Does anyone know an example of a $ \mathbb{Q} $-factorial, normal, Cohen Macaulay, projective, Mori dream space $ Z $ over a field $ k $ of arbitrary characteristic such that the Cox ring of $ Z $ is an integral domain and not Cohen Macaulay?
$\begingroup$
$\endgroup$
4
-
$\begingroup$ I believe that is true of every very general Abelian variety $Z$ of dimension $\geq 2$ (so that the Picard group is cyclic). $\endgroup$– Jason StarrCommented Nov 18, 2023 at 12:10
-
$\begingroup$ Forgive me as I don't know much about Abelian varieties. Do you have a proof that every very general Abelian variety of dimension $ \ge 2$ satisfies these requirements? I am very curious. $\endgroup$– Schemer1Commented Nov 20, 2023 at 1:05
-
1$\begingroup$ This is well-known, perhaps first proved by Bruns and Herzog (perhaps it was known earlier). One reference is Proposition 3.2 of the following article of Axel Staebler: arxiv.org/pdf/1104.5365.pdf $\endgroup$– Jason StarrCommented Nov 21, 2023 at 21:10
-
1$\begingroup$ The picard number might be 1, but surely the Picard group is not cyclic? $\endgroup$– Ennio Mori coneCommented Nov 22, 2023 at 14:16
Add a comment
|