Questions tagged [toric-geometry]

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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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1answer
49 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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1answer
93 views

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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2answers
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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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73 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)...
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1answer
109 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=\...
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84 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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40 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 ...
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150 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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147 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 ...
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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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154 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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1answer
268 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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1answer
87 views

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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108 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 ...
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145 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 ...
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1answer
84 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 ...
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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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148 views

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{P}...
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1answer
222 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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141 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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692 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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1answer
483 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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1answer
153 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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74 views

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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19 views

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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115 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 ...
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1answer
102 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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210 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 ...
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44 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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1answer
213 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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1answer
218 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 ...
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1answer
301 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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1answer
173 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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82 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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137 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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294 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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88 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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1answer
405 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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164 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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1answer
314 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 ...
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1answer
99 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 x^\beta\in\...
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1answer
131 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 variety ...
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108 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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31 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 $\{\...
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1answer
140 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 ...
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304 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 $\...
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53 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, \...
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1answer
128 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 \mathbb{Z}...
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126 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 ...