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(EDIT 07/06/11: although the question has not been settled definitely, Sándor's excellent answer and the comments by Algelo Angelo and ulrich have highlighted many potential obstructions to the constructions I wanted. Thank you all! I am still very interested in any leads, so please keep them coming if you have some more.)

I would like to know examples of log-canonical singularities of low codimension which is normal but non-Cohen-Macaulay. A silly way to do it may be just adjoining variables, so here is the precise question:

Fix a number $c$, for what $n$ can one construct an affine variety $X \subseteq \mathbb A^n_{\mathbb C}$ such that: $X$ is indecomposable and normal of codimension $c$, $X$ has at worst log-canonical singularity but $X$ is not Cohen-Macaulay? Given $c$, can we construct such $X$ for all $n$ big enough?

The case $c=1$ is easy, there are no example since hypersurfaces are Cohen-Macaulay, so let's begin with $c=2$.

Motivation/Comments: I am actually looking for $F$-pure rings (i.e., the Frobenius is a pure morphism), but conjecturally my question above is virtually the same. Karl Schwede told me one can also try to look for (projective) Calabi-Yau varieties with some non-vanishing middle cohomolgy and low codimension embedding, then take their cones. But not being a geometer, I do not know how to construct such things.

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(EDIT 07/06/11: although the question has not been settled definitely, Sándor's excellent answer and the comments by Algelo and ulrich have highlighted many potential obstructions to the constructions I wanted. Thank you all! I am still very interested in any leads, so please keep them coming if you have some more.)

I would like to know examples of log-canonical singularities of low codimension which is normal but non-Cohen-Macaulay. A silly way to do it may be just adjoining variables, so here is the precise question:

Fix a number $c$, for what $n$ can one construct an affine variety $X \subseteq \mathbb A^n_{\mathbb C}$ such that: $X$ is indecomposable and normal of codimension $c$, $X$ has at worst log-canonical singularity but $X$ is not Cohen-Macaulay? Given $c$, can we construct such $X$ for all $n$ big enough?

The case $c=1$ is easy, there are no example since hypersurfaces are Cohen-Macaulay, so let's begin with $c=2$.

Motivation/Comments: I am actually looking for $F$-pure rings (i.e., the Frobenius is a pure morphism), but conjecturally my question above is virtually the same. Karl Schwede told me one can also try to look for (projective) Calabi-Yau varieties with some non-vanishing middle cohomolgy and low codimension embedding, then take their cones. But not being a geometer, I do not know how to construct such things.

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I would like to know examples of log-canonical singularities of low codimension which is normal but non-Cohen-Macaulay. A silly way to do it may be just adjoining variables, so here is the precise question:

Given

Fix a number $c$, for what $n$ can one construct an affine variety $X \subseteq \mathbb A^n_{\mathbb C}$ such that: $X$ is indecomposable and normal of codimension $c$, $X$ has at worst log-canonical singularity but $X$ is not Cohen-Macaulay? If we fix Given $c$, can we construct such $X$ for all $n$ big enough?

The case $c=1$ is easy, there are a lot of log-canonical no example since hypersurfaces are Cohen-Macaulay, so let's begin with $c=2$.

Motivation/Comments: I am actually looking for $F$-pure rings (i.e., the Frobenius is a pure morphism), but conjecturally my question above is virtually the same. Karl Schwede told me one can also try to look for (projective) Calabi-Yau varieties with some non-vanishing middle cohomolgy and low codimension embedding, then take their cones. But not being a geometer, I do not know how to construct such things.

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