Normality and rational singularities via Hilbert series 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. 
 A: The general reason why Sam's examples work is that for a flat projective family $f:X\to B$ the Hilbert polynomial of the fibers is constant. So anything that can be read off from the Hilbert polynomial has to be invariant under flat deformation. In particular neither being normal or having rational singularities, or more generally any smoothable singularity have a chance for that.
Edit:
1) Indeed the question was about $\mathrm{Spec}$ and not $\mathrm{Proj}$. Taking the cone over the corresponding projective variety gives you the $\mathrm{Spec}$ and the question becomes slightly different. But for instance a smooth projective family degenerating to a singular fiber gives you an example of having normal and non-normal cones of the fibers in the same family.
2)
To respond to the additional question raised in the remarks regarding Gorenstein versus CM.
First of all, $X$ is Gorenstein = $X$ is CM + $\omega_X$ is a line bundle.
Let $f:X\to B$ be a flat family of CM schemes. Then 
$$\omega_{X_b}\simeq (\omega_{X/B})|_{X_b}$$
for every $b\in B$.
This implies that any small deformation of a Gorenstein fiber remains Gorenstein. On the other hand I am not entirely sure what exactly you mean by being able to tell if something is Gorenstein. I guess the question is which Hilbert series are you looking at?
Another possibility is that I completely misunderstood the question. :)
A: Here's a counterexample: the Hilbert series of a hypersurface is determined by its degree, but hypersurfaces need not be normal. For example, take a hypersurface cut out by a hyperdeterminant of boundary format. For example the hyperdeterminant of a $3 \times 2 \times 2$ tensor (it is an irreducible polynomial, so it also satisfies the no nonzero divisor condition).
