Finding integral points on algebraic varieties

The Vojta’s conjecture establishes geometrical conditions on the degeneracy of the set of S-integral points for an algebraic variety. The goal of the thesis is to prove that for certain algebraic varieties for which such conditions are not verified the set of S-integral points is Zariski-dense. Some effective methods in this respect has been developed by Beukers in his paper “Ternary form Equations” , in which he proved the density of integral solutions of some homogeneous diophantine equations. Following such ideas, the work of the thesis consists on finding the density of S-integral points on some varieties for which it is not known at the moment, imposing the needed arithmetical and geometrical conditions