Moduli of semistable sheaves as quiver moduli

In the 1980s Dr'ezet and Le Potier realized moduli spaces of Gieseker-semistable sheaves on ${\mathbb{P}}^2$ as what are now called quiver moduli spaces. We discuss how this construction can be understood using t-structures and exceptional collections on derived categories, and how it can be extended to a similar result on ${\mathbb{P}}^1\times {\mathbb{P}}^1$.