Fano Symmetric Varieties with Low Rank

Abstract

The symmetric projective varieties of rank one are all smooth and Fano by a classical result of Akhiezer. We classify the locally factorial (respectively smooth) projective symmetric $G$-varieties of rank 2 that are Fano. When $G$ is semisimple we also classify the locally factorial (respectively smooth) projective symmetric $G$-varieties of rank 2 that 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 that are obtainable from a wonderful variety by a sequence of blow-ups along closed orbits.