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Generally Speaking, Cohen-Macaulay condition is an open condition in a moduli. So in general, even if a special fiber is not Cohen Macaulay, we should not expect that generic fiber is not Cohen-Macaulay.

But here is a special case: if the Groebner degeneration of an irreducible subscheme of a product of projective spaces is not Cohen-Macaulay, under what conditions can we say that the original irreducible subscheme of a product of projective spaces is also not Cohen-Macaulay?

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2 Answers 2

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I will answer in the contrapositive. Let $X \subseteq \prod_i {\mathbb P}^{n_i}$ be irreducible of codimension $k$. If whenever $\sum k_i = k$, you can find subspaces $\prod_i {\mathbb P}^{n_i}$ that intersect $X$ in at most one point, then $X$ is called a "multiplicity-free subvariety".

In this case Brion has proven that any degeneration of $X$ must still be Cohen-Macaulay: http://arxiv.org/abs/math/0211028 and in particular, $X$ itself must be!

The case of the diagonal was studied by Cartwright and Sturmfels: http://arxiv.org/abs/0901.0212

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  • $\begingroup$ One small question, just want to makes sure if I understand multiplicity-free right: whenever $\Sigma k_i=k$, one can find $\Pi_i P^{k_i}$ that intersect $X$ at a single point. Does it have to be a single point? If the intersection with $X$ is empty or a single point, is it still a multiplicity-free subvariety? $\endgroup$
    – BLI
    Apr 6, 2013 at 23:01
  • $\begingroup$ Oops, of course you're right, I've edited to reflect that. $\endgroup$ Apr 7, 2013 at 2:50
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Just to add one comment (which may well follow from Allen's answer): if the degeneration is regular in codimension 1 yet has a point of codimension > 1 (i.e., the generic point of an irreducible closed subset of codimension > 1) at which it is both locally reduced and locally disconnected (i.e., removing the irreducible closed subset locally disconnects the degeneration), then the generic fiber should not be Cohen-Macaulay.

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  • $\begingroup$ That is exactly the case I am running into. Thanks for this helpful comment! $\endgroup$
    – BLI
    Apr 3, 2013 at 21:54

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