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