Singular string polytopes and functorial resolutions from Newton–Okounkov bodies

Abstract

The main result of this note is that the toric degenerations of flag varieties associated to string polytopes and certain Bott-Samelson resolutions of flag varieties fit into a commutative diagram which gives a resolution of singularities of singular toric varieties corresponding to string polytopes. Our main tool is a result of Anderson which shows that the toric degenerations arising from Newton-Okounkov bodies are functorial in an appropriate sense. We also use results of Fujita which show that Newton-Okounkov bodies of Bott-Samelson varieties with respect to a certain valuation νmax coincide with generalized string polytopes, as well as previous results by the authors which explicitly describe the Newton-Okounkov bodies of Bott-Samelson varieties with respect to a different valuation ν min in terms of Grossberg-Karshon twisted cubes. A key step in our argument is that, under a technical condition, these Newton-Okounkov bodies coincide.

The Gromov Width of Bott-Samelson Varieties

“…Remark 5.11 The polytope P i,m enjoys further nice properties. For instance, as proved in [19], under the condition (P ) the polytope P i,m coincides with the generalized string polytope introduced in [14]; the latter turns out to be unimodular to the polytope i,m we are considering in Section 4 (by [14,Theorem 8.2] along with Remark 4.2).…”

“…For some appropriate choice of (i, m), Bott-Samelson varieties Z i equipped with an ample line bundle L i,m can be degenerated into Bott manifolds using the theory of Newton-Okounkov bodies. These toric degenerations were derived in [19] from some previous constructions of Grossberg and Karshon (see [16] and [37] also). Next, we recall the main properties of these toric degenerations.…”

“…In Section 5, we consider the polarized Bott-Samelson varieties which can be degenerated into polarized Bott (toric) manifolds. These toric degenerations (and their underlying combinatorics) were thoroughly studied in [16,19,37]. The Gromov widths of polarized generalized Bott manifolds are computed in [21] 2 .…”