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Questions tagged [fans]

A fan is a collection of cones closed under taking intersections and faces.

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3 votes
1 answer
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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 ...
Hugh Thomas's user avatar
  • 6,210
0 votes
1 answer
68 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 ...
JoS's user avatar
  • 681
0 votes
1 answer
366 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 ...
Mathlover's user avatar
2 votes
1 answer
275 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 ...
Leo Herr's user avatar
  • 1,084
2 votes
1 answer
158 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 = \...
user avatar
3 votes
1 answer
250 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 ...
Kazuki  Sato's user avatar
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 ...
THC's user avatar
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2 votes
2 answers
617 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, ...
THC's user avatar
  • 4,333
4 votes
0 answers
270 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 ...
Camilo Sarmiento's user avatar
7 votes
0 answers
361 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 $...
Gil Kalai's user avatar
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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 ...
David E Speyer's user avatar