Questions tagged [fans]
A fan is a collection of cones closed under taking intersections and faces.
11
questions
3
votes
1
answer
103
views
A question related to the strong Oda conjecture
A fan is a collection of strongly convex rational polyhedral cones in $\mathbb Z^n$, which we often think of as contained in $\mathbb Q^n$ or $\mathbb R^n$ for purposes of visualizing it. The defining ...
- 6,125
0
votes
1
answer
63
views
Iterated barycentric subdivision cofinal in system of subdivisions?
Given a rational polyhedral fan $\Sigma$ in $\mathbb{R}^d$ (say with full-dimensional support), its barycentric subdivision $\mathrm{bar}(\Sigma)$ is obtained by performing star-subdivision at the ...
- 681
0
votes
1
answer
352
views
The dimension of the normal cone of a face in a polytope
Let $P$ is a polytope in $\mathbb{R}^n$ if $F$ is one of its faces of dimension $d$ then the dimension of its normal cone $\mathcal{N}(F)$ is $n-d$.\
This seems to be intuitively obvious but I can't ...
2
votes
1
answer
252
views
Is the blowup of a toric variety corresponding to a subdivision normal?
Toroidal Embeddings 1 by KKMS say subdividing the fan of a toric variety yields the fan of a normalized blowup. How do I avoid normalization? Do I need to choose my subdivision carefully, or is it ...
- 1,084
2
votes
1
answer
157
views
Sections of Cartier divisors on toric varieties
Let $X_{\Sigma}$ be a projective toric variety. Consider the total coordinate ring
$$S = \mathbb{C}[x_{\rho}\: | \: \rho\in\Sigma(1)]$$
Define $\deg(x_{\rho}) = D_{\rho}$.
Now, take a divisor $D = \...
user75933
3
votes
1
answer
244
views
Relation between the number of maximal cones in a fan and the geometry of corresponding toric variety
It is well known that a (smooth complete) fan $\Delta$ corresponds to a (smooth proper) toric variety $X= X_\Delta$.
My question is whether there is a relationship between the number of maximal cones ...
- 95
2
votes
0
answers
273
views
Is there a "fundamental theorem of toric geometry"?
I have some questions about toric geometry.
1) Can any toric variety (I mean not necessarily smooth or projective, ...) be constructed from a fan ?
2) Suppose $T_1$ and $T_2$ are toric varieties ...
- 4,353
2
votes
2
answers
604
views
The boundary of toric varieties
Let $\mathcal{X}$ be a toric variety, with $T$ a torus embedded as an open set
in $\mathcal{X}$ (and where the algebraic action of $T$ extends to $\mathcal{X}$). As I am not a toric specialist at all, ...
- 4,353
4
votes
0
answers
269
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 ...
- 882
7
votes
0
answers
360
views
A question about a blue fan and a red fan and their common refinement
Is the following conjecture true?
Conjecture: Let $M_1$ be a red map and let $M_2$ be a blue map drawn in general position on $S^n$, and let $M$ be their common refinement. There is a vertex $w$ of $...
- 24.2k
5
votes
0
answers
502
views
Reference for this polyhedral lemma
Recall the definition of a fan: Let $U$ be a finite dimensional real vector space. Then a fan is a collection $\mathcal{F}$ of cones in $U$ such that
(1) If $\sigma \in \mathcal{F}$ and $\tau$ is a ...
- 151k