I'm reading Alexeev's "Complete moduli in the presence of semiabelian group action" and I just started to study toric geometry.


In chapter 2 in order to obtain an affine toric variety he takes $P:=Spec(k[\omega\cap X])$ where $\omega$ a cone in $X\otimes \mathbb{R}$ and $X$ is a lattice. Then the torus $T=Hom(X,k^\*)$ acts on $P$ through the action on $k[\omega\cap X]$: given $\xi^u\in k[\omega\cap X]$ and $t\in T$, $t\cdot \xi^u=t(u)\xi^u$. Also, if $T'$ is the quotient of $T$ by the stabilizer of the generic point (which in this case I think is $Hom(\mathbb{R}\omega\cap X,k^\*)$), fixing a point $p\in P$ in the dense orbit, sending $1\in T'$ to $p\in P$ we obtain a Torus embedding $T'\subset P$.


Unfortunately, the projective case is quite obscure to me. At page 639 he takes $\delta$ a lattice polytope in $X\otimes \mathbb{R}$ and $Cone\delta\subset \mathbb{X}\_\mathbb{R}:=(X\oplus \mathbb{Z})\otimes\mathbb{R}$ the cone over $\delta$ lying in the hyperplane $(1,X\_{\mathbb{R}})$. Then he claims that $Q:=Proj(k[Cone\delta\cap \mathbb{X}])$ is a $projective$ $torus$ $embedding$ and it carries an action of the torus $T=Hom(X,k^\*)$. He doesn't specify anything more, but I really can't see how the action is defined, so my question is:


How is the action of $T$ on $Q$ defined? Which is the embedded torus in $Q$?


Thank you very much and again I'm sorry if my question is trivial but I'm just starting and I got stuck here..



Alexeev's projective torus embeddings
There are two ways to understand it.


**(1) A projective variety is a quotient of affine one.**
Let $V$ be denote the projective toric variety defined by polytope $\delta$ and $n=\mathrm{dim}(T)=\mathrm{rk}(X)$ be its dimension.


While we have $\mathrm{cone}(\delta)$ in lattice $X\oplus\mathbb Z$ we have an affine toric variety 
$$
V'=\mathrm{spec}(\mathbb C[\mathrm{cone}(\sigma)\cap X\oplus\mathbb Z])
$$
of dimension $n+1$. Now 
$$
V=(V'\smallsetminus 0)/\mathbb C^\*,
$$
where $0$ is $T-$fixed point on $V'$ and $\mathbb C^\*$ is one dimensional subtorus corresponding to $(\mathbb Z)$ in $X\oplus \mathbb Z$ (exercise: write down the action of this $\mathbb C^\*$ on $\mathbb C[\mathrm{cone}(\sigma)\cap X\oplus\mathbb Z]$). The big torus orbit in $V$ is just an image of the big torus orbit in $V'$ (and has dimension $n$).


Example. $\mathbb P^2$ can be defined by a plane polytope with vertices $(0,0),(0,1),(1,0)$. The toric affine variety corresponding to the cone is just $\mathbb C^3=\mathrm{spec}(\mathbb C[x,y,z])$ and $\mathbb P^2=(\mathbb C^3\smallsetminus 0)/\mathbb C^\*$.


**(2) Projective variety is defined by an embedding.**
This way the polytope $\delta$ provides all information and you can forget about $\mathrm{cone}(\delta)$.


While lattice $X$ has a basis $e\_1,\ldots, e\_n$, torus $T$ has coordinates $t\_1,\ldots,t\_n$ ($t\_i$ is just $e\_i:T\to \mathbb C^\*$). Actually, a point $m\in X$ provides a monomial:
$$
\mathrm{exp}(m)=\prod t\_i^{m\_i},
$$
where $m\_i$ are coordinates of $m$ in the basis $(e\_i)$.


Now a set of all (integer) points of the polytope $\delta$ gives a set of $|\delta|$ monomial ($|\delta|$ is the number of points). These $|\delta|$ monomial define a map of $T$ to projective space of dimension $|\delta|-1$ like in the following example. The toric variety is the closure of the image of $T$ and the torus consists of all points with non-zero coordinates.


Example. $\mathbb P^1$ is a toric variety and the torus has one coordinate. The one-dimensional polytope $[0,d]$ has $d+1$ points and defines an embedding
$$
\mathbb C^\*\to\mathbb P^d
$$
$$
t\mapsto (1:t:t^2\ldots:t^d),
$$
or, if $t=y/x$ would be coordinate on $\mathbb P^1$,  

$$
\mathbb P^1\to\mathbb P^d
$$
$$
(x:y)\mapsto (x^d:x^{d-1}y:\ldots:y^d).
$$


