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 degree $d$ polynomials passing through the point $p$). This gives a hyperplane in $$ \tilde{H}_p := H_p\times \mathbb{P}^2 \in \mathcal{D} \times \mathbb{P}^2.$$ Is it true that if $\tilde{H}_p$ is not transverse to $X$, then there exists an open neighborhood $U$ of $p$ in $\mathbb{P^2}$, such that for all points $q\in U$ that are not equal to $p$, the hyperplane $\tilde{H}_q$ does intersect $X$ transversally? This seems to intuitively say that if something is not transverse then you can ``perturb'' it so that it is transverse. Is there some general theorem in differential topology that implies this immediately?