A note on the normal generation of ample line bundles on an abelian surface

Abstract:

Let $L$ be an ample line bundle on an abelian surface $A$. We prove that the four conditions: (1) $L$ is base point free, (2) $L$ is fixed component free, (3) ${L^{ \otimes 2}}$ is very ample, (4) ${L^{ \otimes 2}}$ is normally generated, are equivalent if $({L^2}) > 4$. Moreover we prove that ${L^{ \otimes 2}}$ is not normally generated if $({L^2}) = 4$.