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Complete intersections in \({\mathbb{P}}^ 2:\) Cayley-Bacharach characterizations. (English) Zbl 0556.14025
Complete intersections, Lect. 1st Sess. C.I.M.E., Acireale/Italy 1983, Lect. Notes Math. 1092, 253-269 (1984).
In this article, the authors prove a ”good” solution to the Euler-Cramer problem (ECP). The problem is to find ”good” conditions on a set of points Z in \({\mathbb{P}}^ 2\), which will ensure that Z is a (scheme- theoretic) complete intersection. The condition is in terms of the Cayley-Bacharach property. Let Z be a finite set of points in \({\mathbb{P}}^ 2\). Let \(\alpha =\alpha (Z)\) be the smallest degree of curves containing Z, and \(\delta=\delta(Z)=\) degree of Z=cardinality of Z. Then Z is a complete intersection if and only if \(\alpha^{-1}\delta \in {\mathbb{Z}}\) and Z has \(CB(\alpha +\alpha^{-1}\delta -3).\) The proofs are elementary and in spite of the technical appearance quite lucid. The main tool is the Hilbert function. Less satisfactory results are proved in the case when Z is not reduced.
Reviewer: N.Mohan Kumar

14M10 Complete intersections
14N05 Projective techniques in algebraic geometry