Fano symmetric varieties with low rank

TL;DR

This paper classifies low-rank Fano symmetric varieties, including smooth, locally factorial, and quasi-Fano cases, and explores their construction from wonderful varieties via blow-ups.

Contribution

It provides a comprehensive classification of Fano symmetric varieties of ranks 2 and 3, and extends to arbitrary rank cases obtained through blow-ups from wonderful varieties.

Findings

Classified rank 2 Fano symmetric varieties that are smooth or locally factorial.

Extended classification to quasi-Fano varieties for semisimple groups.

Identified Fano symmetric varieties of arbitrary rank constructed from wonderful varieties.

Abstract

The symmetric projective varieties of rank one are all smooth and Fano by a classic result of Akhiezer. We classify the locally factorial (respectively smooth) projective symmetric $G$-varieties of rank 2 which are Fano. When $G$ is semisimple we classify also the locally factorial (respectively smooth) projective symmetric $G$-varieties of rank 2 which are only quasi-Fano. Moreover, we classify the Fano symmetric $G$-varieties of rank 3 obtainable from a wonderful variety by a sequence of blow-ups along $G$-stable varieties. Finally, we classify the Fano symmetric varieties of arbitrary rank which are obtainable from a wonderful variety by a sequence of blow-ups along closed orbits.