The irregularity of ruled surfaces in P 3

Abstract

In this paper the authors determine which pairs of integers (d,q) appear as the degree and the irregularity of a ruled surface S in P 3 . The proof is based on the study of the base curve C of S in the Grassmannian G(1,3) ⊂P 5 : according to the dimension of the span of C some inequalities on d,q are given, and the existence of a curve C satisfying them follows from previous results of Rathmann and Pasarescu on curves in P 4 and in P 5 .