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 :

$C\nsubseteq \sigma(1)$ for all $\sigma \in \Delta$

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

$2\le \text{codim}\mathcal Z\le \lfloor \frac{n}{2}\rfloor+1$ or

$d=n+1$ 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.