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..