One nonempty parameter open gives smooth-locus avoidance and transversality

Under [A1], use the given family of [F2]. The zero-incidence scheme $I_D$ over $D$ is the total space of the kernel of the surjection ${\mathcal{O}}_D^N\to E{\mathrm{\mid }}_D$. Indeed local splitting identifies its equations with three independent parameter coordinates set to zero. Its dimension is at most $\dim D+N-3\le N-1$, and in pure dimension three it is at most $N-2$. This is a scheme dimension bound even when $D$ is singular.

By [F3], the closures of the images of $I_D$ and $B$ in $T$ have dimension at most $N-1$. Their union is a proper closed subset of the irreducible $N$-dimensional affine space $T$. Let $T^{\circ }$ be its complement. It is nonempty and has a closed $k$-rational point by the Jacobson/residue-field assertion in [F3]. Thus the parameter exists over the original field, not merely over an unspecified extension.

By definition of those image closures, their open complement has empty inverse image in $I_D$ and $B$. This remains true after any field extension. Therefore every corresponding zero lies in $W$ with nonzero derivative determinant. This controls actual incidence schemes and all geometric zeros. No generic-smoothness assertion from the total dimension of $I_W$ is used, and no dominance of $I_W\to T$ is required for this conclusion. The next lemma identifies its actual fibers over $T^{\circ }$, including its generic fiber.