In Example of a $ \mathbb{Q} $-factorial, CM normal, projective, Mori dream space $ Z $ such that $ \operatorname{Cox}(Z) $ is integral and not CM I asked for an example of a Cohen Macaulay, normal, projective, Mori dream space $ Z $ such that $ \operatorname{Cox}(Z) $ is integral and not Cohen Macaulay.
Jason Starr pointed out that if $ Z $ is a very general Abelian variety of dimension $ g>1 $ should have a cyclic class group with ample generator $ \mathcal{L} $ so that $ \operatorname{Cox}(Z) $ should be graded and isomorphic to the section ring $ \oplus_{n \in \mathbb{N}} H^{0}(Z, \mathcal{L}^{\otimes n}) $. Then the result that such Abelian varieties are not Arithmetically Cohen Macaulay should be an example. Does someone have a reference that a) very general Abelian varieties of dimension $ g>1 $ have a cyclic class group with ample generator and b) that such varieties exist.
