Smooth Projective Symmetric Varieties with Picard Number One

We classify the smooth projective symmetric G-varieties with Picard number one (and G semisimple). Moreover we prove a criterion for the smoothness of the simple (normal) symmetric varieties whose closed orbit is complete. In particular we prove that, given a such variety X which is not exceptional, then X is smooth if and only if an appropriate toric variety contained in X is smooth. 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 one and the Fano varieties with Picard number strictly greater than 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 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 parabolic). We classify the smooth projective symmetric G-varieties whose Picard group is isomorphic to Z (with the hypothesis G semisimple). One can easily show that they are all Fano, because the canonical bundle cannot be ample. In [13] there are some partial results regarding the classification of smooth Fano projective symmetric varieties with Picard number greater than one and