Seminar Algebra and Geometry: Amos Turchet (Chalmers Universitet, Göteborg)

Geometric Lang-Vojta Conjecture in P^2

Lang-Vojtas Conjectures are a set of deep and far reaching conjectures, formulated by Paul Vojta using ideas of Lang, which embrace the distribution of solutions to Diophantine equations over number fields, the behaviour of holomorphic maps into complex manifolds and of algebraic curves into algebraic varieties.
In the (split) function field case the conjecture predicts (weak) algebraic hyperbolicity for log-general type varieties.When the completion of the variety is the projective plane the conjecture is known both if the divisor at infinity consits of four lines in general position (Brownawell-Masser and,independently, Voloch) and for a conic and two lines with five singular points (Corvaja and Zannier). With different methods Chen and Pacienza-Rousseau proved that the conjecture holds in the hyperbolic case, i.e. the complement of a very generic curve of degree at least 5.
In the talk, after an introduction to this fascinating subject, we will show how to prove the conjecture in general for the complement of a very generic curve of degree at least four.The proof relies on a deformation argument applied to a conic and two lines and on the theory of logarithmic stable maps as defined by Abramovich-Chen (and independently by Gross and Siebert) which extends usual stable maps to the logarithmic category (in the sense of Kato and Illusie).

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