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A new Castelnuovo bound for two codimensional subvarieties of $ {\bf P}\sp r$


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 $ X$ be a smooth $ n$-dimensional projective subvariety of $ {\mathbb{P}^r}(\mathbb{C}),(r \geq 3)$. For any positive integer $ k,X$ is said to be $ k$-normal if the natural map $ {H^0}({\mathbb{P}^r},{\mathcal{O}_{\mathbb{P}r}}(k)) \to {H^0}(X,{\mathcal{O}_X}(k))$ is surjective. Mumford and Bayer showed that $ X$ is $ k$-normal if $ k \geq (n + 1)(d - 2) + 1$ where $ d = \deg (X)$. Better inequalities are known when $ n$ is small (Gruson-Peskine, Lazarsfeld, Ran). In this paper we consider the case $ n = r - 2$, which is related to Hartshorne's conjecture on complete intersections, and we show that if $ k \geq d + 1 + (1/2)r(r - 1) - 2r$ then $ X$ is $ k$-normal and $ {I_X}$, the ideal sheaf of $ X$ in $ {\mathbb{P}^r}$, is $ (k + 1)$-regular.

About these problems Lazarsfeld developed a technique based on generic projections of $ X$ in $ {\mathbb{P}^{n + 1}}$; our proof is an application of some recent results of Ran's (on the secants of $ X$): we show that in our case there exists a projection such generic as Lazarsfeld requires.

When $ r \geq 6$ we also give a better inequality: $ k \geq d - 1 + (1/2)r(r - 1) - (r - 1)[(r + 4)/2]$ ([] means: integer part); it is obtained by refining Lazarsfeld's technique with the help of some results of ours about $ k$-normality.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1992-1074747-6
Keywords: Projective varieties, $ k$-normality
Article copyright: © Copyright 1992 American Mathematical Society