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A structure theorem for most unions of complete intersections. (English) Zbl 1387.13038

The present paper study the union of two complete intersection schemes \(X_1\) and \(X_2\) of codimension 2 in \(\mathbb P^r\) and the intersection of their defining ideals \(I_{X_1}\cap I_{X_2}\) in the polynomial ring \(R=k[x_0,\dots,x_r]\).
The authors give a structure theorem for schemes of the type \(X_1\cup X_2\) with restriction about the degrees using facts about Pfaffians and determinants of skew-symmetric matrices. They also give a free graded resolution for the ideal \(I_{X_1}\cap I_{X_2}\).
Thanks to those result, they describe all possible Hilbert function for aCM schemes which are the union of two complete intersection schemes of codimension 2 without components.
Finally they explain how the Betti numbers can be obtained by the previous resolution and give some minimal generators for \(I_{X_1}\cup I_{X_2}\).

MSC:

13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
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