# zbMATH — the first resource for mathematics

Partial intersections and graded Betti numbers. (English) Zbl 1033.13004
For arithmetically Cohen-Macaulay (aCM) subschemes of $$\mathbb{P}^r$$ of codimension $$c= 2$$, the possible sets of graded Betti numbers have been completely classified [ see G. Campanella, J. Algebra 101, 47–60 (1986; Zbl 0609.13001) and R. Maggioni and A. Ragusa, Matematiche 42, 195–209 (1987; Zbl 0701.14030)]. For aCM schemes of codimension $$c\geq 3$$ with fixed Hilbert function, there is still a maximum for the graded Betti numbers, but not necessarily a minimum.
The authors develop a construction which allows them to obtain a large part of the possible sets of graded Betti numbers of aCM schemes with fixed Hilbert function. Their machinery is based on the concept of partial intersection subschemes of $$\mathbb{P}^r$$. Those schemes are reduced, aCM, and unions of linear varieties similar to those used by J. Migliore and U. Nagel [ Commun. Algebra 28, No. 12, 5679–5701 (2000; Zbl 1003.13005)] and more general than the $$k$$-configurations used by A. V. Geramita, T. Harima and Y. S. Shin [ Adv. Math. 152, No. 1, 78–119 (2000; Zbl 0965.13011)]. In codimension $$c=3$$, the authors succeed in computing all graded Betti numbers in terms of certain combinatorial data used to construct the partial intersection scheme. In general codimensions $$c\geq 3$$, they determine the Hilbert function, the degrees of the minimal generators of the vanishing ideal, and the degrees of the last syzygies in terms of those data.

##### MSC:
 13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series 13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) 14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
Full Text: