  In this article we mainly consider the positively Z-graded polynomial ring
R=F[X,Y] over an arbitrary field F and Hilbert series of finitely generated
graded R-modules. The central result is an arithmetic criterion for such a
series to be the Hilbert series of some R-module of positive depth. In the
generic case, that is, the degrees of X and Y being coprime, this criterion can
be formulated in terms of the numerical semigroup generated by those degrees.
