Laudal Type Theorems for Algebraic Curves

Laudal’s Lemma states that if C is an integral curve in P3 of degree d > s2 + 1 and Z is its general plane section, then C is contained in a surface of degree s provided that Z is contained in a curve of degree s. The aim of this paper is to extend Laudal’s Lemma to possibly reducible curves proving that, under the unavoidable hypothesis that the Hilbert function of the generic plane section is of decreasing type, the bound s2 + s (not s2 + 1) holds. Moreover we prove that the bound is sharp providing various examples of reducible curves in P3 of degree d = s2 + s. Last we give an example where d = s2 + s − 1, that is a d-degree curve satisfying an intermediate bound.