Return to Answer

Arithmetically Cohen-Macaulay means, depending on the source/context, either:

1. The homogeneous coordinate ring (with respect to a given embedding into $\mathbb{P}^n$) is Cohen-Macaulay. This seems to be more common.
2. The section ring (with respect to a given ample line bundle) of the variety is Cohen-Macaulay.

Of course if you are projectively normal, 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 most 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.

Arithmetically Cohen-Macaulay means, depending on the source/context, either:

1. The homogeneous coordinate ring (with respect to a given embedding into $\mathbb{P}^n$) is Cohen-Macaulay. This seems to be more common.
2. 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.

Arithmetically Cohen-Macaulay means that the section ring (with respect to a given ample line bundle) of the variety is Cohen-Macaulay. 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 most 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).

Arithmetically Cohen-Macaulay means, depending on the source/context, either:

1. The homogeneous coordinate ring (with respect to a given embedding into $\mathbb{P}^n$) is Cohen-Macaulay. This seems to be more common.
2. The section ring (with respect to a given ample line bundle) of the variety is Cohen-Macaulay.

Of course if you are projectively normal, 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 most 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.

Arithmetically Cohen-Macaulay means that the section ring (with respect to a given ample line bundle) of the variety is Cohen-Macaulay. It doesn't imply anything about Gorenstein-ness. In fact, any 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. If I recall correctly, these notions appear most 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.

Arithmetically Cohen-Macaulay means that the section ring (with respect to a given ample line bundle) of the variety is Cohen-Macaulay. 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 most 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).