Geometrical Description of Smooth Projective Symmetric Varieties with Picard Number One

In [Ru2] 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). keywords: Symmetric varieties, Fano varieties. Mathematics Subject Classification 2000: 14M17, 14J45, 14L30 A Gorenstein normal algebraic variety X over C is called a Fano variety if the anticanonical divisor is ample. The Fano surfaces are classically called Del Pezzo surfaces. The importance of Fano varieties in the theory of higher dimensional varieties is similar to the importance of Del Pezzo surfaces in the theory of surfaces. MoreoverMori’s program predicts that every uniruled variety is birational to a fiberspace whose general fiber is a Fano variety (with terminal singularities). Often it is useful to subdivide the Fano varieties in two kinds: the Fano varieties with Picard number equal to one and the Fano varieties whose Picard number is strictly greater of one. For example, there are many results which give an explicit bound to some numerical invariants of a Fano variety (depending on the Picard number and on the dimension of the variety). Often there is an explicit expression for the Fano varieties of Picard number equal to one and another expression for the remaining Fano varieties. We are mainly interested in the smooth projective spherical varieties with Picard number one. The smooth toric (resp. homogeneous) projective varieties with Picard number one are just projective spaces (resp. G/P with G simple and P maximal). B. Pasquier has recently classified the smooth projective horospherical varieties with Picard number one (see [P]). In a previous work we have classified the smooth projective symmetric G-varieties with Picard number one and G semisimple (see [Ru2]). One can easily show that they are all Fano, because the canonical bundle cannot be ample. We have also obtained a partial classification of the smooth Fano complete symmetric varieties with Picard number strictly greater of one (see [Ru1]). Our classification of the smooth