Anti-affine algebraic groups

TL;DR

This paper classifies anti-affine algebraic groups, which have only constant regular functions, and explores their implications for algebraic group structure in positive characteristic and counterexamples to Hilbert's fourteenth problem.

Contribution

It provides a comprehensive classification of anti-affine algebraic groups and applies this to understand algebraic groups in positive characteristic and construct counterexamples to Hilbert's fourteenth problem.

Findings

Classification of anti-affine algebraic groups

Applications to algebraic groups in positive characteristic

Construction of counterexamples to Hilbert's fourteenth problem

Abstract

We say that an algebraic group $G$ over a field is anti-affine if every regular function on $G$ is constant. We obtain a classification of these groups, with applications to the structure of algebraic groups in positive characteristics, and to the construction of many counterexamples to Hilbert's fourteenth problem.