Algebraic hyperbolicity of very general hypersurfaces in products of projective spaces

We study the algebraic hyperbolicity of very general hypersurfaces in ℙ^m × ℙⁿ 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 ℙ^m × ℙⁿ is algebraically hyperbolic, except in ℙ³ × ℙ¹ for the bidegrees (a, b) = (7, 3), (6, 3) and (5, b) with b ≥ 3. As another application of these techniques, we improve the known result that very general hyper-surfaces in ℙⁿ of degree at least 2n − 2 are algebraically hyperbolic when n ≥ 6 to n ≥ 5, leaving n = 4 as the only open case.