# Class of the discriminant of a conic bundle

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 $$L$$?

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 $$[D] = 2c_1(E^\vee) + 3c_1(L).$$