Batyrev-Manin conjecture for Deligne-Mumford stacks


In this talk, I will explain how the Batyrev-Manin conjecture on rational points can be generalized to Deligne-Mumford stacks by using twisted sectors. In the original conjecture, the so-called a- and b-invariants are determined by positions of the ample line bundle in question and the canonical divisor in the NĂ©ron-Severi space relative to the pseudo-effective cone. In generalization to stacks, we introduce orbifold versions of these algebro-geometric notions. Once we define them suitably, the generalized conjecture is formulated more or less in the same way as the original conjecture was. The Malle conjecture on Galois extensions of a number field is then regarded as a special case of it. This is a joint work with Ratko Darda.