On integral points on surfaces

We study the integral points on surfaces by means of a new method, relying on the Schmidt Subspace Theorem. This method was recently introduced in [CZ] for the case of curves, leading to a new proof of Siegel's celebrated theorem that any affine algebraic curve defined over a number field has only finitely many S-integral points, unless it has genus zero and not more than two points at infinity. Here, under certain conditions involving the intersection matrix of the divisors at infinity, we shall conclude that the integral points on a surface all lie on a curve. We shall also give several examples and applications. One of them concerns curves, with a study of the integral points defined over a variable quadratic field; for instance we shall show that an affine curve with at least five points at infinity has at most finitely many such integral points.