Let $\mathcal{D} \approx \mathbb{P}^{\delta_d}$ be the space of homogeneous 
degree $d$ polynomials in three vriables, where 
$\delta_d = \frac{d(d+3)}{2}$. Let 
$$ X \subset \mathcal{D} \times \mathbb{P}^2$$ 

be a smooth embedded complex submanifold, not necessarily closed. 
Given a point $p\in \mathbb{P}^2$, we get a hyperplane 
$$\tilde{H}_p \in  \mathcal{D} \times \mathbb{P}^2.$$
Note that a point $p$ first of all gives a hyperplane $H_p$ 
in $\mathcal{D}$ (which is the space of degree $d$ 
polynomials passing through the point $p$). This gives us a hyperplane 
$$ \tilde{H}_p := H_p\times \mathbb{P}^2 \in \mathcal{D} \times \mathbb{P}^2.$$
Let us further define $H_p^* \subset H_p$ to be the 
space of degree $d$ curves such that $p$ is a smooth point of the 
curve. Similarly define
$$ \tilde{H}_p^* := H_p^*\times \mathbb{P}^2 \in \mathcal{D} \times \mathbb{P}^2.$$

Is it true that for almost all choices of $p \in \mathbb{P}^2$, 
$\tilde{H}^*_p$ is transverse to $X$?

The answer is no.
A fixed point free action of the finite group $A_5$ on a $n$-cell was constructed by Floyd and Richardson in their paper An action of a finite group on an n-cell without stationary points, Bull. Amer. Math. Soc. Volume 65, Number 2 (1959), 73-76.
For some non-existence results, you can see the paper by Parris Finite groups without fixed-point-free actions on a disk, Michigan Math. J. Volume 20, Issue 4 (1974), 349-351.
