# Questions tagged [toric-varieties]

Toric variety is embedding of algebraic tori.

226
questions

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**1**answer

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### Cohomology ring of a hypersurface in toric variety

Let $X^n$ be a smooth projective toric variety corresponding to a simple polytope $P$. It is well known that the cohomology ring $H^*(X)$ can be described in terms of combinatorics of faces of $P$.
...

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### Hodge structure on intersection cohomology of toric varieties

Given a convex polytope with integer vertices, one can construct a complex projective variety $X$ called toric variety. In general $X$ is not smooth. As I have heard, by the work of M. Saito, the ...

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votes

**1**answer

667 views

### How to describe morphisms to a weighted projective space (bundle)?

The case of an usual projective space (bundle) is well known (Grothendiek,EGA II, Publ.Math. IHES, 8, 1961; or Hartshorne, Alg.Geom.).The more general case of toric varieties has been considered by D. ...

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46 views

### Toric resolution in terms of polytopes

Let $P,Q\subset\mathbb{R}^n$ lattice polytopes such that $P$ and $P'=P+Q$ are smooth polytopes. We obtain the birational morphism $f:X_{P'}\to X_Q$ and I am interested in a criterion when this is a ...

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218 views

### An example in symplectic geometry

$\DeclareMathOperator\SU{SU}$Let $M$ be a coadjoint orbit of dimension 6 of $\SU(3)$, and let $T$ be the maximal torus in $\SU(3)$. If we denote $\mu : M \longrightarrow \mathbb{R}^2$ the moment map ...

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### algebraic de Rham cohomology of toric varieties (reference request)

I haven't been able to find anything workable yet, but I'm looking for a reference on the de Rham cohomology of toric varieties, where as many as possible of the following conditions are handled:
...

**4**

votes

**1**answer

224 views

### Polynomial size embeddings of toric varieties from polytopes?

Background: Let $P$ be a integral polytope, and $X_P$ the toric variety associated to the normal fan.
$X_P$ is always projective, because the collection of characters corresponding to the points $\...

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106 views

### Intersection homology of toric resolutions

I'm interested in the intersection homology of toric varieties associated to a polytope $P$ with proper faces F, and a subdivision $P'$ of P. Let $X_P$ be the toric variety associated to the polytope $...

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votes

**1**answer

207 views

### Separating a lattice simplex from a lattice polytope

Let $P\subset\mathbb{R}^n$ be a convex lattice polytope.
Do there always exist a lattice simplex $\Delta\subset P$ and an affine hyperplane $H\subset\mathbb{R}^n$ separating $\Delta$ from the convex ...

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75 views

### Determining a toric GIT quotient

Consider the following $G = (\mathbb{C}^*)^{\times 3}$-action on $\mathbb{A}^6$:
$(\lambda_0,\lambda_1,\lambda_2) \cdot (x_0,x_1,x_2,y_0,y_1,y_2) = (\lambda_0x_0,\lambda_1x_1,\lambda_2x_2,\frac{\...

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### Where to read about the toric variety coming from a principal nilpotent element of a (semi)simple algebraic group?

Given a principal (regular) nilpotent element $e$ in the Lie algebra $\mathfrak g$ of a complex semisimple algebraic group $G$, let $\mathfrak s=(e,f,h)$ be an $\mathfrak{sl}_2$-triple for $e$. Then ...

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126 views

### Gromov-Witten invariants of cocharacter closures in toric varieties

$\require{AMScd}$
Let $X$ be a toric projective variety with dense algebraic torus $\iota:(\mathbb{C}^\times)^n \to X$, and let $u:\mathbb{C}^\times \to X$ be a cocharacter, by which I mean a map ...

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### When are two complex Tori biholomorphic

Let $g \ge 1$ be a natural number and $\mathbb{C}^g$ complex vector space which
is isomorphic to $\mathbb{R}^{2g}$ is real vector space.
An additive subgroup $\Gamma \subset \mathbb{C}^g$ is called a
...

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83 views

### Isomorphic Jacobians for different choices of basis of $1$-forms

In Otto Forster's Lectures on Riemann Surfaces on page 170 Jacobi Variety is defined in 21.6:
Suppose $X$ is a compact
Riemann surface of genus $g$ and $ \omega_1,..., \omega_g $ is a basis
of $\Omega ...

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86 views

### Primitive collections as lattice generators for the Mori cone

I am looking at the following theorem from "Toric Varieties" by Cox, Little and Schenk:
Theorem 6.4.11: If $X_{\Sigma}$ is a simplicial toric variety, then
$\overline{NE}(X_{\Sigma}) = \...

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### Distinguishing ample divisors by minimally intersecting curves on a smooth projective toric variety

My question has an easily formulated generalization, which I will state first. Let $\sigma \subseteq \mathbf{R}^n$ be a full-dimensional strongly convex polyhedral cone. For each lattice point $m \in \...

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94 views

### Toric spin compactification of toric Calabi-Yau's

Let $X$ be a toric Calabi-Yau in complex dimension $2n, n\geq 2$. In particular, this means that it is described by some fan $F$ that is spanned by vectors lying in the hyperplane $H_1 = \{(x_1,\ldots,...

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### Explicit formula for the moment map of toric manifold

Let $P$ be a Delzant polytope in $M\otimes{\mathbb R}\cong \mathbb R^n$, and it is well-known that we can associate to it a toric manifold $X=X_P$ with the moment map $\pi: X\to P$.
I would like to ...

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votes

**2**answers

274 views

### Relationship between fans and root data

A (split) reductive linear algebraic group is equivalently described by combinatorial information called a root datum.
A toric variety is described by combinatorial information called a fan.
Both ...

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**1**answer

65 views

### Coordinate-symmetric convex polytopes with equal Erhart (quasi)-polynomials

Recall that given a nondegenerate polytope $P \subset \mathbb{R}^n$ which is the convex set of some vectors with integral coordinates, the Erhart polynomial $p_P(t)$ a polynomial such that $p_P(t)$ ...

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55 views

### Low rank approximation

Can we solve low rank approximation problem by using concept of Gröbner basis? I was trying to find it by Macaulay2 but didn't find the answer. I was trying to do by toric ideals as for them Gröbner ...

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68 views

### Is toroidalization local?

Let $f:X \to Y$ be a surjective morphism of smooth projective varieties, $D$ be a simple normal crossings divisor on $X$ and $U_Y \subset Y$ be an open subset over which $(X,D)$ is log smooth (in the ...

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67 views

### Connected components of a codimension one fiber for a finite morphism

Let $f:X \to Y$ be a finite surjective morphism from a $\mathbb{Q}$-factorial variety to a smooth variety. Let $D_Y$ be a prime divisor on $X$ and let $\bigcup D_i$ be the inverse image of $D_Y$. Do ...

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82 views

### Log canonical centers of toric (and toroidal) varieties

Q1: Let $(X,B)$ be a toric variety. There exists a toric resolution of singularities $f:(Y,E) \to (X,B)$. Here is my question:
Is any lc center of $(X,B)$ an irreducible component of an intersection ...

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82 views

### Birational model of a log smooth pair

Given a log smooth pair $(X,B)$ with a reduced boundary divisor $B$, consider a birational model $\pi:X' \to X$ and a boundary divisor $B'$ which is given by $K_{X'}+B'=\pi^*(K_X+B)$. Here is my ...

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69 views

### Birational contraction of toric vector bundle

Let $X$ be the toric vector bundle over $\mathbb{P}(1,1,1,2)$ with grading matrix
$$
\left(\begin{array}{cccccc}
1 & 1 & 1 & 2 & -2 & 0 \\
0 & 0 & 0 & 0 & 1 & ...

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### Groebner basis of a toric ideal

I know about toric ideals that it is a sort of binomial ideal i.e. generated by $x^u - x^v$, where $Au = Av $ ( A is the associated matrix). So by finding all integer solutions of $AX = 0$, can we ...

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**1**answer

241 views

### Equivariant cohomology algebra of toric variety

Let $X$ be a complex projective and smooth toric variety of complex dimension $n$. It is acted by the real torus $T=(S^1)^n$.
Is it true that the $T$-equivariant cohomology $H^*_T(X,\mathbb{Z})$ ...

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67 views

### On two different descriptions of Delzant polytopes

I have seen two different ways of describing a Delzant polytope:
From Canna Da Silva https://people.math.ethz.ch/~acannas/Papers/toric.pdf, a Delzant polytope is a polytope in $\mathbb{R}^{n*}$ ...

**2**

votes

**1**answer

198 views

### When is the blow-up of a closed sub-variety of a toric variety a toric variety?

If $ X $ is a toric variety and one has a closed sub-variety $ Y \subseteq X $, is the blow-up $ \operatorname{Bl}_{Y}(X) $ a toric variety as well? I suspect not, but wanted to check with other ...

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880 views

### Hamiltonian $S^1$ actions with isolated fixed points

I have in mind the following question for some time. Is there an example of a compact symplectic manifold with a Hamiltonian $S^1$-action with isolated fixed points, that does not admit a compatible $...

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95 views

### (Implemented) algorithm for Hodge numbers

Let $X$ be a smooth projective toric variety. Do any of the math computer algebra systems have an algorithm implemented to compute the Hodge numbers of a generic complete intersection in $X$? Say in ...

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147 views

### Quotients of toric varieties

This is a follow up of this question. Given a toric variety $X$ with a fan $\Sigma$ and a finite group $G$ acting on $X$, we know that the GIT quotient $X/G$ exists. However, as stated in the answer ...

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**1**answer

136 views

### Lines on a toric cubic surface with a line of nodes

Consider a cubic surface cut out by equations $x^2y - z^2w$ inside $\mathbb{P}^3$. This gives a cubic surface with a line of nodes, it is toric and has normalisation $\mathbb{F}_1$, a Hirzebruch ...

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### Local toric varieties and tropicalization

Let $K$ be a valued field, and consider the ring $R=K((x_1,\dots,x_m))$ of formal Laurent series. This is "the germ of the torus at $0$". Is there a theory of "local toric varieties" where $R$ ...

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### Is the boundary divisor of a smooth projective toric variety an snc divisor?

Let $X$ be a smooth toric projective variety.
Let $T$ be the big torus acting on $X$.
Let $D=X\backslash T$ be the boundary divisor.
Question 1. Will $D_i$ be a smooth toric projective variety for ...

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### Have these polynomials been studied? (Perhaps as generalizations of Schur polynomials in vector variables?)

For $n\geq 3$, let $\mathbf{a}_1,\ldots,\mathbf{a}_n \in \mathbb{Z}^2$ be a collection of points in the plane with integer coordinates $\mathbf{a}_i = (a_{1i},a_{2i})$ where each $a_{1i} > 0$. For ...

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**1**answer

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### Cohomology of toric blowup

Let $n\geq2$. Let $G$ be a linear automorphisms group of prime order on $\mathbb{C}^n$. We assume that $0$ is the unique fixed point of $G$.
I consider the quotient $\mathbb{C}^n/G$. It is a toric ...

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**1**answer

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### A characterisation of faces of rational polyhedral cones

This is about a (seemingly) basic lemma about rational polyhedral cones that is sometimes used when working with toric varieties and is usually "left to the reader". Unfortunately, I could ...

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votes

**1**answer

126 views

### Sections of Cartier divisors on toric varieties

Let $X_{\Sigma}$ be a projective toric variety. Consider the total coordinate ring
$$S = \mathbb{C}[x_{\rho}\: | \: \rho\in\Sigma(1)]$$
Define $\deg(x_{\rho}) = D_{\rho}$.
Now, take a divisor $D = \...

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**0**answers

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### Toric Fan for the Du Val's singularities D_n and E_n

Let us consider the Du Val's singularities.
i.e. https://en.wikipedia.org/wiki/Du_Val_singularity.
It is well known that they are classified by ADE, because the exceptional divisors arising in the ...

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128 views

### Hyperplane sections of non-singular projective toric varieties

Let $X^n\subset \mathbb{P}^N$ be an embedding of a non-singular projective toric variety (where variety stands
for a reduced irreducible scheme over $\mathbb{C}$, and toric means normal variety ...

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186 views

### When is vector bundle over toric variety a toric variety?

Is it true that a vector bundle over a toric variety is also a toric variety if and only if it splits? if so, how do we prove it?
This seems to be the content of a remark in Oda's Tata's lectures on ...

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votes

**1**answer

236 views

### Is the minimal Chern number of a toric manifold at least 2?

I would like to show that the minimal Chern number $N_M$ of a toric manifold $M$ is at least $2$, where
$$
N_M := \underset{l>0}{\min} \lbrace \exists \ A \in H_2(M;\mathbb{Z}) \ : \ \langle c, A \...

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### Inferring properties of toric manifolds through Delzant's description

Let $(M,\omega, \mathbb{T})$ be a symplectic toric manifold. It is well-known that the properties of $M$ can be retrieved by looking at the moment polytope $\Delta$ image of the momentum map
$$
\mu : ...

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### Secondary fan and KN strata

Let $\mathbb{G}_m^r$ act on the affine space $\mathbb{A}^n$ through an embedding into the open dense torus. Is there a way to calculate the 1-parameter subgroups that determine the KN strata from the ...

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### moduli space of toric structures on a fixed toric variety (reference?)

I'm looking for a reference on the following question:
Given a fixed toric variety $V/k$, how to describe the moduli space of all toric structures on $V$?
In addition to the general question, I ...

**3**

votes

**0**answers

60 views

### Hypertoric varieties in dimension 4?

Are the only smooth hypertoric varieties in real dimension 4 obtained
as minimal resolutions of type A simple singularities $\mathbb{C}^2/\mathbb{Z}_{/n}$?

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votes

**1**answer

564 views

### Why only some del Pezzo are toric?

Let us define smooth del Pezzo surfaces $dP_r$ as the blowup of $r$ generic points in $\mathbb{CP}_2$. One can show that if we request $dP_r$ to be Fano, then $r=0,...,8$. In theoretical physics ...

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### A generalization of toric varieties

Let $M$ be a monoid with cancelation whose groupification is $\mathbb Z^d$ ($d$ finite). Even without assuming a finite generation of $M$, it seems to me that
(a) $X=Spec\, \mathbb C M$ contains the ...