The toric-varieties tag has no wiki summary.

**0**

votes

**1**answer

104 views

### Moment map coordinates in tours action

I am trying to understand the proof of lemma 3.1, in this paper
In proof, they say that $g(dz_i,d\tau_k)=dz_i(\nabla\tau_k)=0$ I don't understand first and second equality.In second they say, ...

**15**

votes

**0**answers

579 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 ...

**12**

votes

**0**answers

429 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
$$
...

**8**

votes

**0**answers

428 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 ...

**8**

votes

**0**answers

298 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
$$
...

**7**

votes

**0**answers

353 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 ...

**5**

votes

**0**answers

235 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 ...

**5**

votes

**0**answers

439 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 ...

**4**

votes

**0**answers

136 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 ...

**3**

votes

**0**answers

135 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 ...

**3**

votes

**0**answers

139 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 ...

**3**

votes

**0**answers

251 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 ...

**3**

votes

**0**answers

362 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 ...

**2**

votes

**0**answers

64 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{ ...

**2**

votes

**0**answers

127 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 ...

**2**

votes

**0**answers

160 views

### 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 ...

**2**

votes

**0**answers

52 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?

**1**

vote

**0**answers

84 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 ...

**1**

vote

**0**answers

144 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 ...

**1**

vote

**0**answers

198 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 ...

**1**

vote

**0**answers

370 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 ...

**1**

vote

**0**answers

227 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 ...

**0**

votes

**0**answers

112 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 ...

**0**

votes

**0**answers

22 views

### What's a toric mirror for a determinated action?

I wonder what are the toric mirrors for the action of a Weyl group of a root system (let's say of type $E_7$) acting on the complex torus $\mathbb{T}=\mathrm{Hom}(Q,\mathbb{C}^*)$ with $Q$ the ...

**0**

votes

**0**answers

178 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 ...

**0**

votes

**0**answers

169 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 ...

**0**

votes

**0**answers

127 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 ...

**0**

votes

**0**answers

178 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 ...

**0**

votes

**0**answers

531 views

### Help with Fulton's Toric Varieties Book

Greetings.
In Fulton's 'Introduction to Toric Varieties' in section 2.5, there is an exercise (top of p44 in my book) which reads 'If $d \geq 4$ then there must be 2 opposite vectors in the ...