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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.83.

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Geometric invariant theory and flips
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by Michael Thaddeus PDF
J. Amer. Math. Soc. 9 (1996), 691-723 Request permission


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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Additional Information
  • Michael Thaddeus
  • Affiliation: St. John’s College, Oxford, England
  • Address at time of publication: Department of Mathematics, Harvard University, 1 Oxford St., Cambridge, Massachusetts 02138
  • Email:
  • Received by editor(s): November 11, 1994
  • Received by editor(s) in revised form: March 23, 1995
  • © Copyright 1996 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 9 (1996), 691-723
  • MSC (1991): Primary 14L30, 14D20
  • DOI:
  • MathSciNet review: 1333296