The toric-varieties tag has no wiki summary.

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### global sections of structure sheaf on the toric Calabi-Yau

Let P be a lattice polytope and lying in $ N \times {1} \subset N \times \mathbb{R}$. Let $\sigma$ be the cone over this polytope and $X_\sigma$ be the corresponding toric variety, which is an ...

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621 views

### 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 ...

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439 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
$$
...

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434 views

### Proving that a variety is not (isomorphic to) a toric variety

Is there an algorithmic (or other) way to prove that a (projective)
variety is not isomorphic to a toric variety?
I'd be happy with an algebraic answer (for affine or projective varieties),
using the ...

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301 views

### 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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381 views

### 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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249 views

### 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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459 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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145 views

### 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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57 views

### Relation between linear independence of lattice vectors and the toric variety defined by that lattice

I have been reading some basic and elementary work on toric varieties, but even though people assured me that toric varieties are very well understood, several questions remained.
Setup. Let ...

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79 views

### 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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52 views

### 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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65 views

### 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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150 views

### What is the combinatorial data classifying non-normal affine toric varieties?

Recall that a toric variety is a variety $V$ containing an open dense algebraic torus. Here an algebraic torus means a finite product of copies of the multiplicative group of the ground field (which I ...

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142 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 ...

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280 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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390 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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69 views

### Explicit equations for conormal bundle to an affine toric variety

Let $L \subset \mathbb{Z}^n$ be a lattice and let $X_L$ be the closed toric subvariety of $\mathbb{C}^n$ cut out by the lattice ideal $I_L = \{x^{l_+} - x^{l_-} \,| \, l_+, l_- \in \mathbb{N}^n \text{ ...

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147 views

### 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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53 views

### Associating an ideal to a subdivision

Given a coherent subdivison of a fan, how does one find a torus invariant ideal sheaf whose (normalized) blowup is the given subdivision?

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221 views

### An example of threefold

Its description is a little bit complicated but it would be great if anyone can give an example.
I tried to construct it as a toric variety (See the previous question) but did not succeed.
I am ...

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118 views

### A question on the secondary fan

I am studying the secondary fan decomposition of the effective cone of a projective variety $X$. Let as assume that $X$ is a Mori Dream Space. As far as I understand passing from a cone of maximal ...

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103 views

### How to Calculate Minimal Log Discrepancy on a Toric Variety?

Let $\sigma$ be a $3$ dimensional strongly convex rational polyhedral cone on $\mathbb{R}^3$ and $X_\sigma$, the corresponding affine toric $3$-fol. Also, assume that $\Delta$ is an anti-boundary ...

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154 views

### About Alexeev and Nakamura's paper “on Mumford's construction of degenerating abelian varieites”

Is there anyone familiar with this paper? It seems to me it contains "some" typos and even some small elementary mistakes, which makes my reading very slow. Of course the key reason for my slow ...

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200 views

### Non-toric affine variety containing a torus

Part of the definition of an affine toric variety is that the action of the torus sitting as an open dense subset of the variety extends algebraically to the whole variety. Is there an easy example ...

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403 views

### Toric Kahler Manifolds

One definition of Toric Kahler manifolds that I know is the following - $(M,J,\omega)$ is a Kahler manifold of dimension n with an $\textit{effective}$ $\textbf{T}^n$ action which preserves the ...

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235 views

### What is known about the Picard scheme of a complete toric variety over C?

Let $X$ be a complete toric variety over a field $k$. Its Picard scheme is defined to be the scheme representing the functor $Pic_{(X/k)(\text{fppf})}$, where $Pic_{X/k}: Sch_k^{op}\to Set$ sends a ...

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39 views

### center 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 ...

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### What is the symplectic manifold whose Delzant polytope is a trapezoid?

What is the symplectic form on the manifold whose associated Delzant polytope is a trapezoid? I am trying to find it by using the Marsden–Weinstein theorem, but I have been unable to do so. If ...

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52 views

### How exactly do we construct the $T^2\times \mathbb{R}$ toric Calabi-Yau three-fold?

I am trying to understand why and how the functions $r_{a}(z) = |z_1|^2 - |z_3|^2$, $r_{b}(z)=|z_2|^2 - |z_3|^2$ and $r_{c}(z)=\Im(z_1z_2z_3)$ "generate" the toric CY threefold $T^2 \times \mathbb{R}$ ...

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122 views

### Finite resolution by sums of line bundles on toric varieties

I hope I wasn't searching wrong keywords or overlooking some easy arguments to prove/disprove it. What I'm asking is the following:
Let $X$ be a smooth complete toric variety. $\mathcal F$ a coherent ...

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124 views

### A question on a toric singularity

Recently I was told that the threefold singularity
$$
x^2+y^2+z^2+w^{2n}=0
$$
in $\mathbb{C}^4$ is a toric variety. How can I see this? What are the generators of the toric fan? Moreover, is it true ...

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196 views

### Connectedeness of toric varieties

Hi,
I would like to understand when a toric variety is connected. Given $\Delta$ a fan (possibly with infinitely many cones) in $\mathbb{R}^n$, $n\geq 2$ denote with $X_{\Delta}$ the associated toric ...

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176 views

### Variation the definition of toric varieties

Let us stick to affine toric variety.
By definition, a toric variety is a variety containing a torus $T \cong (\mathbb{C}^*)^n$, with the torus action on $T$ extend to the whole variety. The torus ...

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134 views

### Relation between the index of (the sum of) lattices in euclidean space, and their orthogonal complement.

I am trying to figure out something concerning the index of lattices.
The question came about after reading the paper of W.Fulton and B.Sturmfels, ("Intersection theorey on toric varieties"). To ...

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196 views

### tangent bundle of the toric variety of the wonderful compactification.

Let G be a adjoint group over $k$,algebraically closed of caracteristic zero.
Let $\overline{G}$ be its wonderful compactification.
I denote by $\overline{T}$ the closure of the torus $T$ in ...