Abstract
Let $H$ be an ample line bundle on a non-singular projective surface $X$, and $M(H)$ the coarse moduli scheme of rank-two $H$-semistable sheaves with fixed Chern classes on $X$. We show that if $H$ changes and passes through walls to get closer to $K_X$, then $M(H)$ undergoes natural flips with respect to canonical divisors. When $X$ is minimal and $\kappa(X)\geq 1$, this sequence of flips terminates in $M(H_X)$; $H_X$ is an ample line bundle lying so closely to $K_X$ that the canonical divisor of $M(H_X)$ is nef. Remark that so-called Thaddeus-type flips somewhat differ from flips with respect to canonical divisors.
Citation
Kimiko Yamada. "Flips and variation of moduli scheme of sheaves on a surface." J. Math. Kyoto Univ. 49 (2) 419 - 425, 2009. https://doi.org/10.1215/kjm/1256219165
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