Effective and Big Divisors on a Projective Symmetric Variety

TL;DR

This paper characterizes the effective and big divisor cones on projective symmetric varieties, providing combinatorial criteria for bigness and geometric interpretations in specific cases, and describes the spherical closure of symmetric subgroups.

Contribution

It introduces a combinatorial criterion for the bigness of nef divisors on projective symmetric varieties, with explicit geometric interpretation in toroidal, G-stable cases.

Findings

Effective and big cones are explicitly described.

A necessary and sufficient combinatorial criterion for bigness is established.

The spherical closure of a symmetric subgroup is characterized.

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 variety is toroidal and the divisor is $G$-stable, such criterion has an explicit geometric interpretation. Finally, we describe the spherical closure of a symmetric subgroup.