# Constructing spherical embeddings from colored fans

The gist: how does one produce a spherical embedding from a colored fan?

The setup: $G$ is a reductive group, $B$ a borel subgroup, and $G/H$ a spherical homogeneous space; that is, it has an open $B$-orbit. A spherical embedding for $G/H$ is a normal $G$-variety with $G/H$ as an open orbit.

Some Background

Much like in the case of toric varieties, spherical embeddings can be described by combinatorial data: a colored fan. From $X$ you get a colored fan roughly by looking at $B$-semi invariants of $k(G/H)$. The characters of these semi invariants form a lattice; the associated vector space and its dual $N(G/H)$ are where the combinatorial data lives. There is a map you can define from divisors in $X$ to elements of $N(G/H)$ using discrete valuations. You get a fan by looking at cones generated by the images of certain $G$- and $B$-stable divisors in $X$. The colors come from keeping track of which divisors are $B$-stable but not $G$-stable.

More detailed Question

I've learned this from reading (some of) Knop's paper `The Luna-Vust Theory of Spherical Embeddings' and also from some very nice lectures notes of Guido Pezzini: http://ccirm.cedram.org/cedram-bin/article/CCIRM_2010__1_1_33_0.pdf

I have a rough understanding of how one can be given a spherical embedding $X$ and then produce a colored fan $F(X)$, and its a theorem that a map $X \mapsto F(X)$ gives a bijection from iso classes of spherical embeddings and strictly convex colored fans. I would like to understand better how to go in the other direction.

I see that if you have a colored fan $F$ each cone in $F$ should correspond to a $G$-orbit. As I understand it, if you take a rational point of the interior of $C$, then this should correspond to a valuation on $K(G/H)$ and the center of this valuation should be the local ring of the generic point of a $G$-stable subvariety: the desired $G$-orbit; but this is still a little abstract to me. I can't see how to actually produce a $G$-orbit from this.

Further, the colors in $C$ are the $B$-stable prime divisors; these are determined by their intersection $G/H$ so in particular you only need $G/H$ to describe them. The extremal rays of $C$ should correspond to $G$-stable prime divisors that contain $Y_C$; in particular, these don't intersect $G/H$ and live in the desired, but a present unconstructed, embedding $X$. So again, I'm not sure how I would go about producing these $G$-stable prime divisors.

I would appreciate any clarifying thoughts or references to books or papers that might discuss these things in more detail.

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