All Questions
16 questions
20
votes
5
answers
1k
views
From convex polytopes to toric varieties: the constructions of Davis and Januszkiewicz
One of the most useful tools in the study of convex polytopes is to move from polytopes (through their fans) to toric varieties and see how properties of the associated toric variety reflects back on ...
- 24.7k
19
votes
2
answers
1k
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 $v$ ...
- 1,037
17
votes
2
answers
3k
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
2
answers
665
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, ...
- 85.6k
12
votes
0
answers
529
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,...
- 181
6
votes
2
answers
781
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 ...
- 38.6k
5
votes
1
answer
528
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 affine,...
4
votes
1
answer
189
views
2-faces of reflexive Delzant polytopes
Question 1. Can a reflexive Delzant polytope of some dimension contain a $2$-face with more than $11$ edges?
Motivation. I would like more generally to get an answer to the following question:
...
- 14.3k
4
votes
0
answers
87
views
Toric Bézier patches
Toric Bézier patches (as described in https://arxiv.org/abs/0706.2116) are maps from a lattice polytope $P$ to the positive part of its associated toric variety $X_P$. While they are not the inverse ...
- 1,435
3
votes
1
answer
336
views
Cohomology ring of a hypersurface in toric variety
Let $X^n$ be a smooth projective toric variety corresponding to a simple polytope $P$. It is well known that the cohomology ring $H^*(X)$ can be described in terms of combinatorics of faces of $P$.
...
- 21.8k
3
votes
0
answers
232
views
When is a wonderful compactification a toric variety?
Given a (projective) hyperplane arrangement $\mathcal{A}$ in $\mathbb{C}^n$, in which we assume the intersection of all hyperplanes is $0$, and a building set $G$, De Concini and Procesi define the ...
- 518
3
votes
0
answers
189
views
Resolutions of configuration space of the projective line where the complement is of "Tate type"
I would like to find a nice compactification $X_n$ of $F(\mathbb P^1,n)$ (considered as a scheme over $\mathbb Z$), the $n$-fold configuration space of the projective line with the property that the $...
- 7,746
3
votes
0
answers
179
views
Polytope algebra and toric vareties
Let $\Pi$ denote the McMullen polytope algebra (over the field of rationals $\mathbb{Q}$) generated by convex polytopes with rational vertices in $\mathbb{R}^n$.
For a simple polytope $P$ let us ...
- 21.8k
3
votes
0
answers
105
views
Structure of fibers of (complex) moment map of hypertoric variety
I am primarily interested in the hypertoric variety $\mathfrak M(\mathcal B_d)$ associated to the braid arrangement.
Any hypertoric variety $X$, say of complex dimension $2n$, comes equipped with an ...
- 71
0
votes
1
answer
390
views
Moment maps and flat degenerations of toric varieties
We have a flat family of projective varieties with a torus $T$ action, over $\mathbb{A}^1$.
How do the moment map images of the fibers change when we pass from the generic fiber to the special fiber ...
- 1,719
0
votes
0
answers
91
views
How to compute the multiplicity of a strongly convex, rational, polyhedral cone $ \sigma $?
In David Cox, John Little and Hal Schenck's book Toric Varieties page 302, Chapter 6, Section 4, Proposition 6.4.4, the authors state the following proposition. If $ \Sigma $ is a simplicial fan of ...
- 912