Hostname: page-component-7479d7b7d-8zxtt Total loading time: 0 Render date: 2024-07-10T13:00:31.168Z Has data issue: false hasContentIssue false
  • English
  • Français

Flops

Published online by Cambridge University Press:  22 January 2016

János Kollár
János Kollár*
Affiliation:
Institute for Advanced Study, Princeton, New Jersey, U.S.A.
*
University of Utah, Salt Lake City UT 84112, U.S.A.
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The aim of this note is to study birational transformations of three-folds with nef canonical classes. One would like to write any such map as a composite of certain simple and basic transformations.

A special case was first considered by Kulikov [Ku] and later extensively studied by several authors. The next main conceptual step was Reid’s study [R2] of small resolutions of terminal threefold singularities. He obtained the elementary birational transformations as follows: find a copy of such a small resolution inside the threefold and then replace it with another resolution. This approach was extended by Kawamata [K3] to canonical singularities.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 113 , March 1989 , pp. 15 - 36
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1989

References

[A1] Artin, M., Algebraisation of formal moduli: II Existence of modifications Ann. of Math., 91 (1970), 88–135.Google Scholar
[A2] Artin, M., Néron models, in Arithmetic Geometry, Springer 1986, pp. 213–230.CrossRefGoogle Scholar
[Be] Benveniste, X., Sur l’anneau canonique de certaines variétés de dimension trois, Invent. Math., 73 (1983), 157–164.CrossRefGoogle Scholar
[Bi] Bingener, J., On the existence of analytic contractions, Invent. Math., 64 (1981), 25–67,CrossRefGoogle Scholar
[F] Flenner, H., Divisorenklassengruppen quasihomogener Singularitaten J.f.r.u.a. Math., 328 (1981), 128–160.Google Scholar
[G-M] Goresky, M.–MacPherson, R., Intersection homology II. Invent. Math., 72 (1983), 77–129.Google Scholar
[G] Grothendieck, A., Cohomologie locale et théorèmes de Lefschetz locaux et globaux, North Holland, 1968.Google Scholar
[H] Hanamura, M., On the birational automorphism groups of algebraic varieties, Compositio Math., 63 (1987) 123–142.Google Scholar
[K1] Kawamata, Y., On the finiteness of generators of a pluricanonical ring for a threefold of general type, Amer. J. Math., 106 (1984), 1503-1512.CrossRefGoogle Scholar
[K2] Kawamata, Y., The cone of curves of algebraic varieties, Ann. of Math., 119 (1984), 603–633.CrossRefGoogle Scholar
[K3] Kawamata, Y., Crepant blowing ups of three dimensional canonical singularities, and applications to degenerations of surfaces, Ann. of Math., 127 (1988), 93–163.CrossRefGoogle Scholar
[K-M-M] Kawamata, Y., -Matsuda, K.–Matsuki, K., Introduction to the minimal model problem, Adv. Stud. Pure Math., vol. 10, Alg. Geom., Sendai 1985 Oda, T. ed., 283–360.Google Scholar
[K-M] Kawamata, Y., -Matsuki, K., The number of minimal models for a threefold of general type is finite, Math. Ann., 276 (1987), 595–598.CrossRefGoogle Scholar
[Ko] Kollár, J., The cone theorem, Ann. of Math., 120 (1984), 1–5.CrossRefGoogle Scholar
[K-S-B] Kollár, J., -Shepherd-Barron, N., Threefolds and deformations of surface singularities, Invent. Math., 91 (1988), 299–338.Google Scholar
[Ku] Kulikov, V., Degenerations of KS surfaces and Enriques surfaces, Math. USSR Izv., 11 (1977), 957–989.CrossRefGoogle Scholar
[Ma-Mu] Matsusaka, T.–Mumford, D., Two fundamental theorems on deformations of polarised varieties, Amer. J. Math., 86 (1964), 668–684.CrossRefGoogle Scholar
[Mi-Mo] Miyaoka, Y.–Mori, S., A numerical criterion of uniruledness, Ann. of Math., 124 (1086), 65–69.CrossRefGoogle Scholar
[Mo1] Mori, S., On three dimensional terminal singularities, Nagoya Math. J., 98 (1985), 43–66.CrossRefGoogle Scholar
[Mo2], Flip theorem and the existence of minimal models for threefolds, J. Amer. Math. Soc., 1 (1988),Google Scholar
[M] Morrison, D., The birational geometry of surfaces with rational double points, Math. Ann., 271 (1985), 415–438.Google Scholar
[P-P] Persson, U.–Pinkham, H., Degeneration of surfaces with trivial canonical bundle, Ann. of Math., 113 (1981), 45–66.CrossRefGoogle Scholar
[P] Pinkham, H., Factorisation of birational maps in dimension three, Proc. Symp. Pure Math., 40 (1983), 343–371.CrossRefGoogle Scholar
[R1] Reid, M., Canonical threefolds, in Geometrie Algebrique, Angers, 1980, Sijthoff and Noordhoff, 273–310.Google Scholar
[R2] Reid, M., Minimal models of threefolds, in Algebraic Varieties and Analytic Varieties, Adv. Stud, in Pure Math., 1 (1983), Kinokuniya-North-Holland, 131–180.Google Scholar
[R3] Reid, M., Projective morphisms according to Kawamata, preprint, 1983.Google Scholar
[R4] Reid, M., Young persons guide to canonical singularities, Proc. Symp. Pure Math. vol. 46, Alg. Geom., Bowdoin 1985, 345–414.Google Scholar
[Sa] Saito, M., Modules de Hodge polarisable, to appear.Google Scholar
[S-B1] Shepherd-Barron, N., Degeneration with numerically effective canonical divisor, in The birational theory of degenerations, Birkhauser, 1983.Google Scholar
[S-B2] Shepherd-Barron, N., The topology of rational threefold singularities, manuscript, 1986.Google Scholar
[Sh] Shokurov, V. V., Non vanishing theorem, Math. USSR Izv., 19 (1985),Google Scholar