Geometric Lang-Vojta conjecture in P^2

Lang-Vojta conjecture is one of the most celebrated conjec- tures in Diophantine Geometry. Stated independently by Paul Vojta and Serge Lang the conjecture pre- dicts degeneracy of S-integral points in algebraic varieties of log-general type for a finite set of places S of a number field κ containing the infinite ones, provided that the divisor “at infinity” is a normal crossing divisor. This deep conjecture and his analogous formulations are among the main open problems in Number Theory, Complex Analysis and Arithmetic Algebraic Geometry. This thesis contains the work of the author during his Ph.D. studies at the University of Udine under the supervision of Prof. Pietro Corvaja (and, partially, during his visit to Brown University under the supervision of Prof. Dan Abramovich), and it is centered around the function field version of Lang- Vojta conjecture for complements of curves in P2, with at most normal crossing singularities. The main part contains the proof of two cases of this conjecture, namely the non-split case for complements of degree four and three components divisors and the split case for very generic divisors of degree four with simple normal crossing