On some smooth projective two-orbits varieties with Picard number 1

TL;DR

This paper classifies smooth projective horospherical varieties with Picard number 1, analyzing their automorphism groups and orbit structures, and characterizes two-orbits varieties with specific automorphism actions.

Contribution

It provides a complete classification of certain smooth projective varieties with Picard number 1 and describes their automorphism group actions and orbit structures.

Findings

Automorphism groups act with at most two orbits on these varieties.

The automorphism group acts with two orbits on the blow-up at the closed orbit.

Characterization of two-orbits varieties satisfying specific automorphism properties.

Abstract

We classify all smooth projective horospherical varieties with Picard number 1. We prove that the automorphism group of any such variety X acts with at most two orbits and that this group still acts with only two orbits on X blown up at the closed orbit. We characterize all smooth projective two-orbits varieties with Picard number 1 that satisfy this latter property.