Birational geometry old and new

Grassi, Antonella

doi:10.1090/S0273-0979-08-01233-0

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ISSN 1088-9485(online) ISSN 0273-0979(print)

Birational geometry old and new

Author: Antonella Grassi
Journal: Bull. Amer. Math. Soc. 46 (2009), 99-123
MSC (2000): Primary 14E30; Secondary 14J99
Posted: October 27, 2008
MathSciNet review: 2457073
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Abstract: A classical problem in algebraic geometry is to describe quantities that are invariants under birational equivalence as well as to determine some convenient birational model for each given variety, a minimal model. One such quantity is the ring of objects which transform like a tensor power of a differential of top degree, known as the canonical ring. The histories of the existence of minimal models and the finite generation of the canonical ring are intertwined; minimal models and canonical rings constitute the major building blocks for the birational classification of algebraic varieties. In this paper we will discuss some of the ideas involved, recent advances on the existence of minimal models, some applications, and the (algebraic-geometric proof of the) finite generation of the canonical ring. These results have been long standing conjectures in algebraic geometry.

1. Arnaud Beauville, Complex algebraic surfaces, London Mathematical Society Lecture Note Series, vol. 68, Cambridge University Press, Cambridge, 1983. Translated from the French by R. Barlow, N. I. Shepherd-Barron and M. Reid. MR 732439 (85a:14024)
2. E. Bombieri, Canonical models of surfaces of general type, Inst. Hautes Études Sci. Publ. Math. 42 (1973), 171–219. MR 0318163 (47 #6710)
3. C. Birkar, P. Cascini, C. Hacon, J. M^cKernan, Existence of minimal model for varieties of log general type, http://math.mit.edu/ mckernan/Papers/papers.html, July 2008.
4. F. Catanese, Canonical rings and “special” surfaces of general type, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 175–194. MR 927956 (89f:14039)
5. A. Corti, P. Hacking, J, Kollár, R. Lazarsfeld, Lectures on Flips and Minimal Models, ArXiv:math.AG/0706.0494, 1-28, 2007.
6. Olivier Debarre, Higher-dimensional algebraic geometry, Universitext, Springer-Verlag, New York, 2001. MR 1841091 (2002g:14001)
7. Federigo Enriques, Le Superficie Algebriche, Nicola Zanichelli, Bologna, 1949 (Italian). MR 0031770 (11,202b)
8. Osamu Fujino and Shigefumi Mori, A canonical bundle formula, J. Differential Geom. 56 (2000), no. 1, 167–188. MR 1863025 (2002h:14091)
9. Mikhail M. Grinenko, Birational models of del Pezzo fibrations, Surveys in geometry and number theory: reports on contemporary Russian mathematics, London Math. Soc. Lecture Note Ser., vol. 338, Cambridge Univ. Press, Cambridge, 2007, pp. 122–157. MR 2306142 (2008d:14022), http://dx.doi.org/10.1017/CBO9780511721472.005
10. C. Hacon, J. M^cKernan, On the existence of flips, ArXiv:math.AG/0507597, 2005.
11. C. Hacon, J. M^cKernan, Existence of minimal models for variety of log general type II, preprint, July 2008.
12. Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157 (57 #3116)
13. Y. Kawamata, Flops connect minimal models, ArXiv:math.AG/0704.1013 2007, 1-5.
14. Y. Kawamata, Finite generation of a canonical ring, ArXiv:math.AG/0804.315 2008, 1-45.
15. Yujiro Kawamata, Katsumi Matsuda, and Kenji Matsuki, Introduction to the minimal model problem, Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, North-Holland, Amsterdam, 1987, pp. 283–360. MR 946243 (89e:14015)
16. János Kollár, Flops, Nagoya Math. J. 113 (1989), 15–36. MR 986434 (90e:14011)
17. J. Kollár et al., Flips and Abundance for Algebraic Threefolds, Asterisque 211 (1992), 21-45
18. János Kollár, Karen E. Smith, and Alessio Corti, Rational and nearly rational varieties, Cambridge Studies in Advanced Mathematics, vol. 92, Cambridge University Press, Cambridge, 2004. MR 2062787 (2005i:14063)
19. János Kollár and Shigefumi Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998. With the collaboration of C. H. Clemens and A. Corti; Translated from the 1998 Japanese original. MR 1658959 (2000b:14018)
20. K. Kodaira Collected Works, vol. III, Princeton University Press, 1975.
21. Robert Lazarsfeld, Positivity in algebraic geometry. I, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 48, Springer-Verlag, Berlin, 2004. Classical setting: line bundles and linear series. MR 2095471 (2005k:14001a)
22. Robert Lazarsfeld, Positivity in algebraic geometry. II, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 49, Springer-Verlag, Berlin, 2004. Positivity for vector bundles, and multiplier ideals. MR 2095472 (2005k:14001b)
23. Kenji Matsuki, Introduction to the Mori program, Universitext, Springer-Verlag, New York, 2002. MR 1875410 (2002m:14011)
24. J. M^cKernan, The Sarkisov Program, http://www.mfo.de/programme/schedule/2007/40/OWR
25. Shigefumi Mori, Threefolds whose canonical bundles are not numerically effective, Ann. of Math. (2) 116 (1982), no. 1, 133–176. MR 662120 (84e:14032), http://dx.doi.org/10.2307/2007050
26. Shigefumi Mori, Flip theorem and the existence of minimal models for 3-folds, J. Amer. Math. Soc. 1 (1988), no. 1, 117–253. MR 924704 (89a:14048), http://dx.doi.org/10.1090/S0894-0347-1988-0924704-X
27. D. Mumford, The canonical ring of an algebraic variety, Appendix to Zariski's paper ``The theorem of Riemann-Roch for high multiples of an effective divisor on an algebraic surface'', Ann. of Math. (2) 76 (1962), 612-615.
28. Miles Reid, Canonical 3-folds, Journées de Géometrie Algébrique d’Angers, Juillet 1979/Algebraic Geometry, Angers, 1979, Sijthoff & Noordhoff, Alphen aan den Rijn, 1980, pp. 273–310. MR 605348 (82i:14025)
29. Miles Reid, Twenty-five years of 3-folds—an old person’s view, Explicit birational geometry of 3-folds, London Math. Soc. Lecture Note Ser., vol. 281, Cambridge Univ. Press, Cambridge, 2000, pp. 313–343. MR 1798985 (2002b:14001)
30. Y.-T. Siu A general non-vanishing theorem and an analytic proof of the finite generation of the canonical ring, arXiv:math.AG/0610740
31. I. R. Shafarevich, Basic algebraic geometry, Springer-Verlag, New York, 1974. Translated from the Russian by K. A. Hirsch; Die Grundlehren der mathematischen Wissenschaften, Band 213. MR 0366917 (51 #3163)
32. V. V. Shokurov, Numerical geometry of algebraic varieties, (Berkeley, Calif., 1986) Amer. Math. Soc., Providence, RI, 1987, pp. 672–681. MR 934269 (89g:14006)
33. V. V. Shokurov, Prelimiting flips, Tr. Mat. Inst. Steklova 240 (2003), no. Biratsion. Geom. Linein. Sist. Konechno Porozhdennye Algebry, 82–219; English transl., Proc. Steklov Inst. Math. 1 (240) (2003), 75–213. MR 1993750 (2004k:14024)
34. Kenji Ueno, Classification theory of algebraic varieties and compact complex spaces, Lecture Notes in Mathematics, Vol. 439, Springer-Verlag, Berlin, 1975. Notes written in collaboration with P. Cherenack. MR 0506253 (58 #22062)
35. P. M. H. Wilson, On the canonical ring of algebraic varieties, Compositio Math. 43 (1981), no. 3, 365–385. MR 632435 (83g:14014)
36. Oscar Zariski, The theorem of Riemann-Roch for high multiples of an effective divisor on an algebraic surface, Ann. of Math. (2) 76 (1962), 560–615. MR 0141668 (25 #5065)

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