Hilbert curves of conic fibrations over smooth surfaces

Let (X, L) be a complex polarized threefold which is a conic fibration over a smooth surface. The complex affine cubic Γ representing the Hilbert curve of (X, L) is studied, paying special attention to its reducibility. In particular, Γ contains a specific line ℓ_0 if and only if X has no singular fibers. This leads to characterize the existence of a triple point simply in terms of numerical invariants of X. Other lines may cause the reducibility of Γ, which in this case depends also on the polarization. This situation is analyzed for a special class of conic fibrations.