Blowing-ups describing the polarization change of moduli schemes of semistable sheaves of general rank

Abstract

Let $H$ and $H^{\prime }$ be two ample line bundles over a smooth projective surface $X$, and $M(H)$ (resp. $M(H^{\prime })$) the coarse moduli scheme of $H$-semistable (resp. $H^{\prime }$-semistable) sheaves of fixed type $(r,c_1,c_2)$. We construct a sequence of blowing-ups which describes how $M(H)$ differs from $M(H^{\prime })$ not only when $r=2$ but also when $r$ is arbitrary. Means we here utilize are elementary transforms and the notion of a sheaf with flag.