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It is well-known that $Hilb^n(X)$, the hilbert scheme of $n$ points on a smooth projective surface $X$, is isomorphic to $M_X(1,\mathcal O_X,n)$, the moduli space of rank one semistable sheaves with trivial determinant and second chern class $n$. The canonical morphism in one direction sends a subscheme $Z\subset X$ to it's ideal sheaf $\mathcal I_Z$. I was wondering how to go in the other direction. Namely, given a semistable sheaf of rank 1, trivial determinant, and with second chern number $n$, how do I get an injection into $\mathcal O_X$?

A similar result holds for example for Hilbert schemes of curves on Calabi-Yau 3-folds, so an explanation which takes into account this case as well is preferable.

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By definition semistable sheaves are torsion-free. Any torsion-free $F$ includes into its double dual, $F\to F^{\ast\ast}$. The double dual is a reflexive sheaf, so any singularities occur in codimension 3. In the surface case, we conclude $F^{\ast\ast}$ is a line bundle with trivial determinant, so must be $\mathcal O_X$.

More generally, in "Vector Bundles on Complex Projective Spaces" (Okenek-Schneider-Spindler) it is shown that any rank one reflexive sheaf on a smooth variety $X$ is necessarily a line bundle (Lemma 2.1.1.15). So even in higher dimensional cases, it follows that $F^{\ast\ast} = \mathcal O_X$.

(In fact, OSS defines the determinant of a torsion-free sheaf $F$ of rank $r$ by $$\det F = (\Lambda^r F)^{\ast\ast},$$ and notes that the determinant is always a line bundle by the cited lemma. In the rank one case, this is just the double dual, so we get a map $$F \to F^{\ast\ast} = \det F = \mathcal O_X.)$$

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  • $\begingroup$ How do you know that this works in families? If F is a flat family of sheaves, why is the double dual of F still flat? $\endgroup$
    – euklid345
    Commented Jul 23, 2022 at 21:09
  • $\begingroup$ @euklid345 You need to be a bit careful here. For example, if F is a vector bundle then F and its double dual will have the same Chern classes. But if F has a simple singularity at a point (e.g. if there is an exact sequence $0\to F\to F^{**}\to O_p \to 0$) then F and its double dual will have different Chern classes. So if your moduli space parameterizes both vector bundles and sheaves with singularities then there is no flat "double dual" family. In the case here I don't see an issue though, because the double dual of the family should just be the constant family of $\mathcal{O}_X$'s. $\endgroup$
    – Jack Huizenga
    Commented Jul 25, 2022 at 3:50

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