Projective curves: multisecant schemes and lifting problems

Let Y subset of or equal to P" be a pure one-dimensional locally Cohen-Macaulay closed subscheme such that for every general hyperplane H there is a zero-dimensional subscheme Z(H) of Y boolean AND H and a subscheme C(H) of H with Z(H) subset of or equal to C(H). We prove, under certain assumptions on Y, length(Z(H)) and C(H), the existence of a subcurve Y' subset of or equal to Y and a scheme S subset of or equal to P" with Y' boolean AND H = Z(H) and S boolean AND H = C(H), for general H. The main cases we study are: C(H) rational normal curve and C(H) linear space. We prove also a lifting theorem for the property arithmetically Gorenstein, which generalizes a lifting theorem by Strano and Huneke-Ulrich. (C) 2003 Elsevier B.V. All rights reserved.