(Localisation at a face)
Let \(\tau \leq \sigma\) be a face, written \(\tau = \sigma \intersect u_\tau^\perp\) for some \(u_\tau \in S_\sigma = \dualof{\sigma} \intersect M\). Then \[ S_\tau = S_\sigma + \NN \cdot (-u_\tau) , \] so \[ k[S_\tau] = k[S_\sigma]\big[ \chi^{-u_\tau} \big] = k[S_\sigma]_{\chi^{u_\tau}} , \qquad U_\tau = D\big( \chi^{u_\tau} \big) \subseteq U_\sigma , \] a principal open subset.
(This is the gluing)
The construction of \(X_\Sigma\) says the \(U_\sigma\) are glued along the charts of their common faces, and this proposition is what makes that meaningful. If \(\sigma_1 \intersect \sigma_2 = \tau\) is a common face, then \(U_\tau\) sits inside each of \(U_{\sigma_1}\) and \(U_{\sigma_2}\) as a principal open subset, and the gluing is the identity on \(U_\tau\). Both inclusions are open immersions, so the result is a variety and not just a topological quotient, and the compatibility on triple overlaps is automatic because every chart is a localisation of the coordinate ring of the smallest cone.
The inclusion is order-reversing on cones: a smaller cone has a larger dual, hence a larger semigroup and a larger ring, hence a smaller variety. \[ \tau \leq \sigma \implies \dualof{\sigma} \subseteq \dualof{\tau} \implies U_\tau \subseteq U_\sigma . \] The zero cone gives \(S_{\ts{0}} = M\) and \(U_{\ts{0}} = T\), which is therefore contained in every chart: that is why the torus is dense in \(X_\Sigma\).
(The plane, in two charts)
Take \(\sigma_1 = \Cone(e_1, e_2)\) and the face \(\tau = \Cone(e_1)\), cut out by \(u_\tau = \dualof{e_2}\). Then \(S_{\sigma_1} = \NN^2\) and \[ S_\tau = \NN^2 + \NN \cdot (0,-1) = \NN \oplus \ZZ , \qquad U_\tau = \Spec k[x, y^{\pm 1}] = D(y) \subseteq \AA^2 . \] For \(\PP^1\), the two maximal cones \(\RR_{\geq 0}\) and \(\RR_{\leq 0}\) share the face \(\ts{0}\), whose chart is \(\Spec k[x^{\pm 1}] = \GG_m\), inverted from \(k[x]\) on one side and from \(k[\inverseof{x}]\) on the other; the gluing \(x \mapsto \inverseof{x}\) is forced.
Dependencies
PR-FULFACETFacet normals and the half-space presentation of a coneusesD-Q7Q2NFrom a fan to a toric varietyuses