Helly dimension of algebraic groups

TL;DR

This paper establishes a bound on the Helly dimension for linear algebraic groups over characteristic zero fields, linking the intersection properties of cosets to algebraic group actions.

Contribution

It introduces a finite Helly dimension (G) for algebraic groups, providing a new combinatorial bound for intersection problems in algebraic geometry.

Findings

Existence of a natural number (G) for linear algebraic groups over characteristic zero.

Bound on the size of subsystems with empty intersection of Zariski closed cosets.

Application to algebraic group actions on product varieties.

Abstract

It is shown that for a linear algebraic group G over a field of characteristic zero, there is a natural number \kappa(G) such that if a system of Zariski closed cosets in G has empty intersection, then there is a subsystem consisting of at most \kappa(G) cosets with empty intersection. This is applied to the study of algebraic group actions on product varieties.