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The fact that this gives you an example that you want is non-trivial. It is log canonical by inversion of adjunction see the main result of Kawakita's paper and it is non-CM at every point of the singular set by Corollary 1.3 of this paper.

The fact that this gives you an example that you want is non-trivial. It is log canonical by inversion of adjunction see the main result of Kawakita's paper and it is non-CM at every point of the singular set by Corollary 1.3 of this paper.

First, let $X$ be an example as required, of dimension $d=\dim X$ and of codimension $c$. Let $s=\mathrm{Sing} X$ the dimension of the singular set of $X$ and assume that $\mathrm{Sing} X$ is irreducible. Take a general complete intersection of codimension $s$. This will have dimension $d-s$, codimension $c$ and it is an isolated non-CM log canonical singularity.

Next, let $X$ be a codimension $c$ and it is an isolated non-CM log canonical singularity and consider a $\mathbb Q$-Gorenstein deformation of $X$ over a base of dimension $s$. (A $\mathbb Q$-Gorenstein deformation means that the relative dualizing sheaf of the family is a $\mathbb Q$-line bundle and its line bundle powers restrict to the appropriate power of the dualizing sheaf of the members of the family). The total space of the deformation will be an example of the kind you want. (For an indecomposable example you need a deformation without a trivial component).

First, let $X$ be an example as required, of dimension $d=\dim X$ and of codimension $c$. Let $s=\dim\mathrm{Sing} X$ the dimension of the singular set of $X$ and assume that $\mathrm{Sing} X$ is irreducible. Take a general complete intersection of codimension $s$. This will have dimension $d-s$, codimension $c$ and it is an isolated non-CM log canonical singularity.

Next, let $X$ be a codimension $c$ isolated non-CM log canonical singularity and consider a $\mathbb Q$-Gorenstein deformation of $X$ over a base of dimension $s$. (A $\mathbb Q$-Gorenstein deformation means that the relative dualizing sheaf of the family is a $\mathbb Q$-line bundle and its line bundle powers restrict to the appropriate power of the dualizing sheaf of the members of the family). The total space of the deformation will be an example of the kind you want. (For an indecomposable example you need a deformation without a trivial component).

As ulrich points out, there exist abelian surfaces in $\mathbb P^4$, so those give you $c=2$ with $3$-dimensional singularities, that is, $n=5=2c+1$. For $c=2$ this is the smallest $n$ you can get. Here is why: For $n\leq 4$ anything of codimension $c=2$ would be of dimension at most $2$ and hence if it is normal, it is $S_2$ and in particular CM.

The problem you run into is that you need non-trivial deformations of these singularities and they have finite dimensional versal deformation spaces, so you can't get too far with this idea. (Angelo will correct me if this is wrong, since he is one of the ultimate experts on this. See also Artin's extended work on this topic.)

This suggests that given your restriction of being indecomposable, regarding your "every big enough $n$" question, it seems that in order for that to happen you really need low codimensional isolated examples, which will be hard to construct since you can't get too far with cones (cf. ulrich's and Angelo's comments).

As ulrich points out, there exist abelian surfaces in $\mathbb P^4$, so those give you $c=2$ with $3$-dimensional singularities, that is, $n=5=2c+1$. For $c=2$ this is the smallest $n$ you can get. Here is why: For $n\leq 4$ anything of codimension $c=2$ would be of dimension at most $2$ and hence if it is normal, it is $S_2$ and in particular CM.

The problem you run into is that you need non-trivial deformations of these singularities and they have finite dimensional versal deformation spaces, so you can't get too far with this idea. (Angelo will correct me if this is wrong, since he is one of the ultimate experts on this. See also Artin's extended work on this topic.)

This suggests that given your restriction of being indecomposable, regarding your "every big enough $n$" question, it seems that in order for that to happen you really need low codimensional isolated examples, which will be hard to construct since you can't get too far with cones (cf. ulrich's and Angelo's comments).