Questions tagged [toric-geometry]

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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 $U_\sigma := \{x\in \mathbb{C}^{\Sigma(1)} : x^{\hat{\sigma}}\neq0\}$ ...
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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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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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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 ...
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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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What is an example of singular( non-smooth) cone?

I am learning toric geometry and theory of cone and fan. I encountered a lot of examples of smooth cone, for example, cone which is a subset of n-dimmensional affine space of cone, 1-dimmensional cone ...
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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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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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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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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 ...
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2 votes
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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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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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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
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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)...
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1 answer
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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=\...
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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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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 ...
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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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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 ...
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4 votes
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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 ...
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2 votes
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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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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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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, ...
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1 vote
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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 ...
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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 ...
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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 ...
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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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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 answer
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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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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
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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
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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
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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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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 ...
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1 vote
0 answers
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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 ...
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1 vote
1 answer
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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
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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 ...
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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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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
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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 ...
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4 votes
1 answer
365 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 answer
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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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1 vote
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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
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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
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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
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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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