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

2004/07/18 by Jun-Muk Hwang, Hwang, Jun-Muk
Computer Science · Mathematics · #14J40 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Cryptography and Residue Arithmetic #FOS: Mathematics #Polynomial and algebraic computation #math.AG #msc:14J40

paper · pdf · doi:10.48550/arxiv.math/0407311

to appear in Compositio Math

arxiv created 2004/07/18 · openalex publication_date 2004/07/18 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 \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 ≥ 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.

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