On the Mori theory and Newton-Okounkov bodies of Bott-Samelson varieties (Q2069620)

The paper deals with some questions concerning numerical classes of divisors and line bundles on Bott-Samelson varieties. In particular, it is shown that every movable divisor on a Bott-Samelson variety is base-point-free. This enables the authors to consider Zariski decompositions of effective divisors, and to provide a description of the decomposition into Mori chambers of the effective cone of a Bott-Samelson variety. These results are then applied to the study of the Newton-Okounkov bodies of line bundles on Bott-Samelson varieties. With respect to a particular flag of subvarieties introduced by \textit{M. Harada} and \textit{J. J. Yang} [Mich. Math. J. 65, No. 2, 413--440 (2016; Zbl 1352.14036)], the authors show that the associated global Newton-Okounkov cone is rational polyhedral.