Given a quasismooth toric variety $X$ in the sense of GelfandKapranovZelevinsky, i.e. a (not necessarily normal) toric variety $X$ whose normalization is a $\mathbb{Q}$factorial toric variety and such that the normalization morphism is bijective. It is known that such nonnormal toric varieties share nice properties with smooth toric varieties. What kind of singularities can we expect from $X$? Are they CohenMacaulay?

1$\begingroup$ CohenMacaulay schemes are S2, hence satisfy the S2 extension property. So, beginning with $\mathbb{P}^n$, $n\geq 2$, "pinch" it infinitesimally at one point. The resulting scheme is not CohenMacaulay, even though the normalization morphism is universally bijective on points. $\endgroup$ – Jason Starr Jun 16 '16 at 7:46

$\begingroup$ That's a good point. More generally, Serre's criterion says normal=R1+S2. Since CM implies S2 this means that $$\text{quasismooth toric}+\text{CM}+\text{smooth in codimension one}\Rightarrow\text{normal}$$ In other words, quasismooth nonnormal toric varieties are almost never CM. $\endgroup$ – Friedrich Knop Jun 17 '16 at 7:51
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Counterexample to CM: Let $M$ be the monoid $\{x^iy^j\mid i,j\ge0\}\setminus\{x,y\}$ and $X=\text{Spec}\ \mathbb C[M]$ the corresponding toric variety. Then $x^2,y^2$ is a system of parameters of $X$. But $\mathbb C[M]$ is not a free $\mathbb C[x^2,y^2]$module since the fiber $\mathbb C[M]/(x^2,y^2)=\langle 1,xy,x^3,x^2y,xy^2,y^3\rangle$ is of dimension $6$ instead of $4$.