Effective algebraic degeneracy

TL;DR

This paper establishes an effective degree bound ensuring that any nonconstant entire holomorphic curve into a generic projective hypersurface of dimension n is algebraically degenerate, advancing understanding in complex algebraic geometry.

Contribution

It provides the first explicit degree bound guaranteeing algebraic degeneracy of entire curves on generic hypersurfaces of arbitrary dimension.

Findings

Proves algebraic degeneracy for entire curves when degree exceeds a specific bound

Derives an explicit lower bound for the degree of hypersurfaces

Advances the effective aspects of algebraic degeneracy theory

Abstract

We prove that any nonconstant entire holomorphic curve from the complex line C into a projective algebraic hypersurface X = X^n in P^{n+1}(C) of arbitrary dimension n (at least 2) must be algebraically degenerate provided X is generic if its degree d = deg(X) satisfies the effective lower bound: d larger than or equal to n^{{(n+1)}^{n+5}}.