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

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

2004-07-18

arxiv.org/abs/math/0407311v1

Mathematics

Algebraic Geometry

to appear in Compositio Math

Scientific paper

Let $X$ be an irreducible projective variety of dimension $n$ in a projective space and let $x$ be a point of $X$. Denote by ${\rm Curves}_d(X,x)$ the space of curves of degree $d$ lying on $X$ and passing through $x$. We will show that the number of components of ${\rm Curves}_d(X,x)$ 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-K\"uchle-Lazarsfeld's work on Seshadri numbers.

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