On the Homogeneous Ideal of Unreduced Projective Schemes

Here we give upper bounds for the degrees of generators of the homogeneous ideal of a suitable unreduced scheme Z ⊂ P n. Here we take as Z a disjoint union of fat points or of multiple structures on linear subspaces or we take as Zred a subscheme of a smooth curve or a scroll. The actors in this paper are unreduced subschemes (or ”fattening”) of P n with connected components which are fattening of very simple building blocks: points or rational normal curves or linear spaces. For these subschemes of P n in this paper we consider the postulation (i.e. the Hilbert function), the degree of the generators of the homogeneous ideal and of higher syzygies. Most of the methods used This research was partially supported by MURST and GNSAGA of CNR (Italy). Part of the work here was done while the first named author was at Max– Planck–Institut für Mathematik (Bonn) and at SFB 170 (Göttingen). Several stimulating conversations with N. V. Trung were very useful. 132 E. Ballico and A. Cossidente are refinements of [5] (with [8] and [13] very essential here and for [5]). Section 0 contains basic notations, background material and a few conventions which we will use. Section 1 contains the general set up needed to handle the minimal free resolution of fat points with support contained in a fixed subscheme of P . In Section 2 we consider as 0-dimensional subschemes the curvilinear ones (i.e. the one with Zariski tangent space of dimension ≤ 1 at each point of their support) and some fattening of them; these unreduced schemes are the easiest ones to handle and obtain, say, bounds on their postulation. This section depends heavily on the proofs in [8] and [13]. Here we stress also the interest of fattening of higher dimensional disconnected schemes. In Section 3 we prove upper bounds for the regularity index of fattening of disjoint union of certain positive dimensional subvarieties of P , i.e. linear subspaces (see Th. 0.1 stated below), lines (see Th. 3.1) and rational normal curves (see Th. 3.2). Then in Section 4 we consider the case in which the support of the fat points is on a rational normal curve and (very briefly) the case in which it is on an elliptic linearly normal curve (see 4.9 and Prop. 4.10). In 4.11 and Prop. 4.12 we consider briefly the case of fat points with support on a rational norinal scroll (which may be singular, i.e. a cone over a lower dimensional rational normal scroll). In Section 5 we give a very general result on the behaviour of the cohomology of unreduced 0-dimensional subschemes of P n with support on a fixed curve. Just to give to the reader a feeling of the results proven in this paper, now we give the statement of the following theorem which will be proved in Section 3; in the body of the paper the reader will find more details on the notions appearing in its statement. Theorem 0.1. Fix s disjoint proper linear subspaces Ai 1 ≤ i ≤ ≤ s, of P n of any dimension, linear subspaces (of any dimension) Mi, 1 ≤ i ≤ s with Ai ⊆ Mi for every i, integers mi, 1 ≤ i ≤ s with mi > 0 and let U be the union of the (mi − 1)-th infinitesimal neighborhoods of Ai in Mi. Set m := max{mi}. Then h(P , IU (t)) = 0 for every integer t ≥ nm + m1 + · · · + ms. Furthermore, the homogeneous ideal of U is generated by forms of degree ≤ nm + m1 + . . . · · ·+ ms + 1. On the homogeneous ideal of unreduced projective schemes 133 Here are a few motivations. Unreduced schemes arise often and for several different purposes. Fat points in P n arise in the problem of interpolation for homogeneous polynomials; if the fat points are fattening in natural varieties (e.g. scrolls) this is related to the interpolation of polynomials modulo interesting ideals and (in the case of rational scrolls) to the interpolation of weighted homogeneous polynomials. Unreduced subschemes (and very often disjoint union or general disjoint union) are used (thanks to Serre correspondence) to the construction of bundles and the cohomological properties of the unreduced schemes reflects cohomological properties (and even the stability) of the corresponding bundles. Unreduced schemes arise as limits of a flat family of reduced ones and by semicontinuity cohomological properties of the unreduced schemes give cohomological conditions of the general member of the flat family; from this point of view, unreduced structures with support a curve arise often (e.g. ribbons) and their 0-dimensional subschemes are linked to the cohomolgy of their line bundles. For an example in which disjoint unions of lines arise as a technical tool (and a partial motivation for Section 3) see [7] and several related papers. 0. Notations and preliminaries We work over an algebraically closed field K ; we assume char (K) = 0 because some results (e.g. on the conormal bundle of a linearly normal elliptic curve) are not true (in the way we will state them) in positive characteristic. Let R := K[x0, . . . , xn] = ⊕