On strengthening of the Kleiman-Chevalley projectivity criterion

Farnik, Michał

doi:10.1090/S0002-9939-2013-11695-3

On strengthening of the Kleiman-Chevalley projectivity criterion
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by Michał Farnik PDF

Proc. Amer. Math. Soc. 141 (2013), 4005-4013 Request permission

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

A. Białynicki-Birula, Finiteness of the number of maximal open subsets with good quotients, Transform. Groups 3 (1998), no. 4, 301–319. MR 1657520, DOI 10.1007/BF01234530
MichałFarnik and Zbigniew Jelonek, A complete variety with infinitely many maximal quasi-projective open subsets, Demonstratio Math. 43 (2010), no. 2, 277–284. MR 2668476
William Fulton, Introduction to toric varieties, Annals of Mathematics Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993. The William H. Roever Lectures in Geometry. MR 1234037, DOI 10.1515/9781400882526
H. Hironaka, On the theory of birational blowing up, Thesis, Harvard, 1960.
Christopher D. Hacon and James McKernan, Flips and flops, Proceedings of the International Congress of Mathematicians. Volume II, Hindustan Book Agency, New Delhi, 2010, pp. 513–539. MR 2827807
Zbigniew Jelonek, On the projectivity of threefolds, Proc. Amer. Math. Soc. 133 (2005), no. 9, 2539–2542. MR 2146196, DOI 10.1090/S0002-9939-05-07859-7
Steven L. Kleiman, Toward a numerical theory of ampleness, Ann. of Math. (2) 84 (1966), 293–344. MR 206009, DOI 10.2307/1970447
János Kollár, Rational curves on algebraic varieties, 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. 32, Springer-Verlag, Berlin, 1996. MR 1440180, DOI 10.1007/978-3-662-03276-3
G. Kempf, Finn Faye Knudsen, D. Mumford, and B. Saint-Donat, Toroidal embeddings. I, Lecture Notes in Mathematics, Vol. 339, Springer-Verlag, Berlin-New York, 1973. MR 0335518
Igor R. Shafarevich, Basic algebraic geometry. 2, 2nd ed., Springer-Verlag, Berlin, 1994. Schemes and complex manifolds; Translated from the 1988 Russian edition by Miles Reid. MR 1328834
Jarosław Włodarczyk, Embeddings in toric varieties and prevarieties, J. Algebraic Geom. 2 (1993), no. 4, 705–726. MR 1227474
Jarosław Włodarczyk, Maximal quasiprojective subsets and the Kleiman-Chevalley quasiprojectivity criterion, J. Math. Sci. Univ. Tokyo 6 (1999), no. 1, 41–47. MR 1683254