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15 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 ...
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0answers
10 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
16 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 ...
1
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0answers
13 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 ...
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2answers
92 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
30 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 ...
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1answer
24 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 ...
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0answers
32 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 ...
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0answers
14 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
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1answer
34 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 ...
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0answers
19 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
80 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
103 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 ...
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0answers
31 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
108 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 ...
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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
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1answer
70 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 $S_{\sigma}$ ...
4
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1answer
84 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. ...
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0answers
33 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 ...
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0answers
23 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 ...
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34 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 ...
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1answer
40 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 ...
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0answers
31 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
57 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 ...
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0answers
64 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 ...
2
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0answers
33 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
150 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 ...
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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? ...
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0answers
65 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 ...
0
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1answer
40 views

Question about toric ideal

In the proposition 1.2 contained in the following http://www.math.harvard.edu/~jbland/ma232b2_notes.pdf, I can't understand why a monomial satisfying (1.7) exists. Can you help me? Thanks.
4
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0answers
88 views

Good confusion-avoiding notation for gluing toric varieties from fans?

$ \newcommand\R{\mathbb R} \newcommand\C{\mathbb C} \DeclareMathOperator\Cone{Cone} $I'm trying to establish a good notation to avoid confusion when we glue toric varieties from affine pieces. A ...
2
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1answer
101 views

Faces of cones give localizations of affine toric varieties

I'm trying to establish an explicit description of the open subsets of affine toric varieties given by faces of the underlying cone. Background For a rational convex polyhedral cone $\sigma\subseteq ...
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1answer
353 views

Dual of a rational convex polyhedral cone

A rational convex polyehedral cone $\sigma\subseteq\mathbb R^n$ is a set of the form $$ \sigma=\operatorname{Cone}(u_1,\dots,u_k) := \left\{ \sum_{i=1}^k r_i u_i \,\Bigg|\, r_i\ge ...
4
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2answers
119 views

Showing a quotient $\mathbb{Z}$ module is free

In Fulton's "Introduction to Toric Varieties" he repeatedly uses the following fact. Let $\sigma$ be a strongly convex rational polyhedral cone in a lattice $N$ and let $N_{\sigma}$ be the subgroup ...
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2answers
222 views

Prove that $(C \cap D)^\vee = C^\vee + D^\vee$.

Definitions and notations: Let $M$ to be the $n$-dimensional Euclidean space $\mathbb{R}^n$ and $N$ its dual space $M^\ast = \mathrm{Hom}_\mathbb{R}(M, \mathbb{R}).$ A subset $C \subset N$ is said to ...
4
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1answer
237 views

Motivation and Applications for Toric Varieties

I'm a graduate student of mathematics starting to study algebraic geometry with a focus on toric varieties (along Cox, Little, Schenk). From what I learned so far, I can grasp that toric varieties ...
4
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3answers
399 views

Isomorphism of lattices

This is an exercise that appears in some sheet of exercises in toric geometry. Take two lattices, $N=\mathbb{Z}^{3}$ and $N'=\operatorname{Span}_\mathbb{Z}\{(-1,-1,1), (-1,2,1), (2,-1,1)\}$. The ...
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0answers
67 views

Realizing a list of varieties as the toric variety of a fan

The locus of the following curves define affine varieties. $xy-z^m=0.$ $x^m+xy^2+z^2=0.$ $x^4+y^2+z^2=0.$ $x^2+y^3+z^2=0.$ $x^5+y^3+z^2=0.$ In Fulton's "Toric Varieties" book he constructs toric ...
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0answers
156 views

Procedure to find generators of the dual cone

Page 11 of Fulton's "Toric Varieties" gives the following procedure for finding generators of the dual of a convex polyhedral cone in $\mathbb{R}^d$: For each set of $n-1$ independent vectors ...
3
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2answers
322 views

Reference request: toric geometry

What is a good book on algebraic geometry, with focus on toric varieties, similar both in the philosophy and in the prestige of the autors to Modern Geometric Structures and Fields by Novikov and ...
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0answers
43 views

$H^{1}(O_{F})$ of a surface in a toric variety

I have a surface inside a toric variety $X$ and I would like to compute the first cohomology of its structure sheaf via the Cech complex, since I already know which cones of $X$ it hits (five ...
3
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1answer
206 views

Example of a variety that is not toric

My question is simple, but I haven't seen it to be addressed anywhere: What would be a simple example of an affine variety that is not a toric variety? Toric varieties (the ones I have ...
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0answers
92 views

Injectivity of a rational map

I have a variety $C_{0}\subset\mathrm{Spec}\mathbb{C}[M_{Y}]$ living in some $Y$ (which is toric) and I am guessing, that it is $\mathbb{P}^{1}$. So I have to find a birational function ...
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0answers
145 views

Classification of Cones

I am attempting to classify (convex rational) cones in $\Bbb{R}^2$. We say here that $\sigma\subset\Bbb{R}^2$ is a cone if there exist $u,v\in\Bbb{Z}^2$ which $\Bbb{R}$-span the whole plane ...
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1answer
370 views

Toric Varieties: gluing of affine varieties (blow-up example)

Let $\Delta$ be a fan, consisting of cones $\sigma_0=conv(e_1,e_1+e_2)$ and $\sigma_2=conv(e_1+e_2,e_2)$ and $\tau=\sigma_0\cap\sigma_1=conv(e_1+e_2)$. The dual cones are $\sigma_0^\vee= ...
3
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1answer
131 views

Toric Varieties from Cones

Consider the lattice $N=\Bbb{Z}^d$ spanned by $e_1,\dots,e_d$ and the cone $$\sigma=\text{Cone}\{e_1,\dots,e_k\}, \quad k<d.$$ I am trying to understand why the toric variety $V_\sigma$ obtained is ...
3
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1answer
168 views

Finding a toric variety of a cone

I'm trying to find the toric variety associated to the cone $\sigma_0$ which is the region in the real plane with $x\geq 0$ and $y-x\geq 0.$ I found that it's dual cone is $\check{\sigma_0}$ the ...