Integral points on the complement of the branch locus of projections from hypersurfaces

Abstract

We study the integral points on ${\mathbb{P}}_n\setminus D$, where $D$ is the branch locus of a projection from a hypersurface in ${\mathbb{P}}_{n+1}$ to a hyperplane $H\simeq {\mathbb{P}}_n$. We extend to the general case a result by Zannier (whose approach we follow) and we also obtain a sharper bound that yields, in some cases, the finiteness of integral points. The results presented are effective and the proofs provide a way to actually construct a set containing all the integral points in question. Thus, there are concrete applications to the study of Diophantine equations, more precisely to the problem of finding integral solutions to equations $F(x_0,\dots ,x_n)=c$, where $c$ is a given nonzero value and $F$ is a homogeneous form defining the branch locus $D$, i.e. a discriminant.