We are interested in understanding finite surjective morphisms f: A → X, where A is an abelian variety and X is a smooth projective variety. This leads us to study two complementary constructions. The first is when X is the quotient of A by a finite group acting freely; this connects to complex crystallographic groups, and we construct families of such quotients and study their Albanese and Picard schemes. The second is when f is purely inseparable; this leads to the theory of p-closed foliations. We provide a complete classification for abelian surfaces. This is joint work with Hazem Hassan (Heidelberg) and Fabrizio Andreatta (Milano).

TBA.

The Cox ring of a projective variety X is a graded ring Cox(X) that encodes the global sections of all line bundles over X. If Cox(X) is finitely generated, we say that X is a Mori Dream Space. In this case, we can deduce geometric properties of X from the algebraic properties of Cox(X). On the other hand, not much is know about Cox(X) when it is not finitely generated.

The Kawamata-Morrison cone conjecture, after a work by Gachet, Lin, Stenger and Wang, predicts that the cone of effective divisors of a Calabi-Yau variety X can be described as the orbit of a rational polyhedral subcone, under the action induced by the birational automorphism group of X. In this talk, we will discuss a method to understand and describe the Cox ring of a Calabi-Yau variety, even if this ring is not finitely generated, provided that the variety satisfies the cone conjecture.