On the General Hyperplane Section of a Projective Curve

Here we study cohomological and geometrical properties of the general zero-dimensional section of a nonreduced curve C P n and (in positive characteristic) of an integral variety. 0. Introduction The main actors in this paper will be zero-dimensional subschemes of a projective space P r. This paper consists of three parts. The link between the three parts is given by the common main actors. The content of the last part (section 6) is used in the second part for the proof of Theorems 5.1 and 5.4. In the rst part of this paper (i.e. in the rst three sections) we study the general hyperplane section, X, of a nonreduced curve Y P r+1. This topic was considered in 9]. An interesting question is whether X is in linearly general position (see e.g. 9] and 4] for terminology). The motivation came from Castelnuovo's theory (see 11, Ch. III]) which gives an upper bound on p a (Y) (see 9, Th. 3.2]). If Y is a double structure on Y red , the linear generality of X was proved in 9, Th. 3.1], in characteristic 0 and studied in positive characteristic in 4]. When Y has high multiplicity at a generic point of Y red , the linearly general position of X may fail just because it fails even for the connected components of X. Hence it seems very useful to have a more general setup. This is done in sections 1 and 2 (which are the main motivation for this paper). As a by-product, under suitable assumptions we get several results on the postulation of X and hence upper bounds for p a (Y). Indeed there is a machine which takes as input assumptions on Y and gives as output results on the postulation of X. We choose to write explicitely just a few statements, whose proofs show how to use the machine. In these two sections we stress the general setup and the results 1.3, 2.4, 2.5 and 2.6. In the third section for a given zero dimensional scheme X P r we construct a nice curve Y P r+1 with X as hyperplane section.