ac.commutative algebra - Simple example of a ring which is normal but not CM

Another family of examples is given by the homogeneous coordinate rings of
irregular surfaces (ie 2-dimensional $X$ such that $H^1({mathcal O}_X) neq 0$);
these surfaces cannot be embedded in any way so that their homogeneous coordinate rings
become Cohen-Macaulay. Elliptic scrolls (such as the one in the previous answer)
and Abelian surfaces in P4, made from the sections of the Horrocks-Mumford bundle, are such examples.

The point is that sufficiently positive, complete embeddings of any smooth variety (or somewhat more generally) will have normal homogeneous coordinate rings, and they will be Cohen-Macaulay iff the intermediate cohomology of the variety vanishes. All the examples above fall into this category. It's an interesting general question to ask how positive is "sufficiently positive".