## 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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