Class of the discriminant of a conic bundle

Question

Let $X$ be a smooth projective variety and $E$ a vector bundle of rank $3$ over $X$. Moreover let $L \in Pic(X)$ be a line bundle and $$q:S^2E \rightarrow L$$ a $L-$valued quadratic form. Then we can consider the subvariety $C_q \subset \mathbb{P}(E)$, where $\pi:\mathbb{P}(E) \rightarrow X$ is the projective bundle associated to $E$, given as the zero locus of the section $s_{C_q} \in H^0(\mathbb{P}(E), \mathcal{O}_{\mathbb{P}(E)}(2) \otimes \pi^*L)$ corresponding to $q$. With the restriction projection $\pi:C_q \rightarrow X$ the scheme $C_q$ has a structure of a conic bundle over $X$. Call $D \subset X$ the discriminant divisor, i.e. the locus of points $x \in X$ such that $\pi^{-1}(x)$ is a singular conic in the fiber $\mathbb{P}^2_x$.

How can we express $D \in H^2(X,\mathbb{Z})$ as a function of Chern classes of $E$ and $L$?

Answer 1

The map $q$ induces a morphism $$ E \to E^\vee \otimes L $$ and the discriminant is the zero locus of its determinant $$ \det(E) \to \det(E^\vee) \otimes L^{\otimes 3}. $$ Thus $$ [D] = 2c_1(E^\vee) + 3c_1(L). $$