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 <s (see [HH], [E] for more informations). For this reason this method is called Castelnuovo method.

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.

1. a) If Γ is contained in a curve of H of degree σ and d > σ2-σ+4, then C lies on a surface of degree σ+1.

2. […]