# Questions tagged [toric-varieties]

Toric variety is embedding of algebraic tori.

99
questions with no upvoted or accepted answers

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### Homology classes of subvarieties of toric varieties

Let $X$ be a smooth proper toric variety, $Z\subseteq X$ a smooth subvariety.
Is the fundamental class $[Z] \in H_\ast(X) = A_\ast(X)$ nonzero?
Background
If $X$ is a Kaehler variety, this is of ...

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504 views

### A commutative monoid associated with a finite abelian group

Let $M$ be a finite abelian group, and denote by $e_m$, for $m \in M$, the canonical basis of $\mathbb{Z}^M$. For $m, n \in M$ define elements $v_{m,n} \in \mathbb{Z}^M/\langle e_0\rangle$ as
$$
v_{m,...

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### Hilbert schemes of points on toric surfaces

Let $\mathrm{S}$ be a smooth toric surface. The Hilbert scheme of $n$ points $\mathrm{Hilb}^n(\mathrm{S})$ inherits a torus action, but need not admit the structure of a toric variety itself. For ...

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### Big tangent bundle

Let $X$ be a nonsingular complex algebraic variety whose tangent bundle is $T_X$. I use Lazarsfeld's book for the definition of a big vector bundle. A line bundle $L$ is big if it has Itaka dimension ...

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### Computing Ext for toric divisors

Ok, I have an affine (normal) toric variety $X = \text{Spec} k[\sigma]$. Suppose that $F, G$ are two torus invariant Weil divisors on $X$. Is there any relatively straightforward way to compute
$$
\...

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### How to calculate the top Chern class of a “functorial” vector bundle on a moduli space of sheaves?

Let $\mathcal M$ be a moduli space of sheaves on a nice variety $X$, with fixed rank $r$ and Chern classes $c_i$, semi-stable with respect to an ample bundle $H$. Let $E$ be a "functorial" vector ...

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### Derived category of toroidal varieties

This question comes from the first reduction step of Theorem 4.2 of Kawamata's paper on K-equivalent implies D-equivalent on toroidal varieties. But my question has little to do with this theorem.
A ...

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591 views

### When should a moment polytope have “smooth” faces?

A codimension $d$ face of a polytope is called rationally smooth if it lies on only $d$ facets, because this is exactly the condition for the corresponding toric variety to have only orbifold ...

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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:
...

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748 views

### Picard group of toric varieties

I am trying to understand how to obtain the Picard group for general toric varieties. So far, I have been using information found in https://arxiv.org/pdf/1003.5217.pdf .
Here, a toric variety has ...

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**1**answer

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### 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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### Toric Degenerations and Nearby Cycles

Suppose that $f: X \to \mathbb{A}^1$ is a toric degeneration in the sense of Nishinou-Siebert. In other words let X be a (possibly singular) toric variety equipped with a (not necessarily proper) ...

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### Geodesic rays in a toric variety

Let $\{\alpha_0, \ldots, \alpha_r\} \subset \mathbb{Z}^n$ be a finite subset of lattice points and let $\Phi: (\mathbb{C}^*)^n \to \mathbb{C}\mathbb{P}^r$ be the corresponding map from the algebraic ...

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### Do simplicial toric varieties have “lots” of base point free linear systems?

Question: Let $n$ be a positive integer and let $X$ be a simplicial toric variety. Does every coset of $n\cdot Pic(X)\subseteq Pic(X)$ contain a base point free linear system?
If $X$ is not ...

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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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### 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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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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145 views

### “Reflexive” differentials on Gorenstein affine toric variety

Let $P \subset \mathbb{R}^{n-1}$ be a lattice polytope of dimension $n-1$ and let $\sigma \subset \mathbb{R} \times \mathbb{R}^{n-1}$ be the cone over $1 \times P$.
To the cone $\sigma$, we may ...

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### Principal $G$-bundles on affine toric varieties

Let $X_\sigma$ be an affine toric variety for an action of a torus $T$ and let $\mathcal{P}$ be a toric principal $G$-bundle over $X_\sigma$ where $G$ is an affine algebraic group (here base field $k$ ...

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### Examples of small covers that are not real toric manifolds

Definition of quasi toric manifolds :
The action of $(S^1)^n$ on $\Bbb C^n$ by pointwise multiplication is called the standard representation. Given a manifold $M^{2n}$ with an $(S^1)^n$-action, a ...

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### 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}) = \...

**4**

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**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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110 views

### 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 ...

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165 views

### Local structure of non-normal toric varieties---possible mistake in “Discriminants, Resultants and Multidimensional Determinants”

I believe I may have a counterexample to Theorem 5.3.1 on page 179 from the book book Discriminants, Resultants and Multidimensional Determinants by Gel'fand, Kapranov, and Zelevinsky. To summarize ...

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136 views

### Reference for the notion of polyhedra “degenerations”

Let $P$ be a convex polyhedron and let $P(t)$ be a continuous deformation thereof, such that:
a) $P(0)=P$;
b) for all $t\in[0;1)$ the polyhedron $P(t)$ is strongly combinatorially equivalent to $P$ (...

**4**

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234 views

### Vanishing theorems on toric DM stacks

In chapter 9 of the book Toric varieties by Cox-Little-Schenck several cohomology vanishing theorems for toric varieties are proved or mentioned.
In this question I am interested in references for ...

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### Intersection numbers on blow ups of toric varieties

Suppose we have a smooth, complete toric varietiy $X_{\Sigma}$ of dimension $n$. Let $\sigma_k \in \Sigma(k)$ a smooth $k$-dimensional cone in $\Sigma$ and suppose we make the toric blow up at the ...

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### Intuition about Toroidal Embeddings

I have been trying to understand the very basics of toroidal embeddings, and the definitions on the face of them are not terribly daunting. I've been going with the "locally analytically looks like a ...

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585 views

### Intersection Theory on a toric variety

Hi All,
I'm having some trouble understanding a result about calculating $D.C$ on a toric variety. The proposition I am trying to follow is from Cox, Little, Schenck. Either there is a mistake with ...

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### the map on divisor class groups induced by restriction to a toric subvariety

Let $X$ be a (say, complex) toric variety acted upon by a torus $T$ and defined by a fan $\Sigma$ in the cocharacter lattice $N=\mathrm{Hom}(\mathbb{C}^\times, T)$, and let $M$ be the character ...

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### 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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110 views

### 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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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 $\mathbb{R}^{n*}$ ...

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### (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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### 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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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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### 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 ...

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143 views

### A question about smooth convex lattice polygons

Let $P$ be a smooth convex lattice polygon in $\mathbb{R}^2$ (the lattice being $\mathbb{Z}^2$). Here smooth means that at any vertex of $P$, the two primitive integer vectors (i.e. vectors whose ...

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### Can a toric surface be an elliptic surface?

It is known that a rational elliptic surface is a blow-up of $\mathbb{P}^2$ at 9 points. More precisely it is obtained as the blow-up of the base locus of a pencil of cubic curves in $\mathbb{P}^2$. ...

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### Zeros of Hilbert series of affine toric varieties

Consider a convex rational polyhedral cone $C\subset\mathbb R^m$ with vertex at the origin. Let $X$ be the corresponding affine toric variety, i.e. $\mathbb C[X]=\mathbb C[\mathbb Z^m\cap C^\circ]$. ...

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### Which (polytopal) fans/polytopes are secondary?

Let $P$ be a (d-1)-dimensional polytope with $n$ vertices that sits in an affine hyperplane in $\mathbb{R}^d$.
The secondary fan of $P$ is a polytopal fan in $\mathbb{R}^n$ with $d$-dimensional ...

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207 views

### determine if a toric variety is Gorenstein

Let $G$ a simply connected group over $k$ and $car(k)=0$.
Let $T_{+}=(T\times T)/Z_{G}$ we consider the closure $\overline{T}_{+}$ of the torus $T_{+}$ in $\prod End(V_{\omega_{i}})\times\prod\...

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447 views

### “Step-by-Step” toric resolution process?

WLOG the fan $\Sigma$ of our toric variety $X_{\Sigma}$ is simplicial. (So $X_{\Sigma}$ has at worst orbifold singularities and all cones $\sigma \in \Sigma$ are simplicial).
The classical toric ...

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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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### 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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### 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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### 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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### 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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### 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 ...