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

الأصل · EN

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 ≥ 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-Lazarsfeld's work on Seshadri numbers.