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We know that there are examples of Cox rings of Mori Dream Spaces that are not Cohen-Macaulay, for instance, the Cox ring of a very general algebraic hyper-Kähler 4-fold. But, this Cox ring seems to be Du Bois (please, correct me if this is wrong).

i) Do we know examples of Mori Dream Spaces that are not Du Bois? (I expect the answer to be yes, but I don't have an example).

ii) Do we know examples of Cox rings of Mori Dream Spaces that are not Du Bois?

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  • $\begingroup$ Can you give a reference for the statement about the Cox ring of a very general HK 4-fold? That is surprising to me. $\endgroup$ – potentially dense Oct 3 '16 at 10:50
  • $\begingroup$ A very general HK 4-fold $X$ has Picard rank 1 and $H^2(X,\mathcal{O}_X)\neq 0$, then the second local cohomology at the fixed point of the Cox ring is non-zero, therefore is not CM. $\endgroup$ – Joaquín Moraga Oct 3 '16 at 22:15

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