THE GAMMA CONSTRUCTION AND ASYMPTOTIC INVARIANTS OF LINE BUNDLES OVER ARBITRARY FIELDS

We extend results on asymptotic invariants of line bundles on complex projective varieties to projective varieties over arbitrary fields. To do so over imperfect fields, we prove a scheme-theoretic version of the gamma construction of Hochster and Huneke to reduce to the setting where the ground field is $F$ -finite. Our main result uses the gamma construction to extend the ampleness criterion of de Fernex, Küronya, and Lazarsfeld using asymptotic cohomological functions to projective varieties over arbitrary fields, which was previously known only for complex projective varieties. We also extend Nakayama’s description of the restricted base locus to klt or strongly $F$ -regular varieties over arbitrary fields.

Tame filling invariants for groups

A new pair of asymptotic invariants for finitely presented groups, called intrinsic and extrinsic tame filling functions, is introduced. These filling functions are quasi-isometry invariants that strengthen the notions of intrinsic and extrinsic diameter functions for finitely presented groups. We show that the existence of a (finite-valued) tame filling functions implies that the group is tame combable. Bounds on both intrinsic and extrinsic tame filling functions are discussed for stackable groups, including groups with a finite complete rewriting system, Thompson's group F, and almost convex groups.