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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)
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