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On strengthening of the Kleiman-Chevalley projectivity criterion

Author: Michał Farnik
Journal: Proc. Amer. Math. Soc. 141 (2013), 4005-4013
MSC (2010): Primary 51N35; Secondary 14A10, 14M25
Published electronically: July 10, 2013
MathSciNet review: 3091791
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Abstract: We discuss the possibilities of strengthening the classical Kleiman-Chevalley projectivity criterion by exploring the properties of the Picard number and the maximal quasiprojective open subsets of a variety. We also prove two theorems which give a bound on how much the criterion can be strengthened.

References [Enhancements On Off] (What's this?)

  • 1. A. Białynicki-Birula, Finiteness of the number of maximal open subsets with good quotients, Transform. Groups, Vol. 3, No. 4, 1998, 301-319. MR 1657520 (99m:14089)
  • 2. M. Farnik, Z. Jelonek, A complete variety with infinitely many maximal quasiprojective open subsets, Demonstratio Math., 43 (2010), no. 2, 277-284. MR 2668476 (2011j:14001)
  • 3. W. Fulton, Introduction to toric varieties, Princeton Univ. Press, Princeton, New Jersey, 1993. MR 1234037 (94g:14028)
  • 4. H. Hironaka, On the theory of birational blowing up, Thesis, Harvard, 1960.
  • 5. Ch.D. Hacon, J. McKernan, Flips and flops, Proceedings of the International Congress of Mathematicians, 2010, 513-539. MR 2827807
  • 6. Z. Jelonek, On the projectivity of threefolds, Proc. Amer. Math. Soc., vol. 133, no. 9, 2005, 2539-2542. MR 2146196 (2005m:14069)
  • 7. S. Kleiman, Toward a numerical theory of ampleness, Annals of Math. 84, 1966, 293-344. MR 0206009 (34:5834)
  • 8. J. Kollár, Rational curves on algebraic varieties, Springer-Verlag, Berlin, Heidelberg, 1996. MR 1440180 (98c:14001)
  • 9. G. Kempf, F. Knudsen, D. Mumford, B. Saint-Donat, Toroidal embeddings, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1978. MR 0335518 (49 #299)
  • 10. I.R. Shafarevich, Basic Algebraic Geometry 2, Springer-Verlag, Berlin, Heidelberg, 1994. MR 1328834 (95m:14002)
  • 11. J. Włodarczyk, Embeddings in toric varieties and prevarieties, J. Algebraic Geometry 2, 1993, 705-726. MR 1227474 (94e:14070)
  • 12. J. Włodarczyk, Maximal quasiprojective subsets and the Kleiman-Chevalley quasiprojectivity criterion, J. Math. Sci. Univ. Tokyo 6, 1999, 41-47. MR 1683254 (2000i:14009)

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Additional Information

Michał Farnik
Affiliation: Instytut Matematyczny, Polska Akademia Nauk, św. Tomasza 30, 31-027 Kraków, Poland
Address at time of publication: Instytut Matematyczny, Polska Akademia Nauk, Śniadeckich 8, 00-956 Warszawa, Poland

Keywords: Picard number, maximal open quasiprojective set, Chevalley projectivity criterion, nonprojective varieties, toric varieties.
Received by editor(s): June 15, 2011
Received by editor(s) in revised form: January 7, 2012
Published electronically: July 10, 2013
Additional Notes: The author was partially supported by Polish MNiSW grant N N201 611740, 2011–2012
Communicated by: Lev Borisov
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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