Geometric invariant theory and flips

Thaddeus, Michael

doi:10.1090/S0894-0347-96-00204-4

Geometric invariant theory and flips

Author: Michael Thaddeus
Journal: J. Amer. Math. Soc. 9 (1996), 691-723
MSC (1991): Primary 14L30, 14D20
DOI: https://doi.org/10.1090/S0894-0347-96-00204-4
MathSciNet review: 1333296
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Abstract: We study the dependence of geometric invariant theory quotients on the choice of a linearization. We show that, in good cases, two such quotients are related by a flip in the sense of Mori, and explain the relationship with the minimal model program. Moreover, we express the flip as the blow-up and blow-down of specific ideal sheaves, leading, under certain hypotheses, to a quite explicit description of the flip. We apply these ideas to various familiar moduli problems, recovering results of Kirwan, Boden-Hu, Bertram-Daskalopoulos-Wentworth, and the author. Along the way we display a chamber structure, following Duistermaat-Heckman, on the space of all linearizations. We also give a new, easy proof of the Bialynicki-Birula decomposition theorem.

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