli space $\mathbf{M}(d,1)$ of stable sheaves. We also discuss the wall-crossing when the number of stable components in Jordan-H"{o}lder filtrations is three.