Orbit-cone correspondence

The orbit-cone correspondence assigns to every cone $\sigma \in \Sigma$ a torus orbit of codimension $\dim \sigma$. In particular, maximal cones correspond to torus-fixed points.

At a point $p$ of $\mathrm{Spec}k[C\cap M]$, the exponents $m$ with ${\chi }^m(p)\ne 0$ form a semigroup face. They are the lattice points of one face of $C$. Points with that same support face form a single algebraic torus orbit: their nonzero monomial evaluations extend to homomorphisms from the lattice generated by the face into $k^{\ast }$, and restriction from the full torus is surjective.

// Target: mathematical-optimization.bigb

If $\tau ⪯\sigma$, choose $m\in {\sigma }^{\vee }\cap M$ with $\tau =\sigma \cap m^{\perp }$. Then $U_{\tau }=D({\chi }^m)\subset U_{\sigma }$. Its complement is the union of orbit closures indexed by faces of $\sigma$ that are not faces of $\tau$.

// Target: geometry-and-topology.bigb

For $\sigma \in \Sigma$, the closure of its torus orbit isThus orbit-closure inclusion reverses face inclusion.

On the orbit of a face $\tau ⪯\sigma$, the monomial ${\chi }^{\mu }$, with $\mu \in {\sigma }^{\vee }\cap M$, vanishes exactly when $\mu \notin {\tau }^{\perp }$. Its reduced zero locus is consequently the union of those orbit closures. The principal subscheme need not be reduced: $V(x^2)$ and $V(x)$ have the same support but different ideals.

// Target: geometry-and-topology.bigb

The action of the algebraic torus on a toric variety is by automorphisms, and automorphisms preserve regular local rings. The smooth and singular loci are therefore unions of torus orbits.