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A bound on the number of curves of a given degree through a general point of a projective variety

Published online by Cambridge University Press:  21 April 2005

Jun-Muk Hwang
Affiliation:
Korea Institute for Advanced Study, 207-43 Cheongryangri-dong, Seoul, 130-722, [email protected]
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Abstract

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Let X be an irreducible projective variety of dimension n in a projective space and let x be a point of X. Denote by Curvesd(X, x) the space of curves of degree d lying on X and passing through x. We will show that the number of components of Curvesd(X, x) for any smooth point x outside a subvariety of codimension $\geq 2$ is bounded by a number depending only on n and d. An effective bound is given. A key ingredient of the proof is an argument from Ein, Küchle and Lazarsfeld's work on Seshadri numbers.

Type
Research Article
Copyright
Foundation Compositio Mathematica 2005