Hilbert depth of graded modules over polynomial rings in two variables
  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. 
Julio Jos\'e Moyano-Fern\'andez and Jan Uliczka
math.AC math.CO