On regular ruled surfaces in \({\mathbb{P}}^ 3\)
In the author's joint paper with \textit{E. Arrondo} and \textit{I. Sols} [Algebraic curves and projective geometry, Proc. Conf., Trento/Italy 1988, Lect. Notes Math. 1389, 1-15 (1989)] appears a proof in nowaday's terms, of \textit{F. Severi}'s assertion [Rend. Mat., Univ. Roma, V. Ser. 2, 1-32 (1941; Zbl 0024.34301)] that there are regular ruled surfaces of degree \( d\) and \(irregularity\quad q,\) if and only if \(d\geq 2q+2\), and the corresponding Hilbert scheme is irreducible and of general moduli. We can thus talk about the generic ruled surface of degree \( d\) and \(irregularity\quad q.\) The main purpose of this paper is to prove a maximal rank theorem for regular ruled surfaces in \({\mathbb{P}}^ 3\) by using these results. This fact was taken for granted by the Italian geometers as a ``principio de enumerazione delle constanti'', but an effective proof of it wasn't known in the literature up to now.
scientific article; zbMATH DE number 4123900