Geometrical description of smooth projective symmetric varieties with Picard number one

TL;DR

This paper provides a geometric characterization of smooth projective symmetric varieties with Picard number one, including their automorphism groups and embeddings into larger homogeneous varieties, extending previous classifications.

Contribution

It offers a detailed geometric description of these varieties and identifies their automorphism groups, building on prior classification results.

Findings

Automorphism groups of the varieties are explicitly determined.

A G-equivariant embedding into larger homogeneous varieties is constructed.

The geometric structure of these varieties is thoroughly described.

Abstract

In a previous paper we have classified the smooth projective symmetric G-varieties with Picard number one (and G semisimple). In this work we give a geometrical description of such varieties. In particular, we determine their group of automorphisms. When this group, Aut(X), acts non-transitively on X, we describe a G-equivariant embedding of the variety X in a homogeneous variety (with respect to a larger group).