Resolution of Richardson Varieties, Stable curves, and dual simplicial spheres

These notes were taken during talk given by Allen Knutson.

Initial part was difficult to understand as I am not familiar with combinatorial part. Anyway, initial part was mainly about Simple normal crossings divisor (sncd), Bott-Samelson manifolds, Flag manifolds, brick manifolds, subword complexes.

Moduli Space of stable maps of rational curves

Fix 2-homology class $\beta\in H_2(M)$ and a number n of "marked points. Consider $\gamma:\Sigma\longrightarrow M$, where $\Sigma$ is a tree of smooth $\mathbb{P}^1$s with simple normal crossings and $n$ points (not at nodes?) marked $1,2,\cdots n$.

Definition: Map $\gamma$ is stable if $\Sigma$ has only finitely many automorphisms compatible with $\gamma$. Specifically, each component of $\Sigma$ collapsed by $\gamma$ to a point should have at least three nodes + marked points.

Theorem [Fulton-Pandharipande]: $\overline{\mathcal{M}}_{0,n}(G/P,\beta)$ is smooth proper stack (compact orbifold).

Note: G/P is flag manifold

Definition: A torus action is $d-GKM$ (Goresky- Kottwitz- MacPherson) if it fixes only finitely many subvarieties of $dimension\leq d$ (necessarily toric).

The fixed points and curves in a $1-GKM$ space gives the vertices and edges of a graph.