Big Divisors on a Projective Symmetric Variety

We describe the big cone of a projective symmetric variety. Moreover, we give a necessary and sufficient combinatorial criterion for the bigness of a nef divisor (linearly equivalent to a G-stable divisor) on a projective symmetric variety. When the variety is toroidal, such criterion has an explicitly geometrical interpretation. Finally, we describe the spherical closure of a symmetric subgroup. keywords: Symmetric varieties, Big divisors. Mathematics Subject Classification 2000: 14L30, 14C20, (14M17) Brion give a description of the Picard group of a spherical variety in [Br89]. He also finds necessary and sufficient conditions for the ampleness and global generation of a line bundle. From these conditions follows that a line bundle is nef if and only if it is globally generated. It is natural to ask what are the conditions on a line bundle to be big. It is known that the effective cone is closed, polyhedral and, if the variety is Q-factorial, generated by the classes of the B-stable prime divisors. But in general it is hard to say which are the B-stable prime divisors whose classes generate an extremal ray of the effective cone. In the very special case of projective homogeneous varieties, the big cone coincides with the ample cone. More generally, the case of wonderful varieties is studied in [Br07]. There are also known some conditions for a divisor on a toric variety to be big. See also [FS08] for a study of the big cone of some toric varieties. We are interested to study the bigness of line bundles on symmetric varieties (over which acts a semisimple group). First, we describe explicitly the effective cone; we determine also when the classes of two B-stable prime divisors are proportional. When the variety is Q-factorial, we find the conditions so that the class of a B-stable prime divisor generates an extremal ray of the effective cone (see Theorem 3.1 and Corollary 3.1). In a second time, we restrict ourselves to study the bigness of a nef divisor which is linearly equivalent to a G-stable Q-divisor. In many case, the last condition is satisfied for all divisors. Moreover, the canonical divisor is always linearly equivalent to a G-stable Q-divisor. We prove a necessary and sufficient combinatorial condition on the piecewise linear function associated to a G-stable, nef, Cartier Q-divisor so that the divisor is big (see Theorem 4.1). Each projective toroidal symmetric variety X contains a projective toric variety Z which determines univocally X . We prove that the restriction of a big Cartier divisor of X to Z is always big. If the divisor is G-stable, then this condition is also sufficient. In particular, we prove that, if G is simple, every non-trivial, nef, G-stable, Cartier divisor on X is big (see Proposition 4.1