Vojta's Main Conjecture on Rational Surfaces


13:30:00 - 14:30:00

103 , Mathematics Research Center Building (ori. New Math. Bldg.)

Vojta's Main Conjecture is an inequality of local heights, describing how the arithmetic is controlled by the geometry. Here, the 'arithmetic' is how p-adically close a rational point can be to a divisor, and the 'geometry' is how negative the canonical divisor is. It is a very deep conjecture, implying Mordell's conjecture (Faltings' theorem), Schmidt's subspace theorem, and Bombieri--Lang conjecture among others.
In this talk, I will discuss Vojta's conjecture on multiple blowups of the projective plane. Some cases can be proved unconditionally, while other cases have inter-relations with the abc conjecture, which is usually thought to be a (very deep) arithmetic geometry of dimension 1. Some (presumably new) properties of Farey sequences will be a key to our proofs.