Let $\Delta$ be a fan in the lattice $N\cong\mathbb Z^n$ with $d$ edges $\{\rho\_1,\cdots,\rho\_d\}$. Consider the co-ordinate ring $\mathbb C[x\_1,\cdots,x\_n]$. Let $\mathcal Z=\bigcup\_C\mathcal V(x\_i\ |\ \rho\_i\in C)$ where the union is taken over all primitive collections $C$ of edge vectors in $\Delta$. 


**Definition** - A subset $C\subseteq \Delta(1)$ (the set of edges in $\Delta$) is a primitive collection if :


1. $C\nsubseteq \sigma(1)$ for all $\sigma \in \Delta$
2. For every proper subset $C'$ of $C$ there is a $\sigma\in\Delta$ with $C'\subseteq \sigma(1)$


**Question** - Is there a way in general to check if $\mathbb C^d\setminus\mathcal Z$ is simply connected in the analytical topology?


I know that $2\le\text{codim}\mathcal Z\le d$. 


Suppose further that $\Delta$ is compete and simplicial then we have either 


1. $2\le \text{codim}\mathcal Z\le \lfloor \frac{n}{2}\rfloor+1$ or
2. $d=n+1$ and $\mathcal Z=\{0\}$


It is easy to see that if $\Delta$ is simplicial and 2. is true then $\mathbb C^d\setminus\mathcal Z=\mathbb C^{n+1}\setminus\{0\}$ is simply connected. What about in general? I understand that it will not always be true, but for what codimension of $\mathcal Z$ can one expect $\mathbb C^d\setminus\mathcal Z$ to be simply connected? (Assuming there is a way to relate codimension of $\mathcal Z$ to the simply connectedness of $\mathbb C^d\setminus\mathcal Z$).


Thank you.



When is $\mathbb C^d\setminus\mathcal Z$ simply connected?
There is a simple general fact about k-connectedness of the complement of an affine algebraic set, which seems not to be so well known:



> 
> **Theorem.** If $Z \subseteq \mathbb{C}^d$ is Zariski-closed of codimension $c$, then $\pi\_i(\mathbb{C}^d\smallsetminus Z) = 0$ for $0<i\leq 2c-2$, and if $Z$ is nonempty, then $\pi\_{2c-1}(\mathbb{C}^d\smallsetminus Z) \neq 0$.
> 
> 
> 


This is stated as Proposition 4.1 in [these notes](https://people.math.osu.edu/anderson.2804/eilenberg/appA.pdf) (PDF); the proof is not hard. In particular, I think the sets you're asking about are always simply connected.


