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Quartic surfaces with 16 skew conics. (English) Zbl 0826.14020
For an integer \(d \geq 1\) let \(N_4 (d)\) be the maximal number of smooth rational curves of degree \(d\) that can lie on a smooth quartic surface in \(\mathbb{P}_3\). By classical results of Schur and Segre one has \(N_4 (1) = 64\). Up to now the numbers \(N_4 (d)\), \(d \geq 2\), seem to be unknown. – In a previous joint paper with W. Barth [cf. Manuscr. Math. 85, No. 3-4, 409-417 (1994)] the lower bound \(N_4 (2) \geq 352\) was established by embedding the smooth Kummer surface of a generic polarized abelian surface of type (1,9) into \(\mathbb{P}_3\).
In the present paper a classification of the smooth quartic surfaces in \(\mathbb{P}_3\) containing 16 skew conics is given in terms of their abelian covers. Then principally polarized abelian surfaces with endomorphism ring \(\mathbb{Z} [\sqrt 7]\) are considered. These surfaces carry a natural (1,9)-polarization and their smooth Kummer surfaces are shown to embed into \(\mathbb{P}_3\). The smooth quartic surfaces in the two-dimensional family constructed in this way contain exactly 432 smooth conics. This gives the improved lower bound \(N_4 (2) \geq 432\).
Reviewer: T.Bauer (Erlangen)

14J25 Special surfaces
14H50 Plane and space curves
14N10 Enumerative problems (combinatorial problems) in algebraic geometry
14N05 Projective techniques in algebraic geometry
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