Hilbert curves of quadric fibrations over smooth surfaces

Let (X, L) be a complex polarized n-fold with the structure of a geometric quadric fibration over a smooth projective surface. The Hilbert curve of (X, L) is a complex affine plane curve of degree n, containing n − 3 evenly spaced parallel lines. This paper is devoted to a detailed study of the cubic representing the residual component. Reducibility, existence of triple points, and properties of the irreducible components are analyzed in connection with the structure of (X, L).