Questions tagged [toric-geometry]

For questions related to toric geometry. The objects of study in toric geometry are toric varieties. Toric varieties are called ‘toric’ because they are equipped with a ‘torus action’. By a torus we mean the linear algebraic group $C^∗ ×\dots× C^∗$, not the torus from topology. A toric variety contains a torus as an open subset and this defines the torus action.

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Projection of rational normal curve is still a rational normal curve (of smaller degree)?

We work over $\mathbb{C}$. Let us define the rational normal curve of degree $d$ as the image of the morphism $$\nu_d:\mathbb{P}^1\to \mathbb{P}^d,\quad [x:y]\mapsto [x^d:x^{d-1}y:\ldots:xy^{d-1}:y^d]....
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How to see this assertion about coordinate rings and a localisations

I am studying 'The Homogeneous Coordinate Ring of a Toric Variety' by David Cox, and in the proof of his theorem 2.1 he defines $U_\sigma := \{x\in \mathbb{C}^{\Sigma(1)} : x^{\hat{\sigma}}\neq0\}$ ...
1 vote
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Conditions for evaluation morphisms $H^{0}(X, -K_X) \rightarrow k^n$ to be surjective.

Suppose I have a projective toric variety $X = X(N)$ (over a field $k$) associated to a polytope $N$, and I have $p_1,...,p_n \in D$ generically chosen points in the toric boundary $D$ of $X$. I am ...
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Chow group of disjoint union

I have a disjoint union of open sets $U_1,..., U_k$ on a variety $X$. In Fultons "Introduction to toric varieties", he used $X=X_\Sigma$ a toric variety and $U_i=\mathcal{O}(\sigma_i)$ ...
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What is wrong with this proof of $\operatorname{Cl}(X_\Sigma) \simeq \mathbb{Z}^{r-n}$?

This question is a bit urgent, as I have written this in my Master's thesis and I have to submit it next week! Proposition: Let $X_\Sigma$ be a toric variety without torus factors, meaning that the ...
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1 vote
1 answer
57 views

Why is for an open subset $U\subset X$ of a variety the field of rational functions equal, i.e. $\mathbb{C}(U)=\mathbb{C}(X)$?

Given any variety $X$, we can get the field of rational functions $\mathbb{C}(X)$. If we take any non-empty open subset of $U$, we get another field of rational fractions $\mathbb{C}(U)$. These fields ...
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3 votes
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54 views

Drawing the toric diagram for $\mathcal{L}^{m} \rightarrow T^{2}$ geometries.

I'm having some problems to understand the geometry involved in the section 3.1 of the paper Two Dimensional Yang-Mills, Black Holes and Topological Strings. Concretely, I was wondering to know if it ...
1 vote
0 answers
42 views

Intersection multiplicity on toric varieties using simplicial cones

In the book "Introduction to toric varieties" by William Fulton (p. 100) there is a formula for the intersection product of two subvarieties. Let $\tau,\sigma$ be cones in the fan $\Delta$, ...
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4 votes
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135 views

How does gluing of affine patches of toric varieties at the examples $\mathbb{P}^2$ and $\mathcal{H}_r$ work?

I don't fully understand how the gluing of the affine parts of a toric variety exactly works. I have a hard time developing a common sense or any intuition how to tell the result of a gluing morphism ...
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1 vote
1 answer
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Chow group of UFD is trivial?

In literature I saw the word "Chow group problem" coming up, questioning under what conditions the Chow Group is trivial ($=0$). In general this seems to be quite a difficult question. What ...
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1 vote
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Which blowups are toric?

I am interested in blowups of normal toric varieties (i.e. toric varieties which come from a fan $\Sigma$). I know that blowups along torus-invariant closed subvarieties $V(\sigma), \sigma \in\Sigma$ ...
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1 vote
1 answer
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A semigroup $\mathbb{N}\mathcal{A}$ is saturated in $M$ if and only if $\mathbb{N}\mathcal{A}=\text{Cone}(\mathcal{A})\cap M$

I'll first recall some definitions here for convenience. Given a finite set $S$ in a real vector space $V$, $$\text{Cone}(S)=\{\sum\limits_{u\in S} \lambda_u u\mid \lambda_u\geq 0\}.$$ An affine ...
2 votes
1 answer
199 views

Simplicial Toric Varieties

I am trying to solve an exercise in the well known book "Toric Varieties" by Cox, Little and Schenck: Prop $4.2.7$: Let $X_\Sigma$ be the toric variety of the fan $\Sigma$. Then the ...
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1 answer
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A basis for the Picard group of the blow up at $\mathbb{P}^2$ at two points

(Toric) Intro: (Superflous, just to consider the initial data) From a toric point of view, the blow up of $\mathbb{P}^2$ at two points (which we can assume are $[0:1:0]$ and $[0:0:1]$ is given by a ...
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2 votes
2 answers
283 views

Couple of questions about Picard group of $\mathbb{C}^*$

I'd like to compute the Picard group of $\mathbb{P}^n\times \mathbb{C}^*$. So using toric geometry I've easily found $\text{Cl}(\mathbb{P}^n\times \mathbb{C}^*)\simeq \text{Cl}(\mathbb{P}^n)\oplus\...
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3 votes
0 answers
112 views

How can I determine whether my variety is toric or not?

Assume throughout that $X$ is a normal rational variety over $\mathbb{C}$. Is there a way to determine whether or not $X$ is a toric variety? I am particularly interested in the following examples. 1)...
1 vote
1 answer
190 views

Rational Normal Cone of degree d

Consider the surface in $\mathbb{C}^{d+1}$ parametrized by the map $$\theta:\mathbb{C}^2\rightarrow\mathbb{C}^{d+1}$$ defined by $(s,t)\rightarrow(s^d,s^{d-1}t,\dots,st^{d-1},t^d)$ And $J_d=\...
2 votes
0 answers
160 views

Is the cotangent bundle to $P^2$ a toric variety?

Is the cotangent bundle to the complex projective plane, $T^* P^2$, a toric variety? More generally, is there a criterion for when a vector bundle over a toric variety is a toric variety? For example,...
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1 vote
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45 views

Two equivalent definitions of fibration structure on toric CY $3$-fold

I've been trying to reconcile two seemingly different definitions of what a toric space is, specifically a toric Calabi-Yau 3-fold. The first definition is from the paper Branes and Toric Geometry, by ...
2 votes
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217 views

The idea behind Delzant construction of a toric manifold from a convex polytope

I am trying to understand how to visualize a symplectic toric manifold from its moment polytope, following chapter 29.4 in "Lectures on Symplectic Geometry" by Ana Cannas da Silva: https://people.math....
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2 votes
1 answer
212 views

The dual of a regular polyhedral cone is regular

A rational polyhedral cone in $\mathbb{R}^n$ is a set of the form $$\sigma=\{\lambda_1x_1+\dots+\lambda_kx_k\in \mathbb{R}^n\mid \lambda_i\in \mathbb{R}_{\geq 0} \; \;\forall \, 1\leq i\leq k\}$$ for ...
4 votes
0 answers
117 views

What happens with subdivisions of normal fans in Sage?

I've been trying to compute specific subdivisions of a particular 4D complete fan, to try to speed up computations I have started looking into using Sage. The problem I'm having is that I would like ...
2 votes
0 answers
184 views

Is the quotient of a toric variety by a finite group still toric

Suppose $X$ is a toric variety with fan $\Sigma$, and the lattice of its torus is $N$. Suppose there is a representation of finite group $G$, \begin{equation} \phi:G \rightarrow \text{GL}(N \otimes_{\...
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1 vote
1 answer
495 views

Toric surfaces among rational surfaces

I'm learning about rational surfaces. Precisely, I've just read that all minimal rational projective, smooth surfaces are $\mathbb{P}^2$ and the Hirzebruch surfaces $\mathbb{F}_n$, for $n=0,2,3,\ldots$...
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1 answer
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Recovering data about the toric fan from minimal information

I am a physics grad student studying toric diagrams since they naturally arise in the description of certain gauge theories obtained by compactifying M/F-theory on Calabi-Yau 3-folds. In this context, ...
1 vote
0 answers
133 views

Intersection numbers from matrix of divisors and relations

There is a specific procedure to "read-off" or "compute" intersection products of divisors and curves, and of divisors and exceptional divisiors, from the so called matrix of relations (I don't know ...
3 votes
0 answers
237 views

Question about the relation between the Weierstrass equation and weighted projective space

I am reading a review on Toric Geometry for string theorists by Harald Skarke (arXiv:hep-th/9806059). In section 3, the author says A standard way of describing an elliptic curve is by embedding it ...
0 votes
1 answer
127 views

When is a projective space smooth from the toric perspective?

I am studying the lecture notes on toric geometry (for physicists) by Bouchard (arxiv:hep-th/0702063) and Skarke (arXiv:hep-th/9806059). In these lecture notes, the authors give the following ...
2 votes
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Determining wall crossings in toric geometry

In toric geometry, we can describe manifolds by specifying a fan, which is roughly a collection of cones (of various dimensions) in an $n$-dimensional lattice. Many topological properties can be ...
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2 votes
0 answers
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Is the nodal curve a toric variety?

Let $k$ be a field with $2 \in k^{\times}$. Set $$A:= k[x,y,z]/(y^{2}z = x^{3}+x^{2}z)$$ and let $X := \operatorname{Proj} A$ be the nodal curve. The normalization of $X$ is a morphism $\pi : \mathbb{...
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1 vote
1 answer
425 views

Does a strongly convex cone have a unique set of minimal generators?

I am studying toric varieties from the book "Toric Varieties" by Cox, Little and Schenck. They define rays as the edges of a strongly convex cone, $\sigma$. Each ray, $\rho$ has a unique ray generator ...
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2 votes
0 answers
169 views

twisting sheaf of toric projective bundle

Let $X_{\Sigma} = \mathbb{P}_{X_{\Sigma'}}(\mathcal{L_1}\oplus\mathcal{L_2}),X_{\Sigma'}$ be toric varieties ('good' if it's necessary), $\mathcal{L}_i\in \mathrm{Pic}(X_{\Sigma'})$ and $\pi:X_{\...
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6 votes
1 answer
1k views

Atiyah flop, flip and related toric computation

I am trying to understand the definitions of flips and flops by studying examples in this article of Hacon and McKernan. I would like to ask why the toric varieties constructed in Ex. 1.13 are indeed ...
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4 votes
1 answer
638 views

What is the Newton polyhedron of a toric variety?

First of all I want to say that my knowledge of toric geometry is minimal. A paper I'm reading considers a toric variety $X$ and then claims that in its Newton polyhedron, $\Delta(X)$, the fixed ...
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2 votes
1 answer
237 views

Why is this not a cartier divisor?

Consider the lattice $$N=\{(a,b,c)\in\mathbb{Z}^3\mid a+b+c\equiv 0\mod 2\}$$ and the Cone $$\sigma=Cone(e_1,e_e,e_3)\subset N_{\mathbb{R}}\cong \mathbb{R}^3$$ The associated affine toric variety $X_{\...
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1 vote
0 answers
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toroidal compactifications of moduli spaces of ppav

Are the modular toroidal compactifications of ppav's (second Voronoi) defined by Alexeev without self-intersections? i.e. are the irreducible component of the boundary divisor normal? If not, can one ...
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toric variety associated to schur polynomial?

In a Schur polynomial, which is homogeneous, monomials are added with fixed coefficients. Collecting the indices of monomials seems to define a toric data for a projective variety. But the ...
1 vote
0 answers
126 views

Induced map on cohomology from inclusion of toric varieties

Suppose you have an equivariant closed immersion of toric varieties $Y \subset X$, and suppose further that they are both smooth (meaning that the torus of $X$ restricts to the torus on $Y$). Suppose ...
1 vote
1 answer
134 views

Reference for a result in toric geometry

I want to know where I can find the proof of the following theorem : if $X$ is a smooth toric variety, then $H^{\bullet}(X) \cong CH^{\bullet}(X)$. Thanks in advance !
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3 votes
1 answer
267 views

First Chern class of toric manifolds

I have been reading a Mirror Symmetry monograph, and its physical arguments seem to imply that all toric manifolds have semipositive-definite first Chern class. Is this true, and if yes, how does ...
0 votes
0 answers
52 views

What do we know about the set of all fans?

I know the question is not very rigorous, but I have been trying to prove some facts about toric varieties and I think giving this set some structure would be very helpful. So, suppose you fix a ...
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1 vote
0 answers
510 views

Relation between fan and toric variety

I understand what a fan is, but I don't really know how it associated to a toric variety. Could a non-toric variety has a fan?
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2 votes
1 answer
345 views

Why take the dual cone when constructing toric variety?

I am reading an introduction to toric varieties. I don't understand why we are taking the monoid associated to the dual cone instead of simply taking the monoid associated to the cone. Is there any ...
4 votes
1 answer
415 views

example of toric varieties with nontrivial first cohomology group

If I remember correctly, when a toric variety is smooth or simplical (the moment polytope is simplicial rational), then there is no odd dimensional cohomology group and the dimension of the even ...
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1 vote
1 answer
213 views

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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1 vote
0 answers
122 views

How to find the distinguished points of this cone?

Consider the cone - $\langle e_1,-e_1+2e_2\rangle$ in the lattice $N=\mathbb Z^2$. Then it has the following faces - $0=\{(0,0)\}$ $\rho_1=\langle e_1\rangle$ $\rho_2=\langle-e_1+2e_2\rangle$ $\...
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3 votes
1 answer
243 views

How does one determine the singular points of a toric variety?

Consider the toric surface corresponding to the fan $\Delta$ consisting of $$\sigma_1=\langle e_1,-e_1+2e_2\rangle$$ $$\sigma_2=\langle-e_1+2e_2,-e_1-2e_2\rangle$$ $$\sigma_3=\langle -e_1-2_2,e_1\...
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1 vote
0 answers
411 views

Is this toric variety the blowup of $\mathbb C^2$ at some point?

Let $u_1=e_1,\quad u_0=e_1+2e_2,\quad u_2=e_2$. Consider the fan consisting of the following cones $\sigma_1= \langle u_1,u_0\rangle$, $\sigma_2=\langle u_0,u_2\rangle$ and their faces. Then the toric ...
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1 vote
0 answers
92 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 ...
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8 votes
1 answer
539 views

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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