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

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Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2003, Volume 240, Pages 82–219 (Mi tm384)

This article is cited in 62 scientific papers (total in 63 papers)

Prelimiting Flips

V. V. Shokurov

Johns Hopkins University

Full-text PDF (1423 kB) Citations (63)

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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.

Received in May 2002

Bibliographic databases:

Document Type: Article

UDC: 512.7

Language: English

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

Citation in format AMSBIB

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\by V.~V.~Shokurov

\paper Prelimiting Flips

\inbook Birational geometry: Linear systems and finitely generated algebras

\bookinfo Collected papers

\serial Trudy 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://mathscinet.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

Linking options:

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https://www.mathnet.ru/eng/tm/v240/p82

This publication is cited in the following 63 articles:

Izv. Math., 87:3 (2023), 616–640
Botero A.M., Gil Jose Ignacio Burgos, “Toroidal B-Divisors and Monge-Ampere Measures”, Math. Z., 300:1 (2022), 579–637
Yu. G. Prokhorov, “Equivariant minimal model program”, Russian Math. Surveys, 76:3 (2021), 461–542
Birkar C., “Generalised Pairs in Birational Geometry”, EMS Surv. Math. Sci., 8:1-2 (2021), 5–24
Dang N.-B., Favre Ch., “Spectral Interpretations of Dynamical Degrees and Applications”, Ann. Math., 194:1 (2021), 299–359
Shokurov V.V., “Skrepa Morphisms”, Pure Appl. Math. Q., 16:1, 3, SI (2020), 35–124
Garcia Barroso E.R., Gonzalez Perez P.D., Popescu-Pampu P., Ruggiero M., “Ultrametric Properties For Valuation Spaces of Normal Surface Singularities”, Trans. Am. Math. Soc., 372:12 (2019), 8423–8475
Holmes D., Pixton A., Schmitt J., “Multiplicativity of the Double Ramification Cycle”, Doc. Math., 24 (2019), 545–562
Cascini P., Tanaka H., “Purely Log Terminal Threefolds With Non-Normal Centres in Characteristic Two”, Am. J. Math., 141:4 (2019), 941–979
Oguiso K., “A Criterion For the Primitivity of a Birational Automorphism of a Calabi-Yau Manifold and An Application”, Math. Res. Lett., 25:1 (2018), 181–198
Patakfalvi Z., “Frobenius Techniques in Birational Geometry”, Algebraic Geometry: Salt Lake City 2015, Pt 1, Proceedings of Symposia in Pure Mathematics, 97, no. 1, eds. DeFernex T., Hassett B., Mustata M., Olsson M., Popa M., Thomas R., Amer Mathematical Soc, 2018, 489–517
Yu. G. Prokhorov, “The rationality problem for conic bundles”, Russian Math. Surveys, 73:3 (2018), 375–456
Fulger M., Lehmann B., “Zariski decompositions of numerical cycle classes”, J. Algebr. Geom., 26:1 (2017), 43–106
Birkar C., “Existence of flips and minimal models for 3-folds in char $p$”, Ann. Sci. Ec. Norm. Super., 49:1 (2016), 169–212
Birkar C., “Singularities on the base of a Fano type fibration”, J. Reine Angew. Math., 715 (2016), 125–142
Choi S.R., Park J., “Potentially non-klt locus and its applications”, Math. Ann., 366:1-2 (2016), 141–166
Odaka Yu., “Invariants of Varieties and Singularities Inspired By Kahler-Einstein Problems”, Proc. Jpn. Acad. Ser. A-Math. Sci., 91:4 (2015), 50–55
Kaveh K., Khovanskii A.G., “Note on the Grothendieck Group of Subspaces of Rational Functions and Shokurov'S Cartier B-Divisors”, Can. Math. Bul.-Bul. Can. Math., 57:3 (2014), 562–572
Ambro F., “An Injectivity Theorem”, Compos. Math., 150:6 (2014), 999–1023
Birkar C., Hu Zh., “Log Canonical Pairs With Good Augmented Base Loci”, Compos. Math., 150:4 (2014), 579–592

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

Труды Математического института имени В. А. Стеклова

Proceedings of the Steklov Institute of Mathematics

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