Fan conventions

If nothing stated, we assume set the following.

$N$ is a lattice of rank $n$. $N_ℚ = N ⊗ ℚ$, $N_ℝ = N ⊗ ℝ$, and $M, M_ℚ, M_ℝ$ are the dual spaces.

$Σ$ is a rational simplicial fan (each face is a strongly convex cone) of pure dimension $d$ in $N_ℝ$. Its support is denoted $|Σ|$. ${\underline{0}}$ is its zero face. Adding 1 at the top element, we get a ranked lattice of faces, with operators $∧$ $∨$ and order $≺$. The dimension of a face $σ$ is denoted $|σ|$. If $σ$ covers $τ$ we write $τ\prec\!\!\!\cdot\ σ$. $Σ_k$ denotes the set of faces of $Σ$ of dimension $k$.

In general, $ρ$ denotes a ray, and $τ$ and $σ$ denotes faces which most of the times verifies $τ ≺ σ$.

We note $N_{σ,ℝ}$ the tangent space of $σ$, $M_{σ,ℝ}$ its dual, $N_{σ,ℚ} = N_{σ,ℝ} ∩ N_ℚ$, etc. We denote $N^σ = N/N_σ$, $π^σ$ the corresponding projection, etc. More generally we use subscripts for objects corresponding to the tangent space, and superscript for objects corresponding to the quotient. If $τ ≺ σ$ is a pair of faces, we note $σ/τ$ or $σ^τ$ the image $π^τ(σ)$. The (transversal) star fan $Σ^τ$ is the fan in $N_ℝ^σ$ of faces $\{ σ^τ | σ ≻ τ \}$.

For each face $σ$, we choose a generator $ν_σ$ of the exterior algebra $⋀^{|σ|} N_σ$. If $σ = ρ$ is a ray, we choose $ν_ρ$ to be the unit vector inside $ρ$, and we alternatively denote it by $e_ρ$. For any face $σ$, we denote $ω^σ$ the dual of $ν_σ$. If $τ\prec\!\!\!\cdot\ σ$, then $σ^τ$ is a ray and has a unique unit vector $e^τ_σ$. We denote any preimage of $e^τ_σ$ by $n_{σ/τ}$, and we call it the unit vector of $σ$ normal to $τ$.

We get an orientation: if $τ\prec\!\!\!\cdot\ σ$, $\mathrm{sgn}(τ,σ) = ω_σ(n_{σ/τ} ∧ ν_τ)$

We say that a fan is balanced if for any $τ$ of dimension $d-1$, $$∑_{σ\ \cdot\!\!\!\succ τ} n_{σ/τ} \in N_{ℝ,τ}.$$

Extensions

Extension 1 (Non-simplicial fans)

Everything works if $\Sigma$ is not simplicial.

Extension 2 (Non-rational fans)

If $\Sigma$ is not rational, we can still define $N_{σ,ℝ}$, etc. For unit normal vectors $n_{σ/τ}$, we take any vector such that $n_{σ/τ} ∧ ν_τ = ±ν_σ$ and such that $π^τ(n_{σ/τ}) ∈ σ/τ$. We choose the $ν_σ$ freely, and the sign function is defined by $$\mathrm{sgn}(τ,σ) = \frac{ω_σ(n_{σ/τ} ∧ ν_τ)}{|ω_σ(n_{σ/τ} ∧ ν_τ)|}.$$

Extension 3 (Weighted fans)

One can assign weights to facet $w: Σ_d → ℤ^*$ on each facet. The star fans inherit the weights. The only difference is that the balancing condition now reads, for any $τ$ of codimension 1, $$∑_{σ\ \cdot\!\!\!\succ τ} w(σ) n_{σ/τ} \in N_{ℝ,τ}.$$

Extension 4 (Pseudo fans)

All the notations works for pseudo-fans.

The Chow ring of a fan