Hilbert curves of scrolls over threefolds

Let (X,L) be a complex polarized n-fold with the structure of a classical scroll over a smooth projective threefold Y. The Hilbert curve of such a pair (X,L) is a complex affine plane curve of degree n, consisting of n−3 evenly spaced parallel lines plus a cubic. This paper is devoted to a detailed study of this cubic. In particular, existence of triple points, behavior with respect to the line at infinity, and non-reducedness, are analyzed in connection with the structure of (X,L). Special attention is reserved to the case n=4, where various examples are presented and the possibility that the cubic is itself the Hilbert curve of the base threefold Y for a suitable polarization is discussed.