Projective images of Kummer surfaces
We consider the linear systems of even resp. odd sections in the powers \({\mathcal L}^ d_ 0\), \(d \geq 2\), of an ample symmetric line bundle \({\mathcal L}_ 0\) on an abelian surface \(A\). These systems define rational maps of the smooth Kummer surface associated to \(A\). We prove that, apart from certain exceptions, these maps are birational morphisms of the smooth Kummer surface onto its image. They contract the \((-2)\)-curves associated to the even resp. odd halfperiods of \({\mathcal L}_ 0\). We classify the exceptional cases, i.e. the cases where (1) the linear system has a base component; or (2) where the morphism is of degree two; or (3) where it contracts additional curves.
scientific article; zbMATH DE number 590007
scientific article