Bernt Ivar Utstøl Nødland (UiO): Chow groups and pseudoeffective cones of complexity one T-varieties

Abstract: A normal variety X equipped with an effective action of an algebraic torus T is called a T-variety. We show that the pseudoeffective cone of k-cycles on a complete T-variety of complexity one  (meaning dim X-dim T =1) is rational polyhedral for any k, generated by classes of T-invariant subvarieties. Moreover, when X is also rational, we give a presentation of the Chow groups of X in terms of generators and relations, coming from the (quasi)combinatorial data defining X as a T-variety. I will spend substantial time introducing toric varieties, T-varieties and the relationship between them, as well as discussing the corresponding statements for toric varieties(which are well-known statements in the literature).