Effective and big divisors on a projective symmetric variety

Abstract

We describe the effective and the big cones of a projective symmetric variety. Moreover, we give a necessary and sufficient combinatorial criterion for the bigness of a nef divisor on a projective symmetric variety. When the divisor is G-stable, such a criterion has an explicit geometric interpretation. Finally, we describe the spherical closure of a symmetric subgroup.

Fano Symmetric Varieties with Low Rank

“…Thus, a Cartier divisor on a projective symmetric variety is nef if and only if it is globally generated. Moreover, a nef G-stable Cartier divisor on a projective symmetric variety is big if and only if the associated piecewise linear function h is such that ( C∈∆(l) h C , R ∨ ) = 0 for each irreducible factor R ∨ of R ∨ G,θ (see [Ru09] Theorem 4.2). In particular, when θ is indecomposable every non-zero nef G-stable divisor is big.…”

“…A standard symmetric variety is always Q-factorial; in particular, K X is a Q-Cartier divisor. Moreover, if X is wonderful then also the X j are wonderful and X = X j (see [Ru09] Corollary 2.1). We have the following theorem:…”