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I had few questions regarding varieties admitting embeddings that make them arithmetically Cohen-Macaulay or Gorenstein varieties. A projective variety is called arithmatically Cohen-Macaulay/Gorenstein if its homogenous coordinate ring is so. I was wondering whether any of these properties ascend along faithfully flat finite morphisms, i.e. if $Y\rightarrow X$ is a faithfully flat finite morphism where $X$ is such a variety (with respect to some embedding), is $Y$ the same type of variety? How about descending property? Does any of these properties descend along faithfully flat finite morphisms?

Lastly I was wondering whether there are any nice examples of families of varieties that are arithmetically Gorenstein? Preferably with dimension greater than $1$.

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  • $\begingroup$ These heavily depend on the embedding. The twisted cubic curve in $P^3$ (which is isomorphic to $P^1$) is not arithmetically Gorenstein. $\endgroup$
    – Hailong Dao
    Commented Jan 28, 2021 at 16:28
  • $\begingroup$ Well in that case maybe question should be whether it has an embedding which makes it arithmetically Gorenstein. I'm mostly interested in vanishing of cohomologies of line bundles and want to see whether this gets preserved. $\endgroup$
    – user127776
    Commented Jan 28, 2021 at 16:35
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    $\begingroup$ If that's the case, you can find some helpful previous questions here: mathoverflow.net/questions/235170/… and mathoverflow.net/questions/47594/… $\endgroup$
    – Hailong Dao
    Commented Jan 28, 2021 at 17:01
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    $\begingroup$ To the person who voted to close: there is no need to do that. $\endgroup$
    – Hailong Dao
    Commented Jan 28, 2021 at 17:40
  • $\begingroup$ I think the OP should edit the question so that it makes sense. $\endgroup$
    – abx
    Commented Jan 28, 2021 at 17:45

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