Algebraic hyperbolicity of very general hypersurfaces in products of projective spaces

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 $b\ge 3.$ 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 $n\ge 6$ to $n\ge 5$, leaving $n=4$ as the only open case.