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

TL;DR

This paper explores the geometric and combinatorial properties of Bott-Samelson varieties, establishing nefness of movable divisors, describing Mori chamber decompositions, and proving the finite generation of Newton-Okounkov bodies.

Contribution

It demonstrates that all movable divisors on Bott-Samelson varieties are nef and characterizes their Mori chamber decompositions, also proving the finite generation of associated Newton-Okounkov bodies.

Findings

All movable divisors on Bott-Samelson varieties are nef.

The Mori chamber decomposition of the effective cone is explicitly described.

The global Newton-Okounkov body is rational polyhedral and finitely generated.

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 body 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.