The toroidal embedding arising from an irrational fan (Q1290390)

Using the language of valuations, Ford generalizes the construction of toric varieties to arbitrary (not necessarily rational) fans \(\Delta\). As in the classical case, the ringed space \(T_N\text{emb}(\Delta)\) comes with an open covering \(\{U_\sigma=T_N\text{emb}(\sigma)\}_{\sigma\in\Delta}\), and the usual semigroup rings \(S_\sigma\) appear as the \(U_\sigma\)-sections of the structure sheaf. However, the big difference is that, in the non-rational case, the \(U_\sigma\) are no longer affine. The set \(\text{Spec} S_\sigma\) is only part of \(U_\sigma\). On the other hand, \(T_N\text{emb}(\Delta)\) keeps most of the nice features we already know from the classical case. The construction is functorial, the space comes with a torus action, and the orbit space equals \(\Delta\) with the usual topology. Moreover, the open subsets \(U_\sigma\) behave well under intersections and the orbit decomposition. It is possible to establish a divisor theory dealing with Weil as well as with Cartier divisors. Finally, Ford presents a discussion of Cox's homogeneous coordinate ring construction. In section 4, there is a second construction of a ringed space out of a fan. Without using the lattice structure \(N\subseteq N_\mathbb R\), one obtains a space \(X(\Delta)\) just by taking the semigroup rings \(k[\sigma^\vee]\) instead of the usual \(S_\sigma=k[\sigma^\vee\cap M]\). In particular, there is always a map \(X(\Delta)\to T_N\text{emb}(\Delta)\) which is studied in the last part of the paper.