On the Mori theory and Newton–Okounkov bodies of Bott–Samelson varieties

Abstract

We prove that on a Bott–Samelson variety X every movable divisor is nef. This enables us to consider Zariski decompositions of effective divisors, which in turn yields a description of the Mori chamber decomposition of the effective cone. This amounts to information on all possible birational morphisms from X. Applying this result, we prove the rational polyhedrality of the global Newton–Okounkov cone of a Bott–Samelson variety with respect to the so called ‘horizontal’ flag. In fact, we prove the stronger property of the finite generation of the corresponding global value semigroup.