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.
67
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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 ...
- 287
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279
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How to prove this is a fibre bundle?
Let $M^{2n}$ be $2n$ dimensional toric manifold over a simple polytope $P^n$. Let $\pi : M^{2n} \longrightarrow P^n$ be the orbit map of the torus action. Let $F^k$ be a $k$ dimensional face of $P^n$. ...
- 7,044
6
votes
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202
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Realizing a list of varieties as the toric variety of a fan
The locus of the following curves define affine varieties.
$xy-z^m=0.$
$x^m+xy^2+z^2=0.$
$x^4+y^2+z^2=0.$
$x^2+y^3+z^2=0.$
$x^5+y^3+z^2=0.$
In Fulton's "Toric Varieties" book he constructs toric ...
- 1,955
5
votes
0
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91
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Isomorphism of divisors
Consider the cartier divisor group $CDiv_{T_{N}}(X_{\Sigma})$ defined on the fan $X_{\Sigma}$. I am having trouble proving the following assertion that there is a natural isomorphism $$CDiv_{T_{N}}(X_{...
- 159
4
votes
0
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142
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How does gluing of affine patches of toric varieties at the examples $\mathbb{P}^2$ and $\mathcal{H}_r$ work?
I don't fully understand how the gluing of the affine parts of a toric variety exactly works. I have a hard time developing a common sense or any intuition how to tell the result of a gluing morphism ...
- 2,562
4
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0
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120
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What happens with subdivisions of normal fans in Sage?
I've been trying to compute specific subdivisions of a particular 4D complete fan, to try to speed up computations I have started looking into using Sage. The problem I'm having is that I would like ...
- 300
4
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299
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Weighted Projective Spaces and varieties
Consider the weighted projective space $\mathbb{P}(1,1,2)$ with variables $x_0,x_1,x_2$ of degrees 1,1,2 respectively. Consider the map $$(a_0,a_1,a_2) \to (a_0^2,a_0 a_1,a_1^2,a_2)$$ with $F:\mathbb{...
- 671
4
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146
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Fibers of toric morpisms
Let $f: X(\Delta_1) \to X(\Delta_2)$ be a toric morphism of toric varieties, and let $\sigma \subset \Delta_2$ be a cone, then for any point in the corresponding orbit $x \in O(\sigma)$ the fiber $f^{-...
- 5,928
4
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190
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Good confusion-avoiding notation for gluing toric varieties from fans?
$
\newcommand\R{\mathbb R}
\newcommand\C{\mathbb C}
\DeclareMathOperator\Cone{Cone}
$I'm trying to establish a good notation to avoid confusion when we glue toric varieties from affine pieces. A ...
- 24.3k
4
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0
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843
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Procedure to find generators of the dual cone
Page 11 of Fulton's "Toric Varieties" gives the following procedure for finding generators of the dual of a convex polyhedral cone in $\mathbb{R}^d$:
For each set of $n-1$ independent vectors among ...
- 1,955
4
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0
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62
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$H^{1}(O_{F})$ of a surface in a toric variety
I have a surface inside a toric variety $X$ and I would like to compute the first cohomology of its structure sheaf via the Cech complex, since I already know which cones of $X$ it hits (five two-dim.,...
- 293
4
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507
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Exercise in David Cox "Toric Varieties"
I want to do an exercise in the book Toric Varieties (by David Cox)
Exercise 3.3.5. Let $\overline{\phi}:N \rightarrow N'$ be a surjective $\mathbb{Z}$-linear mapping and let $\widehat{\sigma}$ and $\...
- 1,131
3
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57
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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 ...
- 243
3
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0
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114
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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)...
- 135
3
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0
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241
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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 ...
- 623
3
votes
1
answer
254
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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\...
- 7,044
3
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0
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124
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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.
...
- 7,044
3
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68
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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, \...
- 1,023
3
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534
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Is anticanonical divisor on complete toric variety big?
It seems to me that for any complete toric variety $P_\Sigma$, the anticanonical divisor is big. Moreover, the argument also shows that $P_\Sigma$ is projective. This sounds a bit strange for me, did ...
- 3,885
3
votes
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220
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Centers of divisorial valuations on toric varieties
Suppose we are given a divisorial valuation $\mathcal{v}$ on a smooth toric variety $X_{\Sigma}$ (for a fan $\Sigma$), i.e. $\mathcal{v}$ is the valuation induced by a torus invariant divisor $D_{\tau}...
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3
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121
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Question regarding example of toric variety and generators of cone
Consider the canonical example taking n=2, and taking the cone $\sigma$ generated by the vectors $e_{2}$ and $2e_{1} - e_{2}$. The dual cone $\sigma^{v}$ is defined as the set of vectors in the dual ...
- 671
3
votes
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102
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Open embedding of affine toric varieties implies Cone is face of the other
let $\tau, \sigma \subseteq N_{\mathbb{R}}$ be two rational, strongly convex polyhedral cones with $\tau \subseteq \sigma$. Now we get an inclusion $S_{\sigma} \to S_{\tau}$ inducing an inclusion $\...
- 959
3
votes
1
answer
671
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Perfect pairing induces isomorphism of tensor products
Let $M, N$ be $R$-modules and $(\cdot, \cdot): M \times N \to R$ be a perfect pairing. Wikipedia sais that this means that the map $\varphi: M \to \text{Hom}_R(N, R), m \mapsto (n \mapsto (m, n))$ is ...
- 959
3
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149
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Inclusion of Tori induces surjection of character groups?
Let $k$ be an algebraic closed field. Let $T, T'$ be algebraic Tori in the classical sense, meaning $T \cong \mathbb{A}_k^n \setminus V(X_1 \cdots X_n)$, $T' \cong \mathbb{A}_k^{n'} \setminus V(X_1 \...
- 959
3
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102
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Toric Variety from a fan with "identical" edges
I have this fan that I've been trying to construct a toric variety from. The problem is, it contains certain edges twice. These are the edges:
$$(1,0,1)$$
$$(0,1,1)$$
$$(-1,-1,1)$$
$$(0,0,1)$$
$$(0,0,...
- 417
3
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0
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143
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Transition functions of toric projective bundle (Proposition in [Cox, Toric Varieties])
My reference: David Cox's "Toric Varieties"
My question is the proof of Proposition 7.3.3.
Proposition 7.3.3.
The cones {$\sigma_i$ | $\sigma \in \Sigma$, $i = 0,\dots,r$} and their faces form a ...
- 1,131
3
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0
answers
330
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GIT quotient for a certain torus action on an affine space
I'm reading various books and some notes and here is my question.
Let $(\mathbb{C}^*)^2$ act on $\mathbb{C}^4$ by
$$(\lambda_1,\lambda_2).(x_1, x_2, y_1,y_2)=(\lambda_1 x_1, \lambda_2 x_2, \lambda_1^{-...
- 1,715
2
votes
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answers
165
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,...
- 2,294
2
votes
0
answers
222
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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....
- 680
2
votes
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answers
185
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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_{\...
- 2,589
2
votes
0
answers
63
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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 ...
- 1,419
2
votes
0
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214
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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{...
- 3,404
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votes
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171
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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_{\...
- 1,500
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0
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368
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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 ...
- 1,303
2
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34
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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 $\{\...
- 4,440
2
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58
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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 $...
- 7,044
2
votes
1
answer
488
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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 ...
- 624
2
votes
0
answers
50
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Prove that $\{m \in S_{\sigma} \, | \, \gamma(m) \neq 0\}$ is a face of $\sigma^V \cap M$
I am trying to solve exercise 3.2.6 pag.124 of Cox, Little, Schenck book
http://www.math.colostate.edu/~renzo/teaching/Toric14/CoxLittleShenck.pdf because it is required to prove orbit-cone ...
- 63
2
votes
0
answers
63
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How to show that the inverse image of a face of a simple polytope is a connected manifold?
If $M^{2n}$ is a toric manifold over a simple polytope $P^n$ i.e; the orbit space of the action of the $(S^1)^n$ on $M^{2n}$ is an $n$ dimensional simple polytope $P^n$. Let $\pi : M^{2n} \rightarrow ...
- 7,044
2
votes
0
answers
152
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Injectivity of a rational map
I have a variety $C_{0}\subset\mathrm{Spec}\mathbb{C}[M_{Y}]$ living in some $Y$ (which is toric) and I am guessing, that it is $\mathbb{P}^{1}$. So I have to find a birational function $f:C_0\...
- 293
1
vote
0
answers
40
views
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 ...
- 1,381
1
vote
0
answers
69
views
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 $U_i=\mathcal{O}(\sigma_i)$ ...
- 2,562
1
vote
0
answers
43
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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 ...
- 2,562
1
vote
0
answers
47
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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 $\Delta$, ...
- 2,562
1
vote
0
answers
48
views
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 $V(\sigma), \sigma \in\Sigma$ ...
- 11
1
vote
2
answers
228
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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 ...
- 483
1
vote
0
answers
45
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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 ...
- 623
1
vote
0
answers
133
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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 ...
- 623
1
vote
0
answers
83
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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 ...
- 231
1
vote
0
answers
126
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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 ...
- 19.8k