A fibered power theorem for pairs of log general type

Let $f: (X,D) \to B$ be a stably family with log canonical general fiber. We prove that, after a birational modification of the base $\tilde{B} \to B$, there is a morphism from a high fibered power of the family to a pair of log general type. If in addition the general fiber is openly canonical, then there is a morphism from a high fibered power of the original family to a pair openly of log general type.Comment: Exposition has been greatly improved. Version to appear in Algebra & Number Theor

Around the Chevalley-Weil Theorem

We present a proof of the Chevalley-Weil Theorem that is somewhat different from the proofs appearing in the literature and with somewhat weaker hypotheses, of purely topological type. We also provide a discussion of the assumptions, and an application to solutions of generalized Fermat equations, where our statement allows to simplify the original argument of Darmon and Granville.Comment: 12 pages, comments welcom

Divisibility of polynomials and degeneracy of integral points

We prove several statements about arithmetic hyperbolicity of certain blow-up varieties. As a corollary we obtain multiple examples of simply connected quasi-projective varieties that are pseudo-arithmetically hyperbolic. This generalizes results of Corvaja and Zannier obtained in dimension 2 to arbitrary dimension. The key input is an application of the Ru-Vojta's strategy. We also obtain the analogue results for function fields and Nevanlinna theory with the goal to apply them in a future paper in the context of Campana's conjectures.Comment: 26 pages. Comments welcom

Hyperbolicity of varieties of log general type

These notes provide an overview of various notions of hyperbolicity for varieties of log general type from the viewpoint of both arithmetic and birational geometry. The main results are based on our paper entitled "Hyperbolicity and uniformity of varieties of log general type." They are expanded notes from a minicourse the authors gave as part of the Geometry and arithmetic of orbifolds workshop at UQ\'AM.Comment: Addressed some inaccuracies and typos pointed out by the referees and some readers. Slight change of title. To appear in CRM short courses (Arithmetic Geometry of Logarithmic Pairs and Hyperbolicity of Moduli Spaces: Hyperbolicity in Montr\'eal

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 crossingopenDottorato di ricerca in Matematica e fisicaopenTurchet, Amo

Fibered threefolds and Lang-Vojta’s conjecture over function fields

Using the techniques introduced by Corvaja and Zannier in 2008 we solve the non-split case of the geometric Lang-Vojta Conjecture for affine surfaces isomorphic to the complement of a conic and two lines in the projective plane. In this situation we deal with sections of an affine threefold fibered over a curve, whose boundary, in the natural projective completion, is a quartic bundle over the base whose fibers have three irreducible components. We prove that the image of each section has bounded degree in terms of the Euler characteristic of the base curve