Let $X$ be a smooth projective variety, let $E$ be a vector bundle of rank $4$ on $X$ and let $L$ be a line budnle on $X$. Consider the projectivization $\mathbb{P}_X(E):=\mathrm{Proj}Sym(E^*)$ of the vector bundle $E$. Denote the projection $\mathbb{P}_X(E)\to X$ by $\pi$ and the Serre sheaf of the projectivization by $\mathcal{O}_{\pi}(1)$. A section of the vector bundle $S^3E^*\otimes L$ defines a family of cubic surfaces $p:\mathcal{S}\to X$ as the zero locus of a section the line bundle $\mathcal{O}_{\pi}(3)\otimes \pi^*L$ on $\mathbb{P}_X(E)$. Let $\Delta\subset X$ be the set of points $x \in X$ such that the cubic surface $\mathcal{S}_x:=p^{-1}(x)$ is singular.

The question: Is it true that $\Delta$ is either a divisor in $X$ or empty? Is there a formula for the divisor $\Delta\subset X$ in terms of $E$ and $L$?

Let $X$ be a smooth projective variety, let $E$ be a vector bundle of rank $4$ on $X$ and let $L$ be a line budnle on $X$. Consider the projectivization $\mathbb{P}_X(E):=\mathrm{Proj}Sym(E^*)$ of the vector bundle $E$. Denote the projection $\mathbb{P}_X(E)\to X$ by $\pi$ and the Serre sheaf of the projectivization by $\mathcal{O}_{\pi}(1)$. A section of the vector bundle $S^3E^*\otimes L$ defines a family of cubic surfaces $p:\mathcal{S}\to X$ as the zero locus of a section the line bundle $\mathcal{O}_{\pi}(3)\otimes \pi^*L$ on $\mathbb{P}_X(E)$. Let $\Delta\subset X$ be the set of points $x \in X$ such that the cubic surface $\mathcal{S}_x:=p^{-1}(x)$ is singular.

The question: Assume a general cubic surface $\mathcal{S}_x:=p^{-1}(x)$ is smooth. Is it true that $\Delta$ is either a divisor in $X$ or empty? Is there a formula for the divisor $\Delta\subset X$ in terms of $E$ and $L$?