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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.