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On a criterion for hyperplane sections

Published online by Cambridge University Press:  24 October 2008

Lucian Bǎdescu
Lucian Bǎdescu
Affiliation:
Department of Mathematics, INCREST, 79622 Bucharest, Romania
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Extract

Throughout this paper we shall fix an algebraically closed field k. Consider the following:

Problem. Let (Y, L) be a normal polarized variety over k, i.e. a normal projective variety Y over k together with an ample line bundle L on Y. Then one may ask under which conditions the following statement holds:

(*) Every normal projective variety X containing Y as an ample Cartier divisor such that the normal bundle of Y in X is L, is isomorphic to the projective cone over Y.

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 103 , Issue 1 , January 1988 , pp. 59 - 67
Copyright
Copyright © Cambridge Philosophical Society 1988

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References

REFERENCES

[1]Atiyah, M. F. and Macdonald, I. G.. Introduction to commutative algebra (Adison-Wesley, 1969).Google Scholar
[2]Badescu, L.. On ample divisors. Nagoya Math. J. 86 (1982), 155–171.CrossRefGoogle Scholar
[3]Badescu, L.. On ample divisors II. In Proceedings of the Week of Algebraic Geometry, Bucharest 1980. Teubner-Texte Math., Band 40, (Teubner-Verlag, 1981), 12–32.Google Scholar
[4]Badescu, L.. Hyperplane sections and deformations. In Algebraic Geometry, Bucharest 1982. Lecture Notes in Math. vol. 1056 (Springer-Verlag, 1984), 1–33.CrossRefGoogle Scholar
[5]Beltrametti, M. and Robbiano, L.. Introduction to the theory of weighted projective spaces. Expo. Math. 4 (1986), 111–162.Google Scholar
[6]Buium, A.. Weighted projective spaces as ample divisors. Rev. Roumaine Math. Pures Appl. 26, (1981), 833–842.Google Scholar
[7]Delorme, C.. Espaces projectifs anisotropes. Bull. Soc. Math. France, 103 (1975), 203–223.CrossRefGoogle Scholar
[8]Dolgachev, I.. Weighted projective varieties. In Group Actions and Vector Fields, in Lecture Notes in Math. vol. 956 (Springer-Verlag, 1982), pp. 34–71.CrossRefGoogle Scholar
[9]Fujita, T.. Rational retractions onto ample divisors. Sci. Papers College Art. and Sciences Univ. Tokyo 33 (1983), 33–39.Google Scholar
[10]Grothendieck, A.. Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (North-Holland, 1968).Google Scholar
[11]Hochster, M. and Roberts, J. L.. Rings of invariants of reductive groups acting on regular rings are Cohen–Macaulay. Adv. in Math. 13 (1974), 115–175.CrossRefGoogle Scholar
[12]Mori, S.. On a generalization of complete intersections. J. Math. Kyoto Univ. 15 (1975), 619–646.Google Scholar
[13]Pinkham, H.. Deformations of Algebraic Varieties with G n-action. Astérisque 2o (Sociéte Math. France, 1974).Google Scholar
[14]Schlessinger, M.. Functors on Artin rings. Trans. Amer. Math. Soc. 130 (1968), 208–222.CrossRefGoogle Scholar
[15]Schlessinger, M.. Rigidity of quotient singularities. Invent. Math. 14 (1971), 17–26.CrossRefGoogle Scholar
[16]Schlessinger, M.. On rigid singularities. Rice Univ. Stud. 59 (1973), 147–162.Google Scholar
[17]Scorza, G.. Sopre una certa classe di varietà razionali. R. C. Mat. Palermo 28 (1909), 400–401.CrossRefGoogle Scholar
[18]Scorza, G.. Sulle varietà di Segre. Opere scelte 376–386. (Vol. 1).Google Scholar
[19]Serre, J.-P.. Faisceaux algébriques cohérents. Annals Math. 61 (1955), 197–278.CrossRefGoogle Scholar
[20]Springer, T. A.. Invariant theory. Lecture Notes in Math. vol. 585 (Springer-Verlag, 1977).CrossRefGoogle Scholar