The Hilbert function of generic plane sections of curves of ?3

Abstract

Summary. A characteristic condition is given on a zero-dimensional differentiable 0-sequence H={hi}i_>o, h~<3, in order to be the Hilbert function of a generic plane section of a reduced irreducible curve of ~,3, hence of points of ~,2 with the uniform position property. In this way an answer is given to some question stated by Harris in [Ha2].The result is obtained by constructing a smooth irreducible arithmetically Cohen-Macaulay curve in ~E)3 whose generic plane section has an assigned Hilbert function satisfying that condition.

Regularity bounds by minimal generators and Hilbert function

“…In these cases, there exists an irreducible smooth aCM curve whose generic plane section has H Z as Hilbert function [14]. So, if C is an integral curve satisfying Theorem 3.9,…”

A construction of codimension three arithmetically Gorenstein subschemes of projective space

“…It has been important, for instance, in the classification and construction of smooth curves in P 3 (e.g. [6], [15], [21], [27]), smooth surfaces in P 4 (e.g. [5], [12], [36], [38]) and smooth threefolds in P 5 (e.g.…”

Reduced Gorenstein codimension three subschemes of projective space

“…A precise analog of these results for the integral Gorenstein codimension three case was provided by J. Herzog, N. Trung and G. Valla [17]. From the point of view of the Hilbert function, the necessary and sufficient condition for the existence of smooth aCM curves in P 3 (or irreducible codimension two aCM schemes in P n ) was given by C. Peskine and L. Szpiro [22] and by R. Maggioni and A. Ragusa [19] (see also [13]). This has a precise analog (using Stanley's result above) for integral Gorenstein codimension three schemes, proved by E. De Negri and G. Valla [9].…”