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Double structures on Bordiga surfaces. (English) Zbl 0706.14025
A Bordiga surface S is a projective plane birationally embedded into $$P^ 4$$ by quartics through given 10 points, so that its degree equals 6. The present paper studies when such surface is a set-theoretical complete intersection of a cubic and a quartic. This relates to the existence of a double structure on S, and the latter may relate to a vector bundle of rank 2 on $$P^ 4$$ (which splits in the present case). The author proves that such a double structure exists if and only if S contains a certain curve C of degree 4 and is contained in the secant variety of C. Here C is either a normal rational curve or a union of two conics which intersect at one point. The 10 points on $$P^ 2$$ to be blown up are also determined in these cases.
Reviewer: E.Horikawa

##### MSC:
 14J26 Rational and ruled surfaces 14M10 Complete intersections
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