On generalized Laudal's lemma  pp. 284-293

On generalized Laudal's lemma

By R. Strano

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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.

  • 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.
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