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### convex polyhedral cone and faces, one to one inclusion preserving map from faces of cone to faces of quotient cones.

I'm doing with a problem in 'lectures on toric variety' written by David Cox. In page 15, exercise 1.5, Suppose that $\sigma$ $\subset$ in $\mathbb R^n$ is convex polyhedral cone of dim d, and assume ...
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### Torus-invariant divisor on $\mathbb{CP}^2\#1\mathbb{CP}^2$

I want to confirm that the following result is right: $M=\mathbb{CP}^2\#1\mathbb{CP}^2$. I was told that the torus-invariant divisors on $M$ are $H,H-E,E$, where $E$ and $H$ are the hyperplane ...
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### Zariski closure of $T:= \{(t_1,t_2,t_3,t_1t_2t_3^{-1})|t_i\in \mathbb{C}^*\}\subseteq \mathbb{C}^4$?

Let $V= \mathcal{V}(\langle xy-zw\rangle)\subseteq \mathbb{C}^4$ be an affine variety. The set $T:= \{(t_1,t_2,t_3,t_1t_2t_3^{-1})|t_i\in \mathbb{C}^*\}$ is a torus contained in $V$. I am trying to ...
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### Can the quotient by a nonabelian group yield an abelian singularity?

Let $V$ be a complex vector space with a faithful linear action of a finite group $G$. Viewing $V$ as affine space (with coordinate ring $\mathbb{C}[V]$), the quotient $V/G$ is the affine variety with ...
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### Where do we use that $\sigma$ is a maximal dimension cone?

Let $N$ be a lattice (free $\mathbb Z$ - module of rank $n$) with dual $M=\operatorname{Hom}_{\mathbb Z - \text{mod}}(N,\mathbb Z)$ and let $\sigma$ be a cone in $N\otimes\mathbb R\cong\mathbb R^n$. ...
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### The induced map $\phi: \mathbb{C}^n \to \mathbb{C}^m$ in the construction of toric varieties

Let $\Sigma(1)$ denote the set of one dimensional cones in a fan $\Sigma$. The corresponding vectors in the lattice are denoted $(v_1, \ldots, v_n)$ and to each $v_i$ we associate a homogeneous ...
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### On factorization theorem of toric birational morphisms

Let $X_{\Sigma'} \to X_{\Sigma}$ be a toric birational morphism between smooth and complete toric varieties induced by a regular subdivision $\Sigma' \leq \Sigma$, i.e. every cone in $\Sigma'$ is ...
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### Does every collection of edge vectors of a cone span a face of the cone?

Definition 1 : A polyhedral cone is a subset of a real vector space which is the intersection of finitely many closed half spaces. (The defining planes of these half spaces must pass through $0$.) A ...
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### Fiber product of toric varieties

Let $X$,$Y$ and $Z$ be three toric varieties defined by the fans $\Sigma_X\subset (N_X)_{\mathbb{R}}$, $\Sigma_Y\subset (N_Y)_{\mathbb{R}}$ and $\Sigma_Z\subset (N_Z)_{\mathbb{R}}$, respectively. It ...
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### $\mathbb{Q}$ divisors on a concrete toric variety: contradiction

I got a contradiction. Help me to understand it. It is about answer by SĂˇndor KovĂˇcs of a question http://mathoverflow.net/questions/55526/example-of-a-variety-with-k-x-mathbb-q-cartier-but-not-...
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### Delzant theorem for polyhedra

Delzant theorem says that there is a 1-1 correspondence between compact toric symplectic manifolds (modulo equivariant symplectomorphism) and the Delzant polytopes (modulo lattice isomorphism). The ...
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### Proj construction in toric geometry

Let $X$ be a complete toric variety and let $D \subset X$ be a toric divisor which is nef and big. Consider the graded algebra $R(D) = \bigoplus_k H^0(X, kD)$. This is a finitely generated graded ...
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### Prove that $\{m \in S_{\sigma} \, | \, \gamma(m) \neq 0\}$ is a face of $\sigma^V \cap M$

I am trying to solve exercise 3.2.6 pag.124 of Cox, Little, Schenck book http://www.math.colostate.edu/~renzo/teaching/Toric14/CoxLittleShenck.pdf because it is required to prove orbit-cone ...
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### Smooth Fano Polytopes and Hypersurfaces

This is a rather extended question, so I will try to make it as compact and readable as possible. I am trying to practice with the Macaulay2 software, in particular the polyhedra and ...
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### How to prove this is a fibre bundle?

Let $M^{2n}$ be $2n$ dimensional toric manifold over a simple polytope $P^n$. Let $\pi : M^{2n} \longrightarrow P^n$ be the orbit map of the torus action. Let $F^k$ be a $k$ dimensional face of $P^n$. ...