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.Comment: 31 page