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Smooth rational curves on Enriques surfaces. (English) Zbl 0545.14024

Let F be an Enriques surface over an algebraically closed field of characteristic \(p\geq 0\) embedded into a projective space as a surface of degree d. The main result of this paper asserts that the set of smooth rational curves on F is either empty or contains a curve of degree \(\leq d\). One of immediate corollaries of this result is the fact that polarized Enriques surfaces without smooth rational curves on them are parametrized by a constructible subset of the Hilbert scheme. By a finer argument we prove that this subset is open dense, if \(p\neq 2\). - The main technical tool of this paper is the arithmetic of the Picard lattice of F, an integral even unimodular quadratic form of rank 10 and signature (1,9).

MSC:

14H45 Special algebraic curves and curves of low genus
14J25 Special surfaces
14M20 Rational and unirational varieties
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