Projective curves: multisecant schemes and lifting problems

Let be a pure one-dimensional locally Cohen–Macaulay closed subscheme such that for every general hyperplaneH there is a zero-dimensional subschemeZH of Y∩H and a subschemeCH of H with ZH CH. We prove, under certain assumptions on Y, length(ZH) and CH, the existence of a subcurveY′ Y and a scheme with Y′∩H=ZH and S∩H=CH, for general H. The main cases we study are: CH rational normal curve and CH linear space. We prove also a lifting theorem for the property arithmetically Gorenstein, which generalizes a lifting theorem by Strano and Huneke–Ulrich.