Algebraic hyperbolicity of very general hypersurfaces in products of projective spaces

TL;DR

This paper investigates the algebraic hyperbolicity of very general hypersurfaces in products of projective spaces, providing a complete classification with a few specific exceptions and improving known bounds for hyperbolicity in projective spaces.

Contribution

The authors develop three techniques to determine algebraic hyperbolicity of hypersurfaces in product spaces, fully resolving the question except for certain bidegrees in P^3 x P^1 and refining bounds in P^n.

Findings

Complete classification of algebraic hyperbolicity for hypersurfaces in P^m x P^n

Identification of specific bidegrees where hyperbolicity remains unresolved

Improved bounds for hyperbolicity of hypersurfaces in P^n for n ≥ 5

Abstract

We study the algebraic hyperbolicity of very general hypersurfaces in $\mathbb{P}^m \times \mathbb{P}^n$ by using three techniques that build on past work by Ein, Voisin, Pacienza, Coskun and Riedl, and others. As a result, we completely answer the question of whether or not a very general hypersurface of bidegree $(a,b)$ in $\mathbb{P}^m \times \mathbb{P}^n$ is algebraically hyperbolic, except in $\mathbb{P}^3 \times \mathbb{P}^1$ for the bidegrees $(a,b)= (7,3), (6,3)$ and $(5,b)$ with $b\geq 3.$ As another application of these techniques, we improve the known result that very general hypersurfaces in $\mathbb{P}^n$ of degree at least $2n-2$ are algebraically hyperbolic when $n\geq 6$ to $n \geq 5$, leaving $n=4$ as the only open case.