All Questions
Tagged with toric-varieties convex-geometry
10
questions
4
votes
1
answer
145
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 ...
- 21.3k
2
votes
0
answers
200
views
Toric decomposition of multipartitions
Fix $k \in \mathbb Z_{>0}$. By a $k$-multipartition $\lambda=(\lambda_1,\dots,\lambda_k)$ of $N$, I mean that each $\lambda_i$ is a partition of some $N_i$ and $\sum N_i = N$.
Let's call $\lambda$ ...
- 111
4
votes
1
answer
284
views
Intrinsic definition of a cone in a normal fan
Let $P\subseteq \mathbb{R}^n$ be a full dimensional polytope. Let us assume that $P$ has a facet description with the following inequalities:
$$ \left<x,u_F\right> \geq -a_F$$
where $u_F\in \...
- 1,949
7
votes
1
answer
284
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 ...
- 8,922
4
votes
0
answers
113
views
Primitive collections as lattice generators for the Mori cone
I am looking at the following theorem from "Toric Varieties" by Cox, Little and Schenk:
Theorem 6.4.11: If $X_{\Sigma}$ is a simplicial toric variety, then
$\overline{NE}(X_{\Sigma}) = \...
- 197
5
votes
0
answers
337
views
Distinguishing ample divisors by minimally intersecting curves on a smooth projective toric variety
My question has an easily formulated generalization, which I will state first. Let $\sigma \subseteq \mathbf{R}^n$ be a full-dimensional strongly convex polyhedral cone. For each lattice point $m \in \...
- 197
5
votes
1
answer
253
views
A characterisation of faces of rational polyhedral cones
This is about a (seemingly) basic lemma about rational polyhedral cones that is sometimes used when working with toric varieties and is usually "left to the reader". Unfortunately, I could ...
- 6,690
1
vote
1
answer
106
views
Linear relations between volume of a polytope and its faces
Let $P$ be a polytope. Is anything known about the set of linear relations that hold between the volumes of the (not-necessarily proper) faces of $P$ as $P$ “varies slightly”? By varies slightly I ...
- 3,039
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
9
votes
2
answers
917
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 ...
