Volumes of line bundles on schemes

The volume of a line bundle is an invariant defined in terms of a limit superior. It is a fundamental question whether this limit superior is a limit. It has been shown that this is always the case on generically reduced proper schemes over arbitrary fields. We show that volumes are limits in two classes of schemes that are not necessarily generically reduced: codimension one subschemes of projective varieties such that their components of maximal dimension contain normal points, and projective schemes whose nilradical squared equals zero.