Skip to Main Content

Bulletin of the American Mathematical Society

The Bulletin publishes expository articles on contemporary mathematical research, written in a way that gives insight to mathematicians who may not be experts in the particular topic. The Bulletin also publishes reviews of selected books in mathematics and short articles in the Mathematical Perspectives section, both by invitation only.

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

The 2020 MCQ for Bulletin of the American Mathematical Society is 0.84.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Birational geometry old and new
HTML articles powered by AMS MathViewer

by Antonella Grassi PDF
Bull. Amer. Math. Soc. 46 (2009), 99-123 Request permission

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.
References
  • 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
  • E. Bombieri, Canonical models of surfaces of general type, Inst. Hautes Études Sci. Publ. Math. 42 (1973), 171–219. MR 318163
    • BCHM C. Birkar, P. Cascini, C. Hacon, J. McKernan, Existence of minimal model for varieties of log general type, http://math.mit.edu/ mckernan/Papers/papers.html, July 2008.
  • 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
    • fa A. Corti, P. Hacking, J, Kollár, R. Lazarsfeld, Lectures on Flips and Minimal Models, ArXiv:math.AG/0706.0494, 1–28, 2007.
  • Olivier Debarre, Higher-dimensional algebraic geometry, Universitext, Springer-Verlag, New York, 2001. MR 1841091, DOI 10.1007/978-1-4757-5406-3
  • Federigo Enriques, Le Superficie Algebriche, Nicola Zanichelli, Bologna, 1949 (Italian). MR 0031770
  • Osamu Fujino and Shigefumi Mori, A canonical bundle formula, J. Differential Geom. 56 (2000), no. 1, 167–188. MR 1863025
  • 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, DOI 10.1017/CBO9780511721472.005
    • HM1 C. Hacon, J. McKernan, On the existence of flips, ArXiv:math.AG/0507597, 2005. HM2 C. Hacon, J. McKernan, Existence of minimal models for variety of log general type II, preprint, July 2008.
  • Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
    • ka Y. Kawamata, Flops connect minimal models, ArXiv:math.AG/0704.1013 2007, 1–5. ka2 Y. Kawamata, Finite generation of a canonical ring, ArXiv:math.AG/0804.315 2008, 1–45.
  • 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, DOI 10.2969/aspm/01010283
  • János Kollár, Flops, Nagoya Math. J. 113 (1989), 15–36. MR 986434, DOI 10.1017/S0027763000001240
    • kola J. Kollár et al., Flips and Abundance for Algebraic Threefolds, Asterisque 211 (1992), 21-45
  • 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, DOI 10.1017/CBO9780511734991
  • 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, DOI 10.1017/CBO9780511662560
    • Ko K. Kodaira Collected Works, vol. III, Princeton University Press, 1975.
  • 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, DOI 10.1007/978-3-642-18808-4
  • 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, DOI 10.1007/978-3-642-18808-4
  • Kenji Matsuki, Introduction to the Mori program, Universitext, Springer-Verlag, New York, 2002. MR 1875410, DOI 10.1007/978-1-4757-5602-9
    • Mc J. McKernan, The Sarkisov Program, http://www.mfo.de/programme/schedule/2007/40/OWR
  • Shigefumi Mori, Threefolds whose canonical bundles are not numerically effective, Ann. of Math. (2) 116 (1982), no. 1, 133–176. MR 662120, DOI 10.2307/2007050
  • 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, DOI 10.1090/S0894-0347-1988-0924704-X
    • Mumford 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.
  • 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
  • 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
    • Siu Y.-T. Siu A general non-vanishing theorem and an analytic proof of the finite generation of the canonical ring, arXiv:math.AG/0610740
  • I. R. Shafarevich, Basic algebraic geometry, Die Grundlehren der mathematischen Wissenschaften, Band 213, Springer-Verlag, New York-Heidelberg, 1974. Translated from the Russian by K. A. Hirsch. MR 0366917
  • 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
  • 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
  • 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
  • P. M. H. Wilson, On the canonical ring of algebraic varieties, Compositio Math. 43 (1981), no. 3, 365–385. MR 632435
  • 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, DOI 10.2307/1970376
Similar Articles
  • Retrieve articles in Bulletin of the American Mathematical Society with MSC (2000): 14E30, 14J99
  • Retrieve articles in all journals with MSC (2000): 14E30, 14J99
Additional Information
  • Antonella Grassi
  • Affiliation: Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104
  • Email: grassi@math.upenn.edu
  • Received by editor(s): June 8, 2008
  • Published electronically: October 27, 2008
  • © Copyright 2008 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • 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
  • MathSciNet review: 2457073