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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..

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  • $\begingroup$ I could recommend to look into books about toric varieties, say, Cox et al. "Toric varieties", chapter 2 or Fulton "Introduction to toric varieties". $\endgroup$ – IBazhov Jan 28 '15 at 19:57
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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 $\mathbb P^1$,
$$ \mathbb P^1\to\mathbb P^d $$ $$ (x:y)\mapsto (x^d:x^{d-1}y:\ldots:y^d). $$

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  • $\begingroup$ More generally, (i) a projective(ly embedded) variety is the same thing as an affine cone, invariant under the dilation action, and (ii) giving a torus action on an affine variety is very much the same thing as giving a multigrading of its coordinate ring. What's special about the cone over a polytope (as opposed to just a lattice cone) is that it singles out one grading inside the multigrading to be the "geometric" one ("divide by this guy to projectivize"), holding the others in abeyance to be the "torus action". $\endgroup$ – Allen Knutson Jan 29 '15 at 10:24
  • $\begingroup$ Thank you very much, but I'm still a little bit confused.. how can I see that the vertices of the polytope correspond to the fixed points of the action of $T$? $\endgroup$ – User28341 Jan 30 '15 at 19:42
  • $\begingroup$ In (i) they come from 1-dimensional orbits of the affine cone and for a vertex $\tau\subset\sigma$ it will be described by ideal $\mathbb C[\mathrm{cone}(\sigma)\cap X\oplus \mathbb Z]\smallsetminus\mathbb C[\mathrm{cone}(\tau)\cap X\oplus \mathbb Z]$. In (ii) they look like $(1:0:\ldots:0)$ or $(0:0:\ldots:1)$ and, since $T$ acts by coordinates, they are $T-$fixed. To go from (ii) to (i) you should notice, that zeroes in $(1:0:\ldots:0)$ are the vanishing of monomials in the mentioned ideal. $\endgroup$ – IBazhov Jan 31 '15 at 10:05

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