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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 ...
Gil Kalai's user avatar
  • 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 $v$ ...
Yunhyung Cho's user avatar
  • 1,037
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, ...
Gjergji Zaimi's user avatar
7 votes
1 answer
289 views

Separating a lattice simplex from a lattice polytope

Let $P\subset\mathbb{R}^n$ be a convex lattice polytope. Do there always exist a lattice simplex $\Delta\subset P$ and an affine hyperplane $H\subset\mathbb{R}^n$ separating $\Delta$ from the convex ...
Puzzled's user avatar
  • 8,998
5 votes
1 answer
462 views

Polynomial size embeddings of toric varieties from polytopes?

Background: Let $P$ be a integral polytope, and $X_P$ the toric variety associated to the normal fan. $X_P$ is always projective, because the collection of characters corresponding to the points $\...
Elle Najt's user avatar
  • 1,462
4 votes
1 answer
159 views

Approximation of convex bodies by polytopes corresponding to smooth toric varieties

Let $P\subset \mathbb{R}^n$ be an $n$-dimensional polytope with rational vertices. There is a well known construction which produces an $n$-dimensional algebraic variety $X_P$ called toric variety. In ...
asv's user avatar
  • 21.8k
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: ...
aglearner's user avatar
  • 14.3k
4 votes
0 answers
225 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 ...
cata's user avatar
  • 357
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 $P$. ...
asv's user avatar
  • 21.8k
3 votes
1 answer
153 views

Taking powers of polytopes

I am not sure this is a well framed question but I would like to know if anything like "taking the power" of a polytope is known. Imagine this situation where I want to think of such a thing : say ...
gradstudent's user avatar
  • 2,246
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 ...
asv's user avatar
  • 21.8k
3 votes
0 answers
182 views

Zeros of Hilbert series of affine toric varieties

Consider a convex rational polyhedral cone $C\subset\mathbb R^m$ with vertex at the origin. Let $X$ be the corresponding affine toric variety, i.e. $\mathbb C[X]=\mathbb C[\mathbb Z^m\cap C^\circ]$. ...
Igor Makhlin's user avatar
  • 3,513
2 votes
2 answers
459 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 $(-K_X)^n$ ...
alvarezpaiva's user avatar
  • 13.5k
1 vote
0 answers
73 views

Polytope of a projected toric variety

I was looking for such a result in the book by Cox, Little and Schenck but I'm not able to find a proper reference. All of the following requirements are tacitly assumed to be in the projective ...
gigi's user avatar
  • 1,343