The toric-varieties tag has no wiki summary.

**20**

votes

**3**answers

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 ...

**17**

votes

**4**answers

724 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 ...

**16**

votes

**1**answer

675 views

### About a Delzant polytope. (In particular dodecahedron)

Hi. I have a question.
Definition. Delzant polytope $P$ is a rational convex simple polytope with the smooth condition. Here, "smooth" means that for each vertex $v$, the $n$ edges containing $v$ ...

**15**

votes

**0**answers

618 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 ...

**14**

votes

**2**answers

2k views

### What are some open problems in toric varieties?

In light of the nice responses to this question, I wonder what are some open problems in
the area of toric geometry? In particular,
What are some open problems relating to the algebraic ...

**12**

votes

**0**answers

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
$$
...

**11**

votes

**2**answers

730 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 ...

**9**

votes

**2**answers

479 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, ...

**9**

votes

**1**answer

720 views

### Cox rings of toric varieties over arbitrary fields

The Cox ring of a toric variety X can be viewed as a generalisation of the homogeneous coordinate ring of projective n-space. Over the complex numbers, the theory is outlined in The Homogeneous ...

**8**

votes

**2**answers

573 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 S^1 ...

**8**

votes

**1**answer

477 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 ...

**8**

votes

**0**answers

433 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

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
$$
...

**7**

votes

**0**answers

372 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 ...

**6**

votes

**2**answers

3k views

### The canonical line bundle of a normal variety

I have heard that the canonical divisor can be defined on a normal variety X since the smooth locus has codimension 2. Then, I have heard as well that for ANY algebraic variety such that the canonical ...

**6**

votes

**1**answer

262 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 ...

**5**

votes

**2**answers

853 views

### Counting/constructing Toric Varieties

Given a torus $T$ is there way to classify all the toric varieties it gives rise to? That is, classify all toric varieties $X$ whose torus is isomorphic to $T$. Is there a way to construct these ...

**5**

votes

**2**answers

430 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 ...

**5**

votes

**1**answer

841 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 ...

**5**

votes

**2**answers

652 views

### Software for computing multi-graded Hilbert series

The ring of invariants $S^T$ of $k[a,b,c,d]$ under the algebraic torus action $T = k^{*}$ with weights (1,1,-1,-1)
is $S = k[ac,ad,bc,bd]$ which has multigraded Hilbert series
$\frac{1 - ...

**5**

votes

**1**answer

267 views

### Can one prove that toric varieties are Cohen-Macaulay by finding a regular sequence?

It's well-known that if $P$ is a finitely generated saturated submonoid of $\mathbb{Z}^n$, then the monoid algebra $k[P]$ is Cohen-Macaulay (at the maximal ideal $m = (P-0)k[P]$). However, all proofs ...

**5**

votes

**2**answers

620 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 ...

**5**

votes

**0**answers

247 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

453 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

**3**answers

450 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 ...

**4**

votes

**2**answers

799 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 ...

**4**

votes

**1**answer

759 views

### Why does the Euler characteristic of a toric variety equal the number of vertices in the defining polytope?

In this link, Corollary 3.2.2, page 59 the author claims that: The Euler characteristic of the toric variety $X_K$ associated to a convex polytope $K$ is the number of vertices of $K$.
I want to see ...

**4**

votes

**2**answers

504 views

### Relationship between topological cohomology and $\ell$-adic cohomology

Let $\Delta \subset \mathbb{R}^n$ be an $n$-dimensional integral polytope, let $f$ be a Laurent polynomial in $n$-variables with coefficients in an extension of the integers and Newton polytope ...

**4**

votes

**3**answers

829 views

### relation between toric geometry and log geometry

Hello,
I'm trying to understand the relation between the points of view of
log geometry (monoids) and toric geometry (fans).
Suppose that $k$ is a field and $P$ is a finitely generated monoid.
Then ...

**4**

votes

**2**answers

299 views

### Cohomology of the toric variety $X_\Sigma=\mathbb C^2\sqcup \mathbb C^2\big/\left((x,y)_1\sim(x^{-1},y^{-1})_2\right)_{x,y\neq 0}$

I'm writing a thesis on Chow rings of toric varieties and am looking for a reference on the singular cohomology ring of the blowup of $\mathbb C^2$ at xy=0, i.e. at the coordinate axes. The ...

**4**

votes

**1**answer

221 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 ...

**4**

votes

**1**answer

429 views

### Secondary fans and Stanley Reisner ideals

Consider a collection of points $S \subset \mathbb R^d$. I would like to understand all possible fans $\Sigma$ whose support is the cone over $S$: $|\Sigma| = cone(S)$.
I have heard that the ...

**4**

votes

**1**answer

162 views

### Learning Quantum (Co)Homology and Landau Ginzburg Superpotential

I am learning about Quantum Homology which I have to use in my research, and I see that in many papers (For example in FOOO, "Spectral invariants with bulk, Quasimorphisms and Lagrangian Floer ...

**4**

votes

**2**answers

140 views

### Computational complexity of deciding isomorphism of rational polyhedral cones

Let $C,C'$ be rational polyhedral cones in $\mathbb R^n$ both with non-empty interior. Rational means they are generated by vectors with rational entries. One says that $C,C'$ are isomorphic if there ...

**4**

votes

**1**answer

434 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 / ...

**4**

votes

**0**answers

144 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

**3**answers

691 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 ...

**3**

votes

**3**answers

798 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?

**3**

votes

**3**answers

410 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 ...

**3**

votes

**1**answer

280 views

### Is there extremal metric on toric Fano manifolds which futaki invariant is nonzero?

According the work by Wang & Zhu, on toric Fano manifolds there exists Kahler-Ricci solitons, if futaki=0 there exist CSCK metric, but if futaki invariant is not vanished, what about extremal ...

**3**

votes

**1**answer

262 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 ?

**3**

votes

**1**answer

264 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 ...

**3**

votes

**1**answer

218 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 ...

**3**

votes

**1**answer

86 views

### Singularities induced by the toric ambient spaces

Let $\Delta \subset \mathbb{R}^4$ be a (reflexive) polytope and $X$ be the hypersurfacedefined by a generic section of the any-canonical bundle of the toric variety $\mathbb{P}_{\Delta}$. Are there ...

**3**

votes

**2**answers

632 views

### Moment map for toric actions — online references?

Consider a toric variety, defined as a (normal?) complex projective variety $X$ together with an algebraic action of $(\mathbb C^*)^n$ with finitely many orbits. Now we have two "real symplectic" ...

**3**

votes

**1**answer

142 views

### Toric ideal of slice of a polytope?

Given a collection $A:=\{a_1, \ldots ,a_n \}$ of different integer points in $\mathbb{N}^d$, which span an affine hyperplane when viewed in $\mathbb{R}^d$, one can define a toric ideal $I_A$ from a ...

**3**

votes

**1**answer

252 views

### Toric Fano Kahler manifolds and Delzant polytopes

Let $P$ be a Delzant polytope in $\mathbb R^n$, given by a set of inequalities $\ell_i(x) > 0$ where $\ell_i(x) = \sum_k \mu_k^i x_k - \lambda_i$.
In his paper http://arxiv.org/abs/0803.0985 ...

**3**

votes

**0**answers

75 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 ...

**3**

votes

**0**answers

50 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 ...

**3**

votes

**0**answers

63 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 ...