I have the following question which is in some way related to an application of Randell Isotopy Theorem to complex hyperplane arrangements.

Let $h,k\geq1$ be integer numbers and let $F_{1},\ldots,F_{h}$ be elements of the polynomial ring $\mathbb{C}\left[t_{1},\ldots,t_{k}\right].$ Now, consider the space $$X=\lbrace P\in\mathbb{C}^{k}\mid F_{j}(P)=0\text{ for }1\leq j\leq h\rbrace$$ where $F_{j}(P)$ simply denotes the evaluation of the polynomials $F_{j}$ at the point $P.$

Let us assume $X$ is endowed with the $\textit{classical}$ topology. In general $X$ is not a topological manifold. However, I have the following question regarding the path connectivity of the connected components of $X.$ In fact, $X$ a locally path connected space, so that its connected components coincide with its path connected components. To be more precise, my question regards the smoothness of paths connecting points in $X.$

$\textbf{Definition:}$ A path $\gamma:\left(a,b\right)\longrightarrow\mathbb{C}^{k}$ is called $\textit{smooth}$ if every derivative of any order are continuous.

$\textbf{Remark:}$ Pay attention that this definition of smoothness is different from the one of $\textit{regularity}$ of a parametrized curve. I am NOT asking the tangent vector $\gamma'(t)$ is not vanishing.

$\textbf{QUESTION:}$ Let $P,Q\in X$ be two points of $X$ which are in the same connected component. Then, there exists $\epsilon>0$ and a smooth path $\gamma:\left(-\epsilon,1+\epsilon\right)\longrightarrow\mathbb{C}^{k}$ with $\gamma(t)\in X$ for any $t\in\left(-\epsilon,1+\epsilon\right)$ and such that $\gamma(0)=P$ and $\gamma(1)=Q.$