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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$4m+2$$\end{document} on a smooth quadric surface, (2) the moduli space of semistable sheaves of Hilbert polynomial m2+3m+2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m^{2}+3m+2$$\end{document} on P3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {P}^{3}$$\end{document}, (3) Kontsevich’s moduli space of genus-zero 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.