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1
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0answers
34 views

Does this theorem hold for real part of a toric variety? + Reference request - Real toric varieties.

Let $X$ be a complex toric variety and let $X_\mathbb R$ be its real part, that is the $X_\mathbb R$ consists of all the real valued points of $X$. I would like to learn a little more about the ...
3
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0answers
40 views
+200

The lattice points in the real cone of some semigroups are just the integer cone of that semigroup.

I'm trying to solve an exercise in Fulton's book on toric varieties, and have reduced it to the following: Let $M$ be a lattice of rank $n$ with $M \otimes \mathbb{R} = V$, and $S$ be a finitely ...
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0answers
26 views

whats wrong with this counterexample to closed subgroups of a Torus are a torus

In Cox Little and Schenck, one result that is cited in chapter two is that if $D_n$ is the $n-dimensional$ torus, and $H < D_n$ is a closed subgroup then $H$ is itself a torus. Let the underlying ...
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0answers
28 views

Binomial ideals that are not toric i.e. binomial and not prime?

Let $R$ be a ring. Toric ideal $I$ is binomial (generated by a binomial) and prime (the quetient ring $R/I$ is integral domain). Paper and corollary 1.3 is to determine whether an ideal is binomial. ...
2
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1answer
53 views

Has toric ideal something to do with torus?

I am studying ideals such as toric ideals but I am unable to find a consistent definition, it seems to be very general so please explain the origin of "toric ideal". Is there a geometric ...
0
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0answers
30 views

Fan associated to a rectangle

In this question, there is a claim that the fan corresponding to the square in $\mathbb{R}^2$ consists of the 4 quadrants - but I think this is wrong or I am confused. The faces of the square are ...
0
votes
1answer
20 views

Key reference book on toric ideals: normal or not? Which definition to follow?

I want to understand sum of binomials better in terms of ideals such as binomial ideals, normal ideals and so by toric ideals. Examples about toric ideals contain $$\sum x^\alpha+\sum ...
1
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1answer
43 views

affine variety/space vs. toric variety

I think I'm not quite clear on the meaning of a toric variety... Could someone explain the relation/difference between the affine variety/space and the toric variety? I know that affine ...
0
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0answers
20 views

Toric variety associated to a rotated fan

Let $\Delta$ be a fan in the lattice $\mathbb Z^2$ consisting of SCRAP cones and let $X(\Delta)$ be the associated toric surface. Suppose we rotate fan in the counter-clockwise direction by a rational ...
0
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0answers
7 views

On two-dimensional non-singular complete toric varieties

Let $v_0,v_1,\dots,v_d=v_0$ be a sequence of lattice points in $\mathbb Z^2$, in counterclockwise order (see figure below), such that successive pairs generate $\mathbb Z^2$ as a $\mathbb Z$-module. ...
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0answers
15 views

Realizing $\mathbf{C}$ as 1-dimensional normal toric variety

Take $N = \mathbf{Z}$, $N_\mathbf{R} = \mathbf{R}$. Then I know that the only cones are the intervals $\sigma_+ = [0,\infty)$, $\sigma_- =(-\infty,0]$, and $\tau = \{0\}$. Consider the fans ...
0
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1answer
74 views

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 ...
1
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0answers
41 views

Examples of homogeneous polynomials that define $\mathbb{P}^n$

The complex projective space $\mathbb{P}^n$ is also a projective variety. According to Hartshorne, a projective variety is defined a s the zero set of a subset of homogeneous polynomials defined on ...
3
votes
0answers
23 views

Toric variety corresponding to coordinate axes in $\mathbb{R}^2$

I have just learned how to construct a toric variety from a fan and I am a bit confused. Let $\Sigma$ be the fan that consists of the coordinate axes in $\mathbb{R}^2$, i.e. $\Sigma = \{ \sigma_0, ...
0
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0answers
16 views

Toric varieties

I've started to read about toric varieties and I have a couple of questions about the definition. There is an example that says the following: "Given a lattice $N$, an isomorphism $N \simeq ...
0
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0answers
30 views

What is this action of $\mathbb Z_2$ on $\mathbb C^2$ that gives the following affine toric variety?

Let $\sigma$ be the cone in $\mathbb R^2$ given by $\langle e_1, e_1+2e_2\rangle$. The corresponding affine variety $U_\sigma=\mathcal{Z}(x^2-yz)\subseteq \mathbb C^3$ I am trying to understand the ...
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0answers
18 views

on the real part of a toric variety

Corresponding to a strongly convex rational polyhedral cone in the lattice $\sigma$ in the lattice $N$ we have the affine toric variety $U_\sigma=\operatorname{Hom_{semi group}}(\sigma^\vee\cap ...
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0answers
19 views

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 $N\otimes\mathbb R\cong\mathbb R^n$. ...
1
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0answers
19 views

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 ...
1
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0answers
16 views

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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0answers
20 views

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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0answers
18 views

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 ...
1
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2answers
95 views

$\mathbb{Q}$ divisors on a concrete toric variety: contradiction

I got a contradiction. Help me to understand it. It is about answer by Sándor Kovács of a question ...
0
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0answers
32 views

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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0answers
18 views

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 ...
0
votes
1answer
29 views

Degree of nef toric divisors which are not big

Let $X$ be a complete toric variety of dimension $n$. It is a classical result that if $D$ is a toric nef divisor, then its degree $D^n$ can be computed as the Volume of the corresponding polytope ...
2
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0answers
35 views

Is anticanonical divisor on complete toric variety big?

It seems to me that for any complete toric variety $P_\Sigma$, the anticanonical divisor is big. Moreover, the argument also shows that $P_\Sigma$ is projective. This sounds a bit strange for me, did ...
0
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0answers
15 views

Complete toric varieties with given codimension of the singular locus

Let $N\cong \mathbb{Z}^n$ be a lattice and $\Delta\subseteq N_\mathbb{R}$ be a fan such that $X=X(\Delta)$ is a complete and simplicial (i.e. $\mathbb{Q}-$factorial) toric variety of dimension $n$ ...
3
votes
1answer
37 views

Prove that the closure of an orbit is invariant with respect to the torus action

I refer to the book http://www.math.colostate.edu/~renzo/teaching/Toric14/CoxLittleShenck.pdf. Let $X_{\Sigma}$ be the toric variety of fan $\Sigma$. Let $\sigma$ be a cone in $\Sigma$. We define ...
2
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0answers
21 views

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 ...
4
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1answer
87 views

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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0answers
107 views

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 $P^n$. ...
4
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1answer
66 views

Computing the sheaf of 1-forms on a toric variety

Consider projective space $P^{2}$ and its corresponding fan. We have the affine opens defined by $U_{\sigma_{0}} = Spec(\mathbb{C}[x,y])$, $U_{\sigma_{1}} = Spec(\mathbb{C}[x^{-1},x^{-1}y])$ and ...
2
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0answers
32 views

How to show that the inverse image of a face of a simple polytope is a connected manifold?

If $M^{2n}$ is a toric manifold over a simple polytope $P^n$ i.e; the orbit space of the action of the $(S^1)^n$ on $M^{2n}$ is an $n$ dimensional simple polytope $P^n$. Let $\pi : M^{2n} \rightarrow ...
3
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2answers
109 views

Properties of the functor $(X, \mathcal{O}_X) \mapsto (X(k), \mathcal{O}_{X(k)})$

Let $k$ be an algebraically closed field. In Görtz and Wedhorns book one can read about an equivalence of categories $\{\text{integral schemes of finite type over } k\} \to \{\text{prevarieties over ...
4
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0answers
58 views

Isomorphism of divisors

Consider the cartier divisor group $CDiv_{T_{N}}(X_{\Sigma})$ defined on the fan $X_{\Sigma}$. I am having trouble proving the following assertion that there is a natural isomorphism ...
2
votes
1answer
75 views

Distinguished points of a cone

Sorry, as this is a rather trivial question that I am misunderstanding, but I do not understand how the distinguished point is defined. We define it as a homomorphism from some semigroup $S_{\sigma}$ ...
5
votes
1answer
102 views

Finding generators of toric ideals

Consider the affine toric variety $V \subset k^{5}$ parametrized by $$\Phi(s,t,u) = (s^{4},t^{4},u^{4},s^{8}u,t^{12}u^{3}) \in k^{5}$$ where k is an algebraically closed field of characteristic 2. ...
2
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0answers
34 views

Centers of divisorial valuations on toric varieties

Suppose we are given a divisorial valuation $\mathcal{v}$ on a smooth toric variety $X_{\Sigma}$ (for a fan $\Sigma$), i.e. $\mathcal{v}$ is the valuation induced by a torus invariant divisor ...
0
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0answers
24 views

Uniqueness of supporting hyperplane for a face of a cone

In William Fulton's 'Introduction to Toric varieties' he says - " When $\sigma$ spans $V$ and $\tau$ is a facet of $\sigma$ then there is a $u \in \sigma ^{\vee}$ unique upto multiplication by a ...
3
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0answers
40 views

Question regarding example of toric variety and generators of cone

Consider the canonical example taking n=2, and taking the cone $\sigma$ generated by the vectors $e_{2}$ and $2e_{1} - e_{2}$. The dual cone $\sigma^{v}$ is defined as the set of vectors in the dual ...
1
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1answer
44 views

Are the orbits of a symplectic toric manifold the fibers of its moment map?

A symplectic toric manifold, by definition, carries an effective torus action generated by a moment map. The orbits of the torus action are of course contained in the fibers of the moment map, but ...
3
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0answers
43 views

Open embedding of affine toric varieties implies Cone is face of the other

let $\tau, \sigma \subseteq N_{\mathbb{R}}$ be two rational, strongly convex polyhedral cones with $\tau \subseteq \sigma$. Now we get an inclusion $S_{\sigma} \to S_{\tau}$ inducing an inclusion ...
2
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1answer
71 views

Perfect pairing induces isomorphism of tensor products

Let $M, N$ be $R$-modules and $(\cdot, \cdot): M \times N \to R$ be a perfect pairing. Wikipedia sais that this means that the map $\varphi: M \to \text{Hom}_R(N, R), m \mapsto (n \mapsto (m, n))$ is ...
4
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0answers
68 views

Weighted Projective Spaces and varieties

Consider the weighted projective space $\mathbb{P}(1,1,2)$ with variables $x_0,x_1,x_2$ of degrees 1,1,2 respectively. Consider the map $$(a_0,a_1,a_2) \to (a_0^2,a_0 a_1,a_1^2,a_2)$$ with ...
3
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0answers
34 views

Inclusion of Tori induces surjection of character groups?

Let $k$ be an algebraic closed field. Let $T, T'$ be algebraic Tori in the classical sense, meaning $T \cong \mathbb{A}_k^n \setminus V(X_1 \cdots X_n)$, $T' \cong \mathbb{A}_k^{n'} \setminus V(X_1 ...
2
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1answer
175 views

Blow-up toric varieties.

I have to take a talk of an hour and I have to talk about blow-up of toric varieties. Can you suggest me some interesting examples that I can present? How can I find a good reference for the theory ...
0
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1answer
38 views

CY 3-folds are $T^2 \times \mathbb{R}$ fibrations over the base $\mathbb{R}^3$. What does it mean?

In this article at section 2. Toric geometry and Mirror Symmetry there is the statement that CY 3-folds are $T^2 \times \mathbb{R}$ fibrations over the base $\mathbb{R}^3$. Now, my questions refers to ...
1
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0answers
27 views

Are the following Hamiltonian actions?

Let $(x,y)$ be the coordinate of $\mathbb{C}^2 \subset \mathbb{P}^1\times \mathbb{P}^1$. Is the $S^1$ action on $\mathbb{P}^1\times \mathbb{P}^1$ given by $$ t\cdot(x,y)=(tx,t^{-1}y) $$ Hamiltonian? ...
2
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0answers
68 views

Fibers of toric morpisms

Let $f: X(\Delta_1) \to X(\Delta_2)$ be a toric morphism of toric varieties, and let $\sigma \subset \Delta_2$ be a cone, then for any point in the corresponding orbit $x \in O(\sigma)$ the fiber ...