  In this paper we determine the facets of the polyhedral cone generated by the
exponent set of the monomials defining the base ring associated to some
transversal polymatroid. We need the description of these facets to find the
canonical module of the base ring which is expressed in terms of the relative
interior of the cone. This would allow us to compute the $a$-invariant of those
base rings. The results presented were discovered by extensive computer algebra
experiments performed with {\it{Normaliz}} \cite{BK}.

Commutative Algebra;Combinatorics