Consider the Segre embedding $$ \mathbb{P}^2 \times \mathbb{P}^2 \to \mathbb{P}^8.$$ Denote by $V$ the image of the Segre embedding and by $B$ the locus of triples $(H\_1, H\_2, H\_3)$ with $H\_i \in H^0(\mathcal{O}\_{\mathbb{P}^8}(1))$ a hyperplane section such that $V \cap H\_1 \cap H\_2 \cap H\_3$ is a proper intersection i.e., a curve. Clearly, $B$ is an open subset in $H^0(\mathcal{O}\_{\mathbb{P}^8}(1))^{\oplus 3}$. Denote by $B' \subset B$ the subloci consisting of triples $(H\_1, H\_2, H\_3)$ such that the intersection $V \cap H\_1 \cap H\_2 \cap H\_3$ is a singular curve. My question is: Is $B'$ necessarily of codimension $1$ in $B$ or can it be of higher codimension in $B$? Any hint/reference will be most welcome.



Segre embedding and intersections by hyperplanes
This is a standard projective duality argument.


Let $W = H^0(\mathcal{O}\_{\mathbb{P}^8}(1))$. Consider the variety $X$ of tuples
$$
(P,H\_1,H\_2,H\_3) \in V \times W^{\oplus 3}
$$
such that $V \cap H\_1 \cap H\_2 \cap H\_3$ is singular at $P$. Then it is easy to see that the projection
$$
X \to V,
\qquad
(P,H\_1,H\_2,H\_3) \mapsto P
$$
is flat and its fibers have dimension $22$ (indeed, there are 4 parameters for a 2-dimensional subspace in the tangent space $T\_PV$ and $3 \cdot 6 = 18$ parameters for the $H\_i$ passing through $P$ and containing this tangent space), hence $\dim(X) = 4 + 22 = 26$. It is also easy to see that $X$ is irreducible.


On the other hand, the general fiber of the map
$$
X \to W^{\oplus 3},
\qquad
(P,H\_1,H\_2,H\_3) \mapsto (H\_1,H\_2,H\_3)
$$
over its image is finite (to see this it is enough to find just one triple $(H\_1,H\_2,H\_3)$ such that $V \cap H\_1 \cap H\_2 \cap H\_3$ has a finite number of singular points), hence the dimension of the image of $X$ is 26, hence it is a divisor in $W^{\oplus 3}$.


