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, 14L30A 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. Moreover Mori'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 1 projective symmetric varieties with Picard number one is a combinatorial one, so we are naturally interested to give a geometrical description of such varieties. In particular, we have proved that, given a symmmetric space G/H, there is at most a smooth completion X of G/H with Picard number one and X must be projective (see [Ru2], Theorem 3.1). We will prove that the automorphism group of a such variety X can act non-transitively on X only if the rank of X is 2. It would be interesting to find a reason for such exceptionality of the rank 2 case. Unfortunately, our prove does not explain completely this fact, because there is a part of the proof that it is a case-to-case analysis. The homogeneousness of the rank one varieties was proved first by Ahiezer in [A1].M...

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.

If G is a complex simply connected semisimple algebraic group and if λ is a dominant weight, we consider the compactification X λ ⊂ P End(V (λ)) obtained as the closure of the G × Gorbit of the identity and we give necessary and sufficient conditions on the support of λ so that X λ is normal; as well, we give necessary and sufficient conditions on the support of λ so that X λ is smooth.

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.

Given a semisimple algebraic group G, we characterize the normality and the smoothness of its simple linear compactifications, namely those equivariant G x G-compactifications possessing a unique closed orbit which arise in a projective space of the shape P(End(V)), where V is a finite dimensional rational G-module. Both the characterizations are purely combinatorial and are expressed in terms of the highest weights of V. In particular, we show that Sp(2r) (with r >= 1) is the unique non-adjoint simple group which admits a simple smooth compactification