Let $X \subseteq \mathbb{P}\_{\mathbb{C}}^n$ be an irreducible, projective, Cohen-Macaulay variety of dimension $k$. Let $L \subseteq \mathbb{P}\_{\mathbb{C}}^n$ be a linear space of dimension $n-k-1$ that does not intersect $X$. Then the linear projection $\pi: X \to \mathbb{P}\_{\mathbb{C}}^k$ from center $L$ is a finite morphism. Let $R=\mathbb{C}[x\_0,\ldots,x\_k]$ and let $S$ be the homogeneous coordinate ring of $X$. Then $S$ is a finitely generated graded $R$-module via $\pi$.


My question is: When is $S$ in fact a free $R$-module? In particular, I am interested in sufficient criteria and examples of when $S$ is not free.



When do we get free modules from Noether normalization
$S$ is free if and only if it is a Cohen-Macaulay ring -- see e.g. Bourbaki, Algèbre commutative X, 4, no. 3, Corollaire (I am afraid this is not yet translated in english). In geometric terms, this means that $X$ is *projectively* (aka *arithmetically*) *Cohen-Macaulay* -- that is, $H^i(X,\mathcal{O}\_X(n))=0$ for all $n$ and $0<i<\dim(X)$.


