I have the following question about certain schemes being Cohen-Macaulay.

Let $X$ be the union of all coordinate $k$-planes in ${\mathbb A}^N$. Is it CM?

Let $R$ be a collection of $k$-element subsets of $\{1,\ldots,N\}$ such that for each $J\in R$ there exists $J'\in R$ which differs from $J$ at just ONE element. Such $R$ will be called an admissible collection. Let us denote the union of corresponding k-planes by $A(R)$. Is it $CM$?

The following statement appears in Eisenbud's book: if $X$ and $Y$ are equidimensional CM subschemes, and $X\cap Y$ has codimension 1, and is CM, then $X\cup Y$ is CM. In case X has a few irreducible components, and $Y$ intersects some in higher codimension (but such intersections are embedded into other intersections of codimension 1, so that the total intersection in a sense does have codimension 1), does the statement still hold? How to prove this statement?

Is the intersection of an admissible $A(R)$ with a k-plane $A_J$ corresponding to a subset $J$ reduced?

If all the answers are positive, one can deduce 2 by induction from 3 and 4.