The toric-varieties tag has no usage guidance.

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### Are Grassmannians toric varieties? [closed]

Are Grassmannians $G(k,n)$ toric varieties for all possible $k,n$? If they are toric varieties, are there any descriptions for the fans?

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174 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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**1**answer

229 views

### Why the blowup of a toric variety corresponding to a subdivison of fan?

I heard the statement "the blowup of a toric variety corresponding to a subdivison of fan" many times, but could not find reference in the literature. What is the precised statement (blowup a point? ...

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212 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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186 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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**1**answer

287 views

### Derived categories of toric varieties and convex geometry

Toric varieties and convex polyhedra are intimately connected. Some of this can be found in standard text books (the connection between divisors and mixed volumes seems to be a popular example).
One ...

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170 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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**1**answer

359 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 ...

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312 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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411 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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625 views

### Stanley-Reisner ring of a simplicial complex is a functor?

Let $K$ bea field and $[n]=\{1,\ldots,n\}$ and $K[x]=K[x_1,\ldots,x_n]$. For $\sigma=\{i_1,\ldots,i_k\}\subseteq [n]$, denote $x_\sigma=x_{i_1}\cdots x_{i_k}=\prod_{i\in\sigma}x_i\in K[x]$. Let ...

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175 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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140 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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484 views

### Toric Fano manifolds with Picard number 1

As far as I know, toric Fano manifolds are classified only up to dimension 4. In dimension one the projective line is the only example. In dimension two we have five examples: $\mathbb P ^2$, $\mathbb ...

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**1**answer

151 views

### Recommendations for binomial system solver

I am interested in solving binomial systems of the form
$$
\begin{cases}
a_1 x_1^{d_{11}} x_2^{d_{12}} \cdots x_n^{d_{1n}} +
b_1 x_1^{d_{11}} x_2^{d_{12}} \cdots x_n^{d_{1n}} &= 0 \\\\
...

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**1**answer

230 views

### Smooth projective toric varieties which are quotients of product of spheres and torii by a free torus action?

My question is just as in the box. Is every smooth projective toric variety diffeomorphic to a quotient of $\prod_i S^{n_i} \times T^k$ (I know torus is a one-sphere but I just wanted to make clear I ...

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**2**answers

244 views

### Computing rational cohomology of smooth (not necessarily compact) toric varieties

The title pretty much says it all. I am looking for references (lecture notes, books, readable articles, suggestions), preferably example laden, that explain how to compute the rational cohomology of ...

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248 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 ...

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**1**answer

311 views

### How to recover toric invariants tropically?

My excuses in advance in case my question is too vague (which is mainly due to the fact that I'm not really familiar with tropical/toric geometry, but at least I still believe that the content of the ...

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**1**answer

1k views

### Clean introduction to toric varieties for an undergraduate audience

I will be giving a talk to a (primarily) undergraduate audience on certain relatively concrete computations with toric varieties and their blowups. The talk is short, about 20 mins. As I result I need ...

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671 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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**4**answers

783 views

### The Constructions of Davis and Januszkiewicz

One of the most useful tools in the study of convex polytope is to move from polytopes (through their fans) to toric varieties and see how properties of the associated toric variety reflects back on ...

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57 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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**1**answer

539 views

### Triangulations of polytopes and tilings of zonotopes

Consider a set $A = \{ a_1,a_2,\ldots, a_n \} $ of vectors in $\mathbb{R}^d$, which lie in a common affine hyperplane. Two convex polytopes may be obtained from $A$, namely the convex hull of the ...

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498 views

### Detecting tilings by toric geometry

This is probably a silly question, but I figured that if there is a good answer, this would be a good place to ask.
Ever since I got my hands on the book "Toric Varieties" by Cox, Little and Schenck, ...

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448 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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**1**answer

402 views

### Hypersurfaces in Toric Varieties, Help understand a proof from Mikhalkin's paper

Hello,
in G. Mikhalkin's Papaer "DECOMPOSITION INTO PAIRS-OF-PANTS FOR
COMPLEX ALGEBRAIC HYPERSURFACES":
http://arxiv.org/pdf/math/0205011.pdf
There is a lemma about the relation between intersection ...

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**1**answer

303 views

### When a quotient singularity is toric?

Let $G \subset SL(n,\mathbb{C})$ be a cyclic subgroup of finite order,
Is it true that $\mathbb{C}^n /G$ is toric ? If not then when it is ?

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312 views

### Bound on the (anticanonical) degree of toric Fano varieties

Does there exists a universal constant $C \geq 1$ such that if $X$ is any a smooth, toric, Fano $n$-dimensional manifold admitting a Kähler-Einstein metric, then its anticanonical degree $(-K_X)^n$ ...

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967 views

### nef Cone of a Toric Variety

Is there a systematic/standard way of extracting the nef cone of a toric variety $X$ from its fan (or polytope)? Can I tell from the basis of $A^1(X)$ induced by the one-skeleton, based on ...

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208 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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713 views

### Do projective hypersurfaces contain projective toric varieties?

Is there an example of a smooth projective hypersurface in $\mathbb{P}^n_k$ ($k=\overline{k}$) that does not contain any projective toric varieties (edit: of positive dimension)? Or is it the case ...

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429 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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**1**answer

594 views

### Description of a birational map, Fulton's “Introduction to toric varieties”

This is an exercise of "Introduction to toric varieties" by Fulton, page 71, section 3.4, (the last one in this page.)
The problem is initiated by constructing a complete (toric) variety which is not ...

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368 views

### Toric automorphism of P1 times P1 blown up at four pts

A toric morphism between toric varieties is a morphism that is equivariant w.r.t. to the toric action, see e.g. section 3.2, Notes by H.Verrill and D.Joyner for definitions. In particular, any toric ...

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448 views

### Are the closures of the tori in the decomposition of a torified variety toric varieties?

In "Torified varieties and their geometries over $\mathbb{F}_1$", J. L. Pena and O. Lorscheid define a torified variety as a variety $X$ over $\mathbb{Z}$ along with a family of locally closed ...

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266 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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309 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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493 views

### Number of $(-1)$ curves on toric surfaces

Hello.
My question is:
Is it possible that a smooth complete toric surface has infinitely many $(-1)$-curves. I know that there is a blow-up of $\mathbb P^2$ in 9 points containing infintely many ...

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**4**answers

564 views

### What is the fan of the toric blow-up of $\mathbb{P}^3$ along the union of two intersecting lines?

Is there a good way to find the fan and polytope of the blow-up of $\mathbb{P}^3$ along the union of two invariant intersecting lines?
Everything I find in the literature is for blow-ups along smooth ...

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**1**answer

569 views

### Smooth complete toric surfaces

Hi,
I'm trying to find all the smooth complete toric surfaces, following section 2.5 of Fulton's book. There is one exercise given that I'm hung up on, hopefully someone can help me out a bit. ...

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717 views

### Deformations of Hirzebruch surfaces and toric action

Hi,
the Hirzebruch surface $F_n$ admits a deformation for $0\leq m\leq n$ defined by the equation
$$
\mathcal{M}=\{ ([x_0:x_1],[y_0:y_1:y_2],t) \in \mathbb{P}^1 \times \mathbb{P}^2 \times ...

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1k views

### Is there always a toric isomorphism between isomorphic toric varieties?

Suppose two toric varieties are isomorphic as abstract varieties. Does it follow that there exists a toric isomorphism between them?
Edit: the comments below lead me to believe that I'm using the ...

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433 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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**1**answer

457 views

### Intersection of curves on projective toric surface and some enumerative questions

Reading on the tropical approach to enumerative geometry I have come across the claim:
given a projective toric surface from a polygon P, we can consider a tautological bundle of algebraic / ...

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246 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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517 views

### On homology of Toric varieties

Lets $X$ be a simply connected projective toric variety of dimension $n$.
Lets $\tau_1,\cdots,\tau_k$ be the set of $(n-1)$-dimensional cones of corresponding fan which is in one-to-one ...

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**1**answer

236 views

### Can cones (toric monoids) be built as colimits of their faces?

Suppose $L$ is a lattice (free abelian group) and $\sigma$ is a (pointed) spanning rational cone in $L\otimes\mathbb Q$. Then $M=L\cap \sigma$ is a monoid with $M^{gp}=L$. A monoid of this form is ...

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926 views

### Are projective toric varieties, locally complete intersection?

Let $X^n \subset \mathbb{P}^N$ to be a toric projective variety. Is $X$ a local complete intersection? Is being a local complete intersection an intrinsic property, independent of embedding?

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817 views

### Is an affine “G-variety” with reductive stabilizers a toric variety?

Let $X=Spec(A)$ be a reduced normal affine scheme over an algebraically closed field $k$ of characteristic $0$, with an action of a connected reductive group $G$. Suppose
$x\in X$ is a ...