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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Principal bundle (or torsor) for a diagonalizable group over a torus
Let $T$ be an algebraic torus and $G$ a diagonalizable group; both are over an algebraically closed field $k$ of characteristic $0$ (take $k=\mathbb C$, if you like).
I am trying to understand ...
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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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GIT quotient for a certain torus action on an affine space
I'm reading various books and some notes and here is my question.
Let $(\mathbb{C}^*)^2$ act on $\mathbb{C}^4$ by
$$(\lambda_1,\lambda_2).(x_1, x_2, y_1,y_2)=(\lambda_1 x_1, \lambda_2 x_2, \lambda_1^{-...
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Restriction of locally free sheaf associated projective modules [duplicate]
My question comes from the paper of Tamafumi's "On Equivariant Vector Bundles On An Almost Homogeneous Variety" (it can be downloaded freely in Link).
Question 1
Let $X = X(\Delta)$ be a ...
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If $M$ and $N$ are graded modules, what is the graded structure on $\operatorname{Hom}(M,N)$?
Let $A$ be a graded ring. Note that the grading of $A$ may not be $\mathbb{N}$, for example, the grading of $A$ could be $\mathbb{Z}^n$. Actually, my question comes from the paper of Tamafumi's On ...
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Question about Tamafumi Kaneyama's Paper: "On Equivariant Vector Bundles On An Almost Homogeneous Variety"
My reference: Link
I have two question about Proposition 3.3.:
Proposition3.3. says that $1 \rightarrow Aut(E) \xrightarrow{j} G(E) \xrightarrow{p} T \rightarrow 1 $ is a short exact sequence of ...
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The dual of a regular polyhedral cone is regular
A rational polyhedral cone in $\mathbb{R}^n$ is a set of the form
$$\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\}$$
for ...
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Projection of rational normal curve is still a rational normal curve (of smaller degree)?
We work over $\mathbb{C}$. Let us define the rational normal curve of degree $d$ as the image of the morphism
$$\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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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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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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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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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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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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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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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, Schenck). From what I have learned so far, I can grasp that toric ...
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on the real part of a toric variety
Corresponding to a strongly convex rational polyhedral cone $\sigma$ in the lattice $N$ we have the affine toric variety $U_\sigma=\operatorname{Hom_{semi group}}(\sigma^\vee\cap M,\mathbb C)$ where $...
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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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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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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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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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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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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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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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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 authors to Modern Geometric Structures and Fields
by Novikov and ...
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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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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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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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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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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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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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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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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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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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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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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 ...
CommunityBot
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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_{\...
CommunityBot
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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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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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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 ...
