Complex Projective Geometry
Edited by G. Ellingsrud
Edited by C. Peskine
Edited by G. Sacchiero
Edited by S. A. Stromme
Publisher: Cambridge University Press
Print Publication Year: 1992
Online Publication Date:July 2010
Chapter DOI: http://dx.doi.org/10.1017/CBO9780511662652.021
Subjects: Geometry and Topology
Introduction. Let C be an integral curve in P3 of degree d and let Γ = C∩H be the generic plane section. The idea of using Γ in order to deduce properties of C goes back to Castelnuovo and has been successfully used by many authors especially in the following problem: determine the maximum genus G(d, s) of a smooth curve of degree d not lying on a surface of degree
In this paper we discuss the following problem: given the Hilbert function of Γ, deduce some information about the least degree of a surface containing C.
Section 1 is devoted to recall some known results in this direction; the main result is Laudal's lemma which says that, if C is integral of degree d > σ2+1 and Γ is contained in a curve of H of degree σ, then C is contained in a surface of degree σ (see [L], Corollary p. 147 and [GP], Lemme).
On the other hand very little is known about curves of degree d, such that Γ is contained in a curve of H of degree σ and C is not contained in a surface of degree σ. In the remaining sections we give some partial results in the previous situation. In particular the following two theorems are proved for an integral curve C.