A new Castelnuovo bound for two codimensional subvarieties of

Author:
Alberto Alzati

Journal:
Proc. Amer. Math. Soc. **114** (1992), 607-611

MSC:
Primary 14M07; Secondary 14F05

MathSciNet review:
1074747

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Abstract: Let be a smooth -dimensional projective subvariety of . For any positive integer is said to be -normal if the natural map is surjective. Mumford and Bayer showed that is -normal if where . Better inequalities are known when is small (Gruson-Peskine, Lazarsfeld, Ran). In this paper we consider the case , which is related to Hartshorne's conjecture on complete intersections, and we show that if then is -normal and , the ideal sheaf of in , is -regular.

About these problems Lazarsfeld developed a technique based on generic projections of in ; our proof is an application of some recent results of Ran's (on the secants of ): we show that in our case there exists a projection such generic as Lazarsfeld requires.

When we also give a better inequality: ([] means: integer part); it is obtained by refining Lazarsfeld's technique with the help of some results of ours about -normality.

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DOI:
https://doi.org/10.1090/S0002-9939-1992-1074747-6

Keywords:
Projective varieties,
-normality

Article copyright:
© Copyright 1992
American Mathematical Society