Questions tagged [toric-geometry]
The toric-geometry tag has no usage guidance.
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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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81 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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57 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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51 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 ...
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108 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
92 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
56 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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61 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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83 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
56 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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32 views
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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107 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
106 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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122 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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373 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
331 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
106 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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54 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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14 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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103 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
89 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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1answer
141 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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42 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
98 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
134 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
242 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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87 views
Torus-invariant divisor on $\mathbb{CP}^2\#1\mathbb{CP}^2$
I want to confirm that the following result is right:
$M=\mathbb{CP}^2\#1\mathbb{CP}^2$. I was told that the torus-invariant divisors on $M$ are $H,H-E,E$, where $E$ and $H$ are the hyperplane ...
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1answer
132 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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58 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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94 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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173 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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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
315 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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80 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
264 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
69 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
102 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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98 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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25 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
118 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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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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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
100 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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97 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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35 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 M,\...
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26 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 $N\otimes\mathbb R\cong\mathbb R^n$. ...
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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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36 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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70 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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100 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 ...
