(Star)
Let \(\tau \in \Sigma\) be a cone. Put \(N_\tau \da \ZZ\gens{\tau \intersect N}\), the sublattice generated by the lattice points of \(\tau\), and \[ N(\tau)_\RR \da N_\RR / (N_\tau)_\RR . \] For \(\sigma \geq \tau\) write \(\bar\sigma\) for the image of \(\sigma\) under the quotient map. The star of \(\tau\) is \[ \Star(\tau) \da \ts{ \bar\sigma \subseteq N(\tau)_\RR \st \tau \leq \sigma \in \Sigma } . \]
\(\Star(\tau)\) is a fan in \(N(\tau)_\RR\), and the orbit closure of \(\tau\) is its toric variety: \[ V(\tau) \da \cl \Orb(\tau) = X_{\Star(\tau)} , \qquad \dim V(\tau) = n - \dim \tau . \]
(Orbit closures)
By the orbit-cone correspondence, the torus orbits of \(X_\Sigma\) are the \(\Orb(\tau)\) for \(\tau\in\Sigma\). Every torus-invariant irreducible closed subvariety of \(X_\Sigma\) is \(V(\tau)\) for a unique \(\tau\in\Sigma\), hence is the toric variety of the fan \(\Star(\tau)\), of dimension \(n-\dim\tau\).
A ray \(\rho\) gives \(V(\rho) = D_\rho\), the boundary divisor, and \(\Star(\rho)\) is the image of the fan of cones containing \(\rho\) in \(N_\RR/\RR\rho\). For a complete toric surface, \(\Star(\rho)\) is the complete fan in \(\RR\), so \(D_\rho \cong \PP^1\). For the star subdivision of a smooth \(n\)-dimensional cone along the sum of its ray generators, the new ray \(\rho\) has \(\Star(\rho)\) the fan of \(\PP^{n-1}\), so the exceptional divisor is \(D_\rho \cong \PP^{n-1}\). A maximal cone gives a point.
The boundary \(X_\Sigma \setminus T\) is the union of the divisors \(D_\rho\). For rays \(\rho_1,\ldots,\rho_k\), the intersection \(D_{\rho_1}\cap\cdots\cap D_{\rho_k}\) is \(V(\tau)\) if \(\rho_1,\ldots,\rho_k\) span a cone \(\tau\in\Sigma\), and is empty otherwise; when \(\Sigma\) is smooth, the boundary is a simple normal crossings divisor.
Dependencies
PR-O8V3IThe orbit-cone correspondenceuses