Journal of Mathematics of Kyoto University

Flips and variation of moduli scheme of sheaves on a surface

Kimiko Yamada

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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.

Article information

J. Math. Kyoto Univ., Volume 49, Number 2 (2009), 419-425.

First available in Project Euclid: 22 October 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14J60: Vector bundles on surfaces and higher-dimensional varieties, and their moduli [See also 14D20, 14F05, 32Lxx]
Secondary: 14E05: Rational and birational maps 14D20: Algebraic moduli problems, moduli of vector bundles {For analytic moduli problems, see 32G13}


Yamada, Kimiko. Flips and variation of moduli scheme of sheaves on a surface. J. Math. Kyoto Univ. 49 (2009), no. 2, 419--425. doi:10.1215/kjm/1256219165.

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