Merging divisorial with colored fans

TL;DR

This paper introduces a method to describe embeddings of spherical homogeneous spaces with minimal rank using divisorial fans, linking algebraic geometry and torus actions for simplified classification.

Contribution

It provides a straightforward procedure to describe embeddings as varieties with torus action via divisorial fans, connecting Chow quotients to simple toroidal compactifications.

Findings

Chow quotient equals a blowup of the simple toroidal compactification

In horospherical cases, Chow quotient is a flag variety

Slices of divisorial fan are shifts of the colored fan

Abstract

Given a spherical homogeneous space G/H of minimal rank, we provide a simple procedure to describe its embeddings as varieties with torus action in terms of divisorial fans. The torus in question is obtained as the identity component of the quotient group N/H, where N is the normalizer of H in G. The resulting Chow quotient is equal to (a blowup of) the simple toroidal compactification of G/(H N^0). In the horospherical case, for example, it is equal to a flag variety, and the slices (coefficients) of the divisorial fan are merely shifts of the colored fan along the colors.