Flips and variation of moduli schemes of sheaves on a surface

TL;DR

This paper studies how moduli schemes of rank-two sheaves on a surface change through flips as the polarization varies, especially near the canonical divisor, revealing a sequence of flips terminating at a nef canonical divisor.

Contribution

It demonstrates that moduli schemes undergo a sequence of flips with respect to canonical divisors when the polarization varies, and this sequence terminates under certain conditions.

Findings

Moduli schemes experience flips when changing polarization near the canonical divisor.

The flip sequence terminates for minimal surfaces with positive Kodaira dimension.

The final moduli scheme has a nef canonical divisor.

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 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.