On the Picard group of Enriques surfaces
L. Illusie has shown that the group of divisors of any Enriques surface X modulo numerical equivalence is isomorphic to the unique even unimodular lattice of index of inertia (1,9). The purpose of this paper is to use this isomorphism to study the Picard group of X. We show the existence of certain configurations of irreducible curves of arithmetic genus 0 or 1 and deduce the existence of certain projective models of X. In particular, we prove that every Enriques surface admits a morphism of degree one onto a surface of degree 10 in \({\mathbb{P}}^ 5\) whose singular points are isolated rational double points.- We also give some characterizations of the nodal Enriques surfaces which, by definition, contain some smooth rational curve. For example, it is shown that X is nodal if and only if there exist an elliptic pencil \(| P|\) and a smooth rational curve R such that \(P.R=2.\)
scientific article; zbMATH DE number 3859302
scientific article