All Questions
Tagged with toric-varieties discrete-geometry
9
questions
1
vote
0
answers
67
views
Embedding toric varieties in other toric varieties as a real algebraic hypersurface
In the question On a Hirzebruch surface, I've seen that the $n$-th Hirzebruch surface is isomorphic to a surface of bidegree $(n,1)$ in $\mathbb{P}^1\times \mathbb{P}^2$. I am trying to answer the ...
- 173
2
votes
1
answer
108
views
Coordinate-symmetric convex polytopes with equal Erhart (quasi)-polynomials
Recall that given a nondegenerate polytope $P \subset \mathbb{R}^n$ which is the convex set of some vectors with integral coordinates, the Erhart polynomial $p_P(t)$ a polynomial such that $p_P(t)$ ...
- 512
3
votes
0
answers
75
views
condition on rational polyhedral cone to guarantee dual cone is homogeneous
Let $\sigma\subseteq \Bbb R^d$ be a full-dimensional rational polyhedral cone which is strongly convex (i.e. $\sigma\cap-\sigma=0$).
Definition. The cone $\sigma$ is homogeneous if there are ...
- 3,039
3
votes
1
answer
168
views
Toric Desingularization Algorithms
There are certainly many algorithms to desingularize toric varieties (e.g https://arxiv.org/pdf/math/0411340.pdf). I would imagine in analogy with desingularizing surfaces these all involve blowing up ...
- 528
3
votes
0
answers
175
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. $\mathbb C[X]=\mathbb C[\mathbb Z^m\cap C^\circ]$. ...
- 3,503
4
votes
0
answers
158
views
Reference for the notion of polyhedra "degenerations"
Let $P$ be a convex polyhedron and let $P(t)$ be a continuous deformation thereof, such that:
a) $P(0)=P$;
b) for all $t\in[0;1)$ the polyhedron $P(t)$ is strongly combinatorially equivalent to $P$ (...
- 3,503
5
votes
2
answers
164
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 ...
- 37.8k
10
votes
1
answer
806
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 ...
- 882
6
votes
1
answer
749
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 ...
- 459