Title: Projectively normal complete symmetric varieties and Fano complete symmetric varieties
Abstract: This thesis is subdivided in two parts. In the first one is studied the projective normality of symmetric varieties, while in the second one are proved some partial results toward the classification of Fano symmetric varieties (i.e. with ample anticanonical bundle). In [R. Chirivi, A. Maffei, Projective normality of complete symmetric varieties, Duke Math. J. 122 (1) (2004) 93–123] Chirivi and Maffei have proved that the surjectivity of the multiplication of sections of any two line bundles generated by global sections on a wonderful symmetric variety. We prove two criterions that allow us to reduce the same problem on a smooth complete toroidal symmetric variety to the analogous problem on the corresponding complete (resp. open) toric variety. We have also found some families of complete toroidal symmetric varieties, in particular those of rank 2, such that the product of sections of any ample linear bundle is surjective. In the second part of my thesis, I have first classified the Fano symmetric varieties of arbitrary rank which are obtainable from a wonderful variety by a sequence of blow-ups along closed orbits. When the rank is at most 3, I have proved more precise results. The symmetric projective varieties of rank one are all smooth and wonderful by a classic result of Akhiezer. I have classified the smooth projective toroidal (adjoint) symmetric varieties of rank 2 whose anticanonical bundle is ample, resp. globally generated. Moreover, I have classified the Fano (adjoint) symmetric varieties of rank 3 obtainable from the wonderful variety by a sequence of blow-ups along G-stable varieties. One can remark that any complete symmetric variety is dominated by a variety obtainable from a wonderful variety by a sequence of blow-ups along G-stable subvarieties of codimension 2.