Singularity of torus fixed points from combinatorial data

Question

May I ask what are the relations between the geometry and combinatorics near a torus fixed point? Any references? In particular, let $S$ be a scheme that is torus invariant with finitely many zero and one dimensional orbits. What can you say about the singularity at a fixed point if the number of torus stable curves through this point is equal to the dimension of $S$? Thank you very much!

Answer 1

I would recommend looking at Brion's paper "Rational Smoothness and Fixed Points of Torus Actions" (mostly section 1). A point in a variety is defined to be rationally smooth if local cohomology with constant coefficients is the same as for a point of a smooth variety. This is weaker than smoothness, yet it implies that if you have finitely many torus stable curves, their number will match the dimension of the variety.

For the other direction, suppose $x$ is a torus fixed point with as many torus stable curves passing through it as the dimension at $x$. If in addition you know that a) $x$ is an attractive point (all torus weights on the tangent space at $x$ are contained in a half plane) and b) $x$ admits a rationally smooth punctured neighborhood, then you can guarantee that $x$ is rationally smooth.

Brion gives an example illustrating that these extra conditions are necessary: The hypersurface in $\mathbb A^5$ given by $x^2+yz+xtw$. There is a $T=G_m\times G_m$ action given by $$(u,v)(x,y,z,t,w)=(u^2v^2x,u^3vy,uv^3z,u^2t,v^2w)$$ the origin is an attractive $T$-fixed point, and there are 4 closed irreducible $T$-stable curves: the $y,z,t,w$ axes. However our hypersurface is not rationally smooth at the origin.