Smooth rational curves on Enriques surfaces
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).
smooth rational curves on Enriques surface
scientific article; zbMATH DE number 3867499