Arithmetically Cohen-Macaulay varieties What do we mean by a variety being arithmetically Cohen-Macaulay? Is every such variety also Gorenstein?
 A: Arithmetically Cohen-Macaulay means, depending on the source/context, either:


*

*The homogeneous coordinate ring (with respect to a given embedding into $\mathbb{P}^n$) is Cohen-Macaulay.  This seems to be more common.

*The section ring (with respect to a given ample line bundle) of the variety is Cohen-Macaulay.  


Of course if you are projectively normal in $\mathbb{P}^n$ and the ample line bundle is a very ample line bundle of that embedding, these two definitions coincide.
It doesn't imply anything about Gorenstein-ness.   In fact, any Cohen-Macaulay projective variety with $H^i(X, \mathcal{O}_X) = 0$ for $0 < i < \dim X$ is arithmetically Cohen-Macaulay with respect to some embedding into projective space.  
To see this, take a sufficiently ample line bundle $L$ such that $H^i(X, \omega_X \otimes L^n) = 0$ and $H^i(X, L^n) = 0$ for all $n \geq 1$ and all $0 < i < \dim X$.  In the previous version of this answer, I forgot the Cohen-Macaulay hypothesis on $X$, in which case the first vanishing can't be forced to hold.
If I recall correctly, these notions appear prominently in the study of Linkage (see Eisenbud's book for an introduction).  
A related notion is that of arithmetic Macaulayfication of a ring.  This means that there exists an ideal $I$ such that the Rees algebra of $I$ (the ring you blow-up to get the blow-up of $I$) is Cohen-Macaulay.  These were shown to exist in the last decade by Kawasaki.  If I recall correctly, a corollary of this result is that every ring with a dualizing complex is a quotient of a Gorenstein ring (this was previously a conjecture of Sharp).  Someone correct me if I'm wrong on this.  
EDIT:  Added the CM hypothesis on the variety and added an explanation (thanks to Long).
EDIT2:  Added the two possible definitions (section ring vs coordinate ring).  Thanks to J. C. Ottem.
