Moduli of sheaves, Fourier-Mukai transform, and partial desingularization

We study birational maps among (1) the moduli space of semistable sheaves of Hilbert polynomial 4m + 2 on a smooth quadric surface, (2) the moduli space of semistable sheaves of Hilbert polynomial m(2) + 3m + 2 on P-3, (3) Kontsevich's moduli space of genuszero stable maps of degree 2 to the Grassmannian Gr(2, 4). A regular birational morphism from (1) to (2) is described in terms of Fourier-Mukai transforms. The map from (3) to (2) is Kirwan's partial desingularization. We also investigate several geometric properties of 1) by using the variation of moduli spaces of stable pairs.