Rationality of homogeneous varieties

TL;DR

This paper investigates the conditions under which homogeneous varieties G/H are rational, establishing new results for solvable subgroups, low-dimensional cases, and certain maximal rank subgroups with specific group types.

Contribution

It proves rationality of G/H for solvable H, low-dimensional cases in characteristic zero, and certain maximal rank subgroups with specified group types, extending known results.

Findings

G/H is rational if H is solvable.

G/H is rational when dim(G/H) < 11 and characteristic(k) = 0.

G/H is rational for maximal rank H with specific group type conditions.

Abstract

Let G be a connected linear algebraic group over an algebraically closed field k, and let H be a connected closed subgroup of G. We prove that the homogeneous variety G/H is a rational variety over k whenever H is solvable, or when dim(G/H) < 11 and characteristic(k) = 0. When H is of maximal rank in G, we also prove that G/H is rational if the maximal semisimple quotient of G is isogenous to a product of almost-simple groups of type A, type C (when characteristic(k) is not 2), or type B_3 or G_2 (when characteristic(k) = 0).