Horospherical geometry: combinatorial algebraic stacks and approximating rational points

Abstract

The purpose of this thesis is to explore and develop several aspects of the theory of horospherical geometry. Horospherical varieties are equipped with the action of a reductive algebraic group such that there is an open orbit whose points are stabilized by maximal unipotent subgroups. This includes the well-known classes of toric varieties and flag varieties. Using this orbit structure and representation-theoretic condition on the stabilizer, one can classify horospherical varieties using combinatorial objects called coloured fans. We give an overview of the main features of this classification through a new, accessible notational framework.

There are two main research themes in this thesis. The first is the development of a combinatorial theory for horospherical stacks, vastly generalizing that for horospherical varieties. We classify horospherical stacks using combinatorial objects called stacky coloured fans, extending the theory of coloured fans. As part of this classification, we describe the morphisms of horospherical stacks in terms of maps between the stacky coloured fans, we completely describe the good moduli space of a horospherical stack, and we introduce a special, hands-on class of horospherical stacks called coloured fantastacks.

The second major theme is using horospherical varieties to probe a conjecture in arithmetic geometry. In 2007, McKinnon conjectured that, for a given point on a projective variety, there is a sequence, lying on a curve, which best approximates this point. We verify a version of this conjecture for horospherical varieties, contingent on Vojta’s Main Conjecture, which says that there is a sequence, lying on a curve, which approximates the given point better than any Zariski dense sequence.