Flips and variation of moduli schemes of sheaves on a surface

Details Flips and variation of moduli schemes of sheaves on a surface Flips and variation of moduli schemes of sheaves on a surface

2008-11-21

arxiv.org/abs/0811.3522v2

Mathematics

Algebraic Geometry

Revised; Observations of main theorem are extended to the case where the Kodaira dimension of the underlying surface is positi

Scientific paper

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 its Kodaira dimension is positive, 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.

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