Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 44 (2003), No. 1, 285-302.

Partial Intersections and Graded Betti Numbers Alfio Ragusa Giuseppe Zappalà Dipartimento di Matematica, Università di Catania, Viale A. Doria 6, 95125 Catania, Italy e-mail: ragusa@dmi.unict.it e-mail: zappalag@dmi.unict.it Abstract. It is well known that for 2-codimensional aCM subschemes of ℙ^{r} with fixed Hilbert function H there are all the possible graded Betti numbers between uitable bounds depending on H. For aCM subschemes of codimension c ≥ 3 with ilbert function H it is just known that there are upper bounds for the graded etti numbers depending on H and these can be reached; but what are the graded etti numbers which can be realized is not yet completely understood. The aim f the paper is to construct c-codimensional subschemes of ℙ^{r} which could recover s many graded Betti numbers as possible generalizing both the 2-codimensional ase and the maximal case. SC 2000: 13D40, 13H10 Introduction In the last few years a large number of researchers in algebraic geometry made deep investigations about Hilbert functions and graded Betti numbers of projective schemes. In particular, to try to find all the possible Hilbert functions and all the possible graded Betti numbers of c-codimensional (reduced) schemes of ℙ^{r} is surely one of the most important questions but not so easy to solve. On the other hand, the problem of finding all the possible graded Betti numbers for schemes with an assigned Hilbert function seems just as difficult. Many results are available on this field but they deal always with particular situations as for the aCM schemes of codimension 2 or, more recently, for the arithmetically Gorenstein schemes of codimension 3 (see, for instance, papers by Gaeta [7], Buchsbaum-Eisenbud [1], Stanley [17], Campanella [3], Maggioni-Ragusa [14], Bigatti [2], Hulett [12], De Negri-Valla [5], Diesel [6], Geramita-Migliore [11], Ragusa-Zappalà [16], and many others).

0138-4821/93 $ 2.50 \begin{matrix}c \\ ⊗ \end{matrix} 2003 Heldermann Verlag