[This was first posted on MSE but did not get any answer. I apologize if it is not suited for MO.]


Let $C$ be a curve, i.e. a purely one-dimensional scheme, embedded in a smooth projective threefold $X$. For a coherent sheaf $E$ of codimension $c$ on $X$, let $E^D=\mathscr Ext\_X^c(E,\omega\_X)$ be the Grothendieck dual of $E$. It is a reflexive sheaf of codimension $c$ on $X$. Reflexive means that the natural map to the double dual is an isomorphism. Let $\mathscr O\_C$ be the structure sheaf of $C$, viewed as a torsion sheaf on $X$.



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> **Question**. For which curves $C\subset X$ is $\mathscr O\_C$ reflexive?
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Certainly, when $C$ is smooth. Let us assume $C$ is singular. I was thinking that maybe if $C$ is a *local complete intersection* then the first dual $\mathscr O\_C^D$ could already be isomorphic to $\mathscr O\_C$. Is this true? If so, then in particular dualizing twice does not do anything. But probably someone knows a more convincing (or even more true) statement. Thank you for any help!



Which curves have reflexive structure sheaf?
Indeed, this condition is equivalent to $C$ being CM (Regardless of $X$ as long as it is also CM). This "dual" is called the $\omega$-dual in [Kol13](https://goo.gl/1NwSOJ). You might be interested in reading section 2.5, or just the part directly dealing with this on pp.80-83. In particular, the statement you need is Cor. 2.70. Furthermore, if $X$ is not necessarily CM, then you get the same result using the dualizing complex instead of the dualizing sheaf.


As *Mohan* points out, the first dual will only be isomorphic to $\mathscr O$ if its own dualizing sheaf is trivial. Then again, why would you want that? That's a freak accident and doesn't give you too much mileage. Reflexivity is a (or "the") natural condition.


