Geodesic
Birational models and flips
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 60 (2005) no. 1, pp. 27-94 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

This survey treats two chapters in the theory of log minimal models, namely, the chapter on different notions of models in this theory and the chapter on birational flips, that is, log flips, mainly in dimension 3. Our treatment is based on ideas and results of the second author: his paper on log flips (and also on material from the University of Utah workshop) for the first chapter, and his paper on prelimiting flips (together with surveys of these results by Corti and Iskovskikh) for the second chapter, where a complete proof of the existence of log flips in dimension 3 is given. At present, this proof is the simplest one, and the authors hope that it can be understood by a broad circle of mathematicians. 
@article{RM_2005_60_1_a1,
     author = {V. A. Iskovskikh and V. V. Shokurov},
     title = {Birational models and flips},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {27--94},
     year = {2005},
     volume = {60},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/RM_2005_60_1_a1/}
}
TY  - JOUR
AU  - V. A. Iskovskikh
AU  - V. V. Shokurov
TI  - Birational models and flips
JO  - Trudy Matematicheskogo Instituta imeni V.A. Steklova
PY  - 2005
SP  - 27
EP  - 94
VL  - 60
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/RM_2005_60_1_a1/
LA  - en
ID  - RM_2005_60_1_a1
ER  - 
%0 Journal Article
%A V. A. Iskovskikh
%A V. V. Shokurov
%T Birational models and flips
%J Trudy Matematicheskogo Instituta imeni V.A. Steklova
%D 2005
%P 27-94
%V 60
%N 1
%U http://geodesic.mathdoc.fr/item/RM_2005_60_1_a1/
%G en
%F RM_2005_60_1_a1
V. A. Iskovskikh; V. V. Shokurov. Birational models and flips. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 60 (2005) no. 1, pp. 27-94. http://geodesic.mathdoc.fr/item/RM_2005_60_1_a1/

[1] F. Ambro, “Quasi-log varieties”, Tr. MIAN, 240, 2003, 220–239 | MR | Zbl

[2] V. V. Batyrev, “The cone of effective divisors on threefolds”, Contemp. Math., 131. Part 3 (1992), 337–352 | MR | Zbl

[3] A. A. Borisov, V. V. Shokurov, “Napravlennye ratsionalnye priblizheniya s nekotorymi prilozheniyami v algebraicheskoi geometrii”, Tr. MIAN, 240, 2003, 73–81 | MR | Zbl

[4] I. A. Cheltsov, “O gladkoi chetyrekhmernoi kvintike”, Matem. sb., 191:9 (2000), 139–162 | MR

[5] A. Corti, Semistable 3-fold flips, arXiv:alg-geom/9505035 | MR

[6] A. Corti, “Factoring birational maps of threefolds after Sarkisov”, J. Algebraic Geom., 4:2 (1995), 223–254 | MR | Zbl

[7] A. Corti, 3-fold flips after Shokurov, Preprint, Univ. of Chicago, Chicago, 2003 | MR

[8] V. A. Iskovskikh, “Biratsionalnaya zhestkost giperpoverkhnostei Fano v ramkakh teorii Mori”, UMN, 56:2 (2001), 3–86 | MR | Zbl

[9] V. A. Iskovskikh, “b-Divizory i funktsionalnye algebry po Shokurovu”, Tr. MIAN, 240, 2003, 8–20 | MR | Zbl

[10] V. A. Iskovskikh, “O rabote Shokurova “Prelimiting flips””, Tr. MIAN, 240, 2003, 21–42 | MR | Zbl

[11] Y. Kawamata, “Crepant blowing-up of 3-dimensional canonical singularities and its application to degenerations of surfaces”, Ann. of Math. (2), 127:1 (1988), 93–163 | DOI | MR | Zbl

[12] Y. Kawamata, “Termination of log flips for algebraic 3-folds”, Internat. J. Math., 3:5 (1992), 653–659 | DOI | MR | Zbl

[13] Y. Kawamata, “Abundance theorem for minimal threefolds”, Invent. Math., 108:2 (1992), 229–246 | DOI | MR | Zbl

[14] Y. Kawamata, “Moderate degenerations of algebraic surfaces”, Complex Algebraic Varieties (Bayreuth, 1990), Lecture Notes in Math., 1507, eds. K. Hulek et al, Springer-Verlag, Berlin, 1992, 113–132 | MR

[15] Y. Kawamata, “Semistable minimal models of threefolds in positive or mixed characteristic”, J. Algebraic Geom., 3:3 (1994), 463–491 | MR | Zbl

[16] Y. Kawamata, “On the cone of divisors of Calabi–Yao fiber spaces”, Internat. J. Math., 8:5 (1997), 665–687 | DOI | MR | Zbl

[17] Y. Kawamata, K. Matsuda, K. Matsuki, “Introduction to the minimal model problem”, Algebraic Geometry (Sendai, 1985), Adv. Stud. Pure Math., 10, ed. T. Oda, North-Holland, Amsterdam, 1987, 283–360 | MR

[18] Y. Kawamata, K. Matsuki, “The number of the minimal models for a 3-fold of general type is finite”, Math. Ann., 276:4 (1987), 595–598 | DOI | MR | Zbl

[19] S. Keel, K. Matsuki, J. McKernan, “Log abundance theorem for threefolds”, Duke Math. J., 75:1 (1994), 99–119 ; corrections: ibid. 122:3 (2004), 625–630 | DOI | MR | Zbl | DOI | MR | Zbl

[20] J. Kollár, “Flops”, Nagoya Math. J., 113 (1989), 15–36 | MR | Zbl

[21] J. Kollár (ed.), Flips and Abundance for Algebraic Threefolds, A Summer Seminar at the University of Utah (Salt Lake City, 1991), Astérisque, 211, Soc. Math. France, Paris, 1992 | MR

[22] J. Kollár, S. Mori, with the collaboration of C. H. Clemens, A. Corti, Birational Geometry of Algebraic Varieties, Cambridge Tracts in Math., 134, Cambridge Univ. Press, Cambridge, 1998 | MR | Zbl

[23] J. Kollár, N. I. Shepherd-Barron, “Threefolds and deformations of surface singularities”, Invent. Math., 91:2 (1988), 299–338 | DOI | MR | Zbl

[24] V. S. Kulikov, “Vyrozhdeniya K3 poverkhnostei i poverkhnostei Enrikvesa”, Izv. AN SSSR. Ser. matem., 41:5 (1977), 1008–1042 | MR | Zbl

[25] K. Matsuki, Introduction to the Mori Program, Springer-Verlag, New York, 2002 | MR | Zbl

[26] S. Mori, “Flip theorem and the existence of minimal models for 3-folds”, J. Amer. Math. Soc., 1:1 (1988), 117–253 | DOI | MR | Zbl

[27] N. Nakayama, “The lower semicontinuity of the plurigenera of complex varieties”, Internat. J. Math., 8:5 (1997), 551–590 | MR

[28] U. Persson, On degenerations of algebraic surfaces, Mem. Amer. Math. Soc., 189, 1977 | MR | Zbl

[29] Yu. G. Prokhorov, “O polustabilnykh styagivaniyakh Mori”, Izv. RAN. Ser. matem., 68:2 (2004), 147–158 | MR

[30] M. Reid, “Canonical 3-folds”, Journeés de Géométrie Algébrique d'Angers (Angers, 1979), ed. A. Beauville, Sijthoff and Noordhoff, Alphen, 1980, 273–310 | MR

[31] M. Reid, “Minimal models of canonical 3-folds”, Algebraic Varieties and Analytic Varieties (Tokyo, 1981), Adv. Stud. Pure Math., 1, North-Holland, Amsterdam, 1983, 131–180 | MR

[32] M. Reid, “Young person's guide to canonical singularities”, Proc. Sympos. Pure Math., 46:1 (1987), 345–414 | MR | Zbl

[33] V. V. Shokurov, “Teorema o neobraschenii v nul”, Izv. AN SSSR. Ser. matem., 49:3 (1985), 635–651 | MR

[34] V. V. Shokurov, “Trekhmernye logperestroiki”, Izv. AN SSSR. Ser. matem., 56:1 (1992), 105–203 | MR | Zbl

[35] V. V. Shokurov, “Semistable 3-fold flips”, Izv. AN SSSR. Ser. matem., 57:2 (1993), 165–222 | MR | Zbl

[36] V. V. Shokurov, “3-fold log models”, J. Math. Sci., 81:3 (1996), 2667–2699 | DOI | MR | Zbl

[37] V. V. Shokurov, “Prelimiting flips”, Tr. MIAN, 240, 2003, 82–219 | MR | Zbl

[38] V. V. Shokurov, “Letters of a bi-rationalist. IV. Geometry of log flips”, Algebraic Geometry, eds. M. C. Beltrametti et al., de Gruyter, Berlin, 2002, 313–328 | MR | Zbl

[39] V. V. Shokurov, “Pisma o biratsionalnom. V: M.l.d. i obryv logflipov”, Tr. MIAN, 246, 2004, 328–351 | MR | Zbl

[40] H. Takagi, 3-fold log flips according to V. V. Shokurov, Preprint, 1999

[41] S. Tsunoda, “Degenerations of surfaces”, Internat. J. Math., 8:5 (1997), 755–764 | MR