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 P^3,3)Kontsevichs 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 Kirwans partial desingularization.Also we investigate several geometric properties of 1)by using the variation of moduli spaces of stable pairs.This is joint work with Han-Bom Moon(Fordham univ.).