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Tr. Mat. Inst. Steklova, 2003, Volume 240, Pages 82–219 (Mi tm384)  

This article is cited in 53 scientific papers (total in 54 papers)

Prelimiting Flips

V. V. Shokurov

Johns Hopkins University

Abstract: The paper discusses an inductive approach to constructing log flips. In addition to special termination and thresholds, we introduce two new ingredients: the saturation of linear systems, and families of divisors with confined singularities. We state conjectures concerning these notions in any dimension and prove them in general in dimension $\le 2$. This allows us to construct prelimiting flips (pl flips) and all log flips in dimension 4 and to prove the stabilization of an asymptotically saturated system of birationally free (b-free) divisors under certain conditions in dimension 3. In dimension 3, this stabilization upgrades pl flips to directed quasiflips. It also gives for the first time a proof of the existence of log flips that is algebraic in nature, that is, via f.g. algebras, as opposed to geometric flips. It accounts for all the currently known flips and flops, except possibly for flips arising from geometric invariant theory.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2003, 240, 75–213

Bibliographic databases:
UDC: 512.7
Received in May 2002
Language:

Citation: V. V. Shokurov, “Prelimiting Flips”, Birational geometry: Linear systems and finitely generated algebras, Collected papers, Tr. Mat. Inst. Steklova, 240, Nauka, MAIK Nauka/Inteperiodika, M., 2003, 82–219; Proc. Steklov Inst. Math., 240 (2003), 75–213

Citation in format AMSBIB
\Bibitem{Sho03}
\by V.~V.~Shokurov
\paper Prelimiting Flips
\inbook Birational geometry: Linear systems and finitely generated algebras
\bookinfo Collected papers
\serial Tr. Mat. Inst. Steklova
\yr 2003
\vol 240
\pages 82--219
\publ Nauka, MAIK Nauka/Inteperiodika
\publaddr M.
\mathnet{http://mi.mathnet.ru/tm384}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1993750}
\zmath{https://zbmath.org/?q=an:1082.14019}
\transl
\jour Proc. Steklov Inst. Math.
\yr 2003
\vol 240
\pages 75--213


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    This publication is cited in the following articles:
    1. V. A. Iskovskikh, V. V. Shokurov, “Preface”, Proc. Steklov Inst. Math., 240 (2003), 1–3  mathnet
    2. V. A. Iskovskikh, “b-Divisors and Functional Algebras According to Shokurov”, Proc. Steklov Inst. Math., 240 (2003), 4–15  mathnet  mathscinet  zmath
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    4. Yu. G. Prokhorov, “On the Zariski Decomposition Problem”, Proc. Steklov Inst. Math., 240 (2003), 37–65  mathnet  mathscinet  zmath
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    6. V. V. Shokurov, “Letters of a Bi-rationalist V: Mld's and Termination of Log Flips”, Proc. Steklov Inst. Math., 246 (2004), 315–336  mathnet  mathscinet  zmath
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