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Multiplicity-2 structures on Castelnuovo surfaces. (English) Zbl 0625.14021
Let Y in \({\mathbb{P}}_ 4({\mathbb{C}})\) be a smooth irreducible surface and let Y’ be a non reduced scheme supported on Y and such that Y’ is locally a complete intersection and has multiplicity two, i.e. for every point \(P\in Y\) and a general plane E through P, the multiplicity of the intersection of E and Y’ at P is 2. Y’ is also called multiplicity-two structure on Y, and the authors are interested in the case in which Y’ is a global complete intersection. In particular they deal with the case in which Y is a so-called Castelnuovo surface, namely a surface of maximal geometric genus with respect to the degree. All such surfaces lie on a quadric and are either complete intersection, hence of even degree, or are residual to a plane in a complete intersection, hence of odd degree, and lie on a quadric cone. The main result of this paper is that on a Castelnuovo surface of odd degree there is a multiplicity-two structure Y’ such that Y’ is a complete intersection if and only if the surface is contained in a quadric cone of rank 3.
Reviewer: C.Ciliberto

MSC:
14J25 Special surfaces
14M10 Complete intersections
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