Geometrical description of smooth projective symmetric varieties with Picard number one

Abstract

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...

The Cayley Grassmannian

“…Fano symmetric varieties of Picard number one. Ruzzi proved in [17] that there exist exactly six smooth projective symmetric varieties of Picard number one which are not homogeneous. One of them is a completion of G 2 , considered as the symmetric space (G 2 × G 2 )/G 2 .…”

“…Moreover we can identify ∆ 8 with ∧ • A and ∆ ′ 8 with ∧ • A ′ , where A = e 1 , e 2 , e 3 and A ′ = e 4 , e 5 , e 6 . Now, the restriction of ∆ 7 to G 2 decomposes as C ⊕ V 7 , so that finally [17] is that DG is the (highly non transverse) intersection of…”

Smooth Projective Symmetric Varieties With Picard Number One

“…Indeed, while W ⊕ C1 is a composition algebra if W is in the open G 2 -orbit, when W is in the closed G 2 -orbit, W ⊕ C1 is isomorphic to the exterior algebra * C 2 of C 2 . In particular, the proof of Proposition 1, §2.2 of [13] holds if one deletes the last line.…”

“…In Theorem 2 of [13] it is said that "the smooth completion X of G 2 /(SL 2 ×SL 2 ) with P ic(X) = Z parametrizes the subspaces W of C 7 such that W ⊕ C1 is a subalgebra of O C isomorphic to the complexified quaternions." I would to thank P. Chaput which pointed to me that I have only proved that X parametrizes the subspace such that W ⊕ C1 is an associative subalgebra of O C of dimension four.…”

“…In [13], we give a geometrical description of these varieties; in particular, we determine their group of automorphism. Moreover, if the action of this last group is not transitive, then the variety is a linear section of a generalized flag variety and it is lied to exceptional groups.…”