Effective divisors on Bott-Samelson varieties

TL;DR

This paper computes the effective divisor cones on Bott-Samelson varieties, providing new tools and explicit descriptions that enhance understanding of their geometric and combinatorial properties.

Contribution

It introduces a general method for computing effective cones on Bott-Samelson varieties and applies it to various cycle classes, including effective codimension-two cycles.

Findings

Computed the cone of effective divisors on Bott-Samelson varieties.

Derived criteria for dense B-orbits in Bott-Samelson varieties.

Constructed desingularizations of intersections of Schubert varieties.

Abstract

We compute the cone of effective divisors on a Bott-Samelson variety corresponding to an arbitrary sequence of simple roots. The main tool is a general result concerning effective cones of certain $B$-equivariant $\mathbb{P}^1$ bundles. As an application, we compute the cone of effective codimension-two cycles on Bott-Samelson varieties corresponding to reduced words. We also obtain an auxiliary result giving criteria for a Bott-Samelson variety to contain a dense $B$-orbit, and we construct desingularizations of intersections of Schubert varieties. An appendix exhibits an explicit divisor showing that any Bott-Samelson variety is log Fano.