Hilbert curves of quadric fibrations

Let (X,L) be a quadric fibration over a smooth curve. The explicit equation of the corresponding Hilbert curve \Gamma is obtained. The geometry of \Gamma reflects some structure properties of (X,L); in particular, its special shape allows us to recognize that (X,L) is a quadric fibration. In fact \Gamma is reducible into \dim X -2 parallel lines with prescribed slope, evenly spaced, plus a conic. On the other hand, this conic can itself be regarded as the Hilbert curve of a polarized surface only in very rare circumstances.