Birational geometry old and new

Grassi, Antonella

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

Birational geometry old and new

Author: Antonella Grassi
Journal: Bull. Amer. Math. Soc. 46 (2009), 99-123
MSC (2000): Primary 14E30; Secondary 14J99
DOI: https://doi.org/10.1090/S0273-0979-08-01233-0
Published electronically: October 27, 2008
MathSciNet review: 2457073
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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
2. E. Bombieri, Canonical models of surfaces of general type, Inst. Hautes Études Sci. Publ. Math. 42 (1973), 171–219. MR 318163
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
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
7. Federigo Enriques, Le Superficie Algebriche, Nicola Zanichelli, Bologna, 1949 (Italian). MR 0031770
8. Osamu Fujino and Shigefumi Mori, A canonical bundle formula, J. Differential Geom. 56 (2000), no. 1, 167–188. MR 1863025
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, https://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-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157
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, https://doi.org/10.2969/aspm/01010283
16. János Kollár, Flops, Nagoya Math. J. 113 (1989), 15–36. MR 986434, https://doi.org/10.1017/S0027763000001240
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
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
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
22. 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
23. Kenji Matsuki, Introduction to the Mori program, Universitext, Springer-Verlag, New York, 2002. MR 1875410
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, https://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, https://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—Germantown, Md., 1980, pp. 273–310. MR 605348
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
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-Heidelberg, 1974. Translated from the Russian by K. A. Hirsch; Die Grundlehren der mathematischen Wissenschaften, Band 213. MR 0366917
32. V. V. Shokurov, Numerical geometry of algebraic varieties, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) Amer. Math. Soc., Providence, RI, 1987, pp. 672–681. MR 934269
33. V. V. Shokurov, Prelimiting flips, Tr. Mat. Inst. Steklova 240 (2003), no. Biratsion. Geom. Lineĭn. Sist. Konechno Porozhdennye Algebry, 82–219; English transl., Proc. Steklov Inst. Math. 1(240) (2003), 75–213. MR 1993750
34. Kenji Ueno, Classification theory of algebraic varieties and compact complex spaces, Lecture Notes in Mathematics, Vol. 439, Springer-Verlag, Berlin-New York, 1975. Notes written in collaboration with P. Cherenack. MR 0506253
35. P. M. H. Wilson, On the canonical ring of algebraic varieties, Compositio Math. 43 (1981), no. 3, 365–385. MR 632435
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 141668, https://doi.org/10.2307/1970376