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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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References:
[1] DOI: 10.1112/jlms/s2-19.2.245 · Zbl 0393.14007
[2] DOI: 10.1007/BF01388741 · Zbl 0596.14017
[3] Hartshorne R., Graduate Texts in Mathematics 52 (1977)
[4] DOI: 10.1007/BF01168835 · Zbl 0576.14017
[5] Hulek K., Ann. Scu. Norm. Sup. Pisa pp 427– (1986)
[6] Kleiman S.L., Publ. Math. IHES 36 pp 281– (1969)
[7] DOI: 10.1090/S0002-9947-1967-0209339-1
[8] DOI: 10.1007/BF01168590 · Zbl 0534.14023
[9] DOI: 10.1007/BF01161734 · Zbl 0524.14018
[10] DOI: 10.1007/BF01458322 · Zbl 0595.14005
[11] Spindler H., Rational normal curves and the Geometry of special instanton bundles on P2n+2 · Zbl 0639.14007
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