Algebraic hyperbolicity of generic high degree hypersurfaces (Q1337781)

Here we show (in any characteristic) that for every \(n \geq 3\) there is an explicit integer \(d(n)\) such that for every \(d \geq d(n)\) a generic (i.e. outside countably many proper subvarieties of the parameter space) hypersurface \(S\) of degree \(d\) of \(\mathbb{P}^ n\) is algebraically hyperbolic, i.e. it contains no image of an abelian variety (hence no rational or elliptic curve (even singular)). Recently over \(\mathbb{C}\) a much stronger result (the hyperbolicity of \(S)\) was independently proved by \textit{K. Masuda} and \textit{J. Noguchi} [``A construction of hyperbolic hypersurface of \(\mathbb{P}^ n (\mathbb{C})\)''].