Algebraic Surfaces with Hyperelliptic Sections

Surfaces whose prime (i.e. hyperplane) sections are hyperelliptic were studied and classified by Castelnuovo (2). If the sections have genus p, no surface can have order greater than 4p + 4, and any of lesser order is a projection of a normal surface Ф in a projective space S of 3p + 5 dimensions. There is a pencil of conies, none of them singular, on Ф; through each point of Ф passes one of the conies and their planes generate a threefold V of order 3p + 3 (2, § as the paper was later republished with different pagination, it may be advisable to refer to it by sections).