Flips and variation of moduli scheme of sheaves on a surface (Q2654729)

Let \(H\) be an ample line bundle on a nonsingular projective surface \(X\) over \(\mathbb C\). Let also \(M(H)\) be the coarse moduli scheme for rank 2 \(H\)-semistable sheaves with fixed Chern classes on \(X\). It is proven that if \(H\) changes and passes through walls to get closer to \(K_X\) then \(M(H)\) undergoes flips with respect to canonical divisors. When \(X\) is minimal and \(\kappa (X) \geq 1\) this sequense of flips terminates in \(M(H_X)\). \(H_X\) is ample line bundle defined as follows: it is so close to \(K_X\) that the canonical divisor of \(M(H_X)\) is nef. Flips are understood in the sense of Kollar and Mori, differ from Thaddeus-type flips and are described moduli-theoretically in the case of interest.