Let $A$ be a finitely generated ${\mathbb Z}_{\geq 0}$-graded algebra over a field without zero divisors; assume that all graded components are finite-dimensional and that $Spec(A)$ is smooth outside of a subscheme of codimension 2. Are there conditions on the corresponding Hilbert series which imply that $Spec(A)$ is normal or that it has rational singularities? In particular, if two such algebras have the same Hilbert series and one of them is normal or has rational singularities, will the same be true for the other?

In fact, I would like to show that $A$ is Gorenstein; in this case the Hilbert series $H(t)$ satisfies $$ H(t)=(-1)^d t^q H(t^{-1}) $$ where $d$ is the dimension and $q$ is an integer. It is known that if $A$ is Cohen-Macaulay, then the above identity implies that $A$ is Gorenstein. I would like to know if the Cohen-Macaulayness assumptions can be dropped or weakened.