Hilbert curves of quadric fibrations

Abstract

Let [Formula: see text] be a quadric fibration over a smooth curve. The explicit equation of the corresponding Hilbert curve [Formula: see text] is obtained. The geometry of [Formula: see text] reflects some structure properties of [Formula: see text]; in particular, its special shape allows us to recognize that [Formula: see text] is a quadric fibration. In fact [Formula: see text] is reducible into [Formula: see text] 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.