# Questions tagged [fans]

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

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### 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 ...
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### 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 ...
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 $D = \... 1answer 179 views ### Relation between the number of maximal cones in a fan and the geometry of corresponding toric variety It is well known that a (smooth complete) fan$\Delta$corresponds to a (smooth proper) toric variety$X= X_\Delta$. My question is whether there is a relationship between the number of maximal cones ... 0answers 246 views ### Is there a "fundamental theorem of toric geometry"? I have some questions about toric geometry. 1) Can any toric variety (I mean not necessarily smooth or projective, ...) be constructed from a fan ? 2) Suppose$T_1$and$T_2$are toric varieties ... 2answers 396 views ### The boundary of toric varieties Let$\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}$). As I am not a toric specialist at all, ... 0answers 183 views ### Which (polytopal) fans/polytopes are secondary? Let$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$-dimensional ... 0answers 334 views ### A question about a blue fan and a red fan and their common refinement Is the following conjecture true? Conjecture: Let$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$of$...
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 ...