Consider the complex algebraic torus $(\mathbb{C}^{*})^{n+1}$, where $\mathbb{C}^{*}=\mathbb{C}\smallsetminus 0$. It is a commutative Lie group under multiplication. The 2-form

(2)

$$\frac{1}{2i}\sum\limits_{j=1}^{n+1}\frac{dz}{z}\wedge\frac{d\bar{z}}{\bar{z}}$$

is an invariant symplectic form on $(\mathbb{C}^{*})^{n+1}$. There is an action of the real torus $T^{n+1}=S^{1}\times\dots\times S^{1}$ on $(\mathbb{C}^{*})^{n+1}$ by coordinatewise multiplication (we treat $S^{1}\subset\mathbb{C}^{*}$ as the unit circle). The action of $T^{n+1}$ is Hamiltonian and thus we have a well-defined moment map (we refer to [1] for the general definition or to a textbook, e.g. [2]) $\operatorname{Log}:(\mathbb{C}^{*})^{n+1}\to\mathbb{R}^{n+1}$

(3)

$$\operatorname{Log}(z_{1},\dots,z_{n+1})=(\log|z_{1}|,\dots,\log|z_{n+1}|).$$

Let $\Delta\subset\mathbb{R}^{n+1}$ be a convex polyhedron with integer (from $\mathbb{Z}^{n+1}$) vertices. Recall (see e.g. [4]) that there is a complex toric variety $\mathbb{C}T_{\Delta}\supset(\mathbb{C}^{*})^{n+1}$. One way to construct it is to consider the Veronese embedding $(\mathbb{C}^{*})^{n+1}\to{\mathbb{C}}{\mathbb{P}}^{\#(\Delta\cap\mathbb{Z}^{n+1})-1}$ defined by the linear system of monomials associated to $\Delta\cap\mathbb{Z}^{n+1}$. Here we associate to a point $(p_{1},\dots,p_{n+1})$ a monomial $z^{p_{1}}\dots z_{n+1}^{p_{n+1}}$. We define $\mathbb{C}T_{\Delta}$ as the closure of the image of the Veronese embedding. Note that the standard, Fubini-Study, symplectic form on the ambient space ${\mathbb{C}}{\mathbb{P}}^{\#(\Delta\cap\mathbb{Z}^{n+1})-1}$ defines a symplectic form on $\mathbb{C}T_{\Delta}$ (as long as the variety $\mathbb{C}T_{\Delta}$ is non-singular). In particular, it gives a symplectic form $\omega_{\Delta}$ on $(\mathbb{C}^{*})^{n+1}$ that invariant with respect to the action of $T_{\Delta}$. This gives us a moment map with respect to $\omega_{\Delta}$

$$\mu_{\Delta}:(\mathbb{C}^{*})^{n+1}\to\Delta,\mu_{\Delta}(z)=\frac{1}{\sum\limits_{j\in\Delta\cap\mathbb{Z}^{n+1}}|z^{2j}|}\sum\limits_{j\in\Delta\cap\mathbb{Z}^{n+1}}j|z^{2j}|.$$

The image of this embedding is the interior $\operatorname{Int}\Delta$. The map $\mu\Delta$ can be compactified to the moment map $\bar{\mu}_{\Delta}:\mathbb{C}T_{\Delta}\to\Delta$.

The maps $\operatorname{Log}:(\mathbb{C}^{*})^{n+1}\to\mathbb{R}^{n+1}$ and $\mu_{\Delta}:(\mathbb{C}^{*})^{n+1}\to\operatorname{Int}\Delta$ both have the orbits of $T^{n+1}$ as their fibers. Thus, they define a natural reparametrization

$$\Phi_{\Delta}:\mathbb{R}^{n+1}\to\operatorname{Int}\Delta.$$

We prove this proposition simultaneously with the following proposition describing the behavior of $\Pi$ near infinity. Recall that a supporting vector $\stackrel{{\scriptstyle\to}}{{v}}$ at a face $\Delta^{\prime}\subset\Delta$ is a vector such that $p_{\stackrel{{\scriptstyle\to}}{{v}}}|_{\Delta}$ reaches its maximum precisely over $\Delta^{\prime}$, where $p_{\stackrel{{\scriptstyle\to}}{{v}}}$ is the orthogonal projection in the direction of $\stackrel{{\scriptstyle\to}}{{v}}$.

If $\Pi$ is a maximal dual $\Delta$-complex then it is generic everywhere except at the points of its boundary $\partial\Pi$. The following proposition describes the local topology of $\bar{\Pi}$ near the boundary. It is a corollary of Proposition 1.12.

We call a point with such a neighborhood a $(k,l)$-point of $\bar{\Pi}$.