Fano symmetric varieties with low rank

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. keywords: Symmetric varieties, Fano varieties. MSC 2010: 14M17, 14J45, 14L30 A Gorenstein (projective) 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). Let θ be an involution of a reductive group G (over C) and let H be a closed subgroup of G such that G ⊂ H ⊂ NG(G ). A symmetric variety is a normal G-variety with an open orbit isomorphic to G/H . The symmetric varieties are a generalization of the toric varieties. The toric smooth Fano varieties with rank at most four are been classified. By [AlBr04], Theorem 4.2 there is only a finite number of Fano smooth symmetric varieties with a fixed open orbit. In [Ru07] we have classified the smooth compact symmetric varieties with Picard number one and G semisimple, while in [Ru10] we have given an explicitly geometrical description of such varieties; they are automatically Fano. In this work, we want to classify the Fano symmetric varieties with low rank (and G semisimple). First, we consider a special case of arbitrary rank. We say that a variety X is quasi Q-Fano if −KX is a nef and big Q-divisor. Fixed an open orbit G/H with G semisimple, there is a unique maximal compactification between the ones which have only one closed orbit. Such variety is called the standard compactification. If it is also smooth, it is called the wonderful compactification; this is the case, for example, if H = NG(G ) (see [dCoPr83], Theorem 3.1). We prove that the standard symmetric varieties are all quasi Q-Fano and we describe when they are Fano. We determine also the symmetric Fano varieties obtainable from a wonderful one by a sequence of blow-ups along closed orbits. In particular, we prove that such a variety must be either a wonderful one or the blow-up of a wonderful one along the unique closed orbit.