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

Abstract

Let X be an irreducible projective variety of dimension n in a projective space and let x be a point of X. Denote by 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 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, Küchle and Lazarsfeld's work on Seshadri numbers.