Automorphism group orbits on horospherical varieties and divisor class group

TL;DR

This paper extends criteria for orbit equivalence from toric varieties to affine and projective horospherical varieties using divisor class groups, enhancing understanding of automorphism group actions.

Contribution

It introduces a necessary condition for orbit equivalence on horospherical varieties, generalizing previous criteria from toric cases.

Findings

Established a necessary condition for orbit equivalence on horospherical varieties.

Extended divisor class group criteria from toric to horospherical varieties.

Bridged gap between automorphism group actions on different classes of algebraic varieties.

Abstract

In 2013 Bazhov proved a criterium for two points on a complete toric variety to lie in the same orbit of the neutral component of automorphism group. This criterium is in terms of divisor class group. Arzhantsev-Bazhov (2013) obtained a similar criterium for affine toric varieties. We prove a necessary condition similar this criteria to the cases of affine and projective horospherical varieties.