See Chapter 7 of [Toric Varieties, Cox Little Schneck] or Example 3.6 of [Tropical Geometry and Mirror Symmetry, Gross].
The one-sentence construction is this: start with a polyhedron \(\sigma \subset M_\mathbb R\), take its normal fan \(\Sigma \subset N_\mathbb R\), then the toric variety \(X_\sigma\) is \(X_\Sigma\). However you don’t actually have to move over to the normal fan, you can just do it directly. This corresponds to the cone picture in Gross-Siebert.
Construction without normal fan Let \(\sigma\) be a strictly convex polytope in \(M_\mathbb R\) (the character lattice of a torus tensored with \(\mathbb R\)). For each facet \(\tau\subset \sigma\), form the cone \[C_\tau := \cone\left(\sigma - \tau\right) = \left\{n \in M_\mathbb R ~ \middle | ~ \exists a \in \sigma, b \in \tau, \lambda \in \mathbb R_{\geq 0} \text{ s.t. } n = \lambda a - b \right\}.\]
- for a vertex \(v\), the cone \(C_v\) is obtained by translating \(\sigma\) so \(v\) is at the origin and then taking the \(\mathbb R_{\geq 0}\) span.
- for a facet \(\tau\), do the same thing but at every point in \(\tau\) and then take the union. Equivalently, you can take any point \(p\) in the relative interior of \(\tau\), translate \(\sigma\) so \(p\) is the origin, and then take the cone over that.
Note the following properties of these cones:
- since \(\sigma\) is strinctly convex, \(C_v\) does not contain a line
- the cone \(C_\tau\) contains a copy of \(\mathbb R^d\) where \(d\) is the dimension of the facet \(\tau\)
- when \(\tau \subset \tau'\) we have \(C_\tau \subset C_{\tau'}\).
We then define \(U_\tau = \Spec\mathbb C[C_\tau \cap M]\) and get inclusions \(U_\tau \hookrightarrow U_{\tau'}\) whenever \(\tau \subset \tau'\). The toric variety \(X_\sigma\) is then the colimit of this diagram, the result of gluing along common inclusions.