On integral Points on Surfaces

This paper introduces a completely new method to analyse the integral points on affine algebraic surfaces. In particular it is proved that certain purely numerical conditions on the divisors at infinity guarantees that the integral points are degenerate. A very special corollary is Siegel's celebrated theorem for curves, but several new applications are presented. Another byproduct is the proof that on an affine curve with at least 4 points at infinity there are only finitely many families of integral points over a variable quadratic extension of a number fields. The method stems from constructing suitable embeddings in high dimensional projective spaces, and then applying Schmidt's subspace theorem.