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Let $A=\mathbb{C}[u,x,y,w]/(uy-x^2,xw-y^2,uw-xy)$, $X=Spec A$. $A$ is a Veronese subring and from the answer of Is there a simple method to test a local ring to be Cohen Macaulay?, we can see that $X$ is CM but not Gorenstein. Then the canonical sheaf comes from a reflexive module $M$ of rank 1 over $A$ which is not locally free.

Question. How to compute $M$ concretely (say as a quotient of sum of $A$)?

I know that if $U\subset A$ is the regular locus, i.e. $A$ cut off the unique singular point, then $\tilde{M}$ is the push forward of the canonical bundle of $U$. However I could not gain much from this description.

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  • $\begingroup$ This is described in Section 21.3 of Eisenbud's "Commutative Algebra with a View towards Algebraic Geometry." $\endgroup$ – Jason Starr Feb 2 '17 at 2:00
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For a toric variety, such as this cone over the 3rd Veronese of $\mathbb P^1$, the complement of the open torus orbit is an anticanonical divisor (indeed, that is the one given by the unique $T$-invariant section of the anticanonical bundle). Correspondingly, the canonical module is the submodule of $\mathcal O$ given by functions vanishing off that orbit. In coordinates (given by $T$-weight sections), it has a basis given by the lattice points in the interior of the moment polytope.

In the case at hand, this polytope is a sector $S \subset \mathbb R^2$, the cone over the interval with lattice points $\vec u,\vec x,\vec y,\vec w$ (why'd you use that weird order???), and the lattice points in the interior lie inside $S+\vec x, S+\vec y$. So $M = \langle x,y\rangle \leq A$, or $M \cong (A\oplus A)/\langle (y,-x)\rangle$.

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  • $\begingroup$ Thank you for the answer! I changed the order of variables.... $\endgroup$ – Xuqiang QIN Feb 2 '17 at 19:46

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