Let $X$ be a Gorenstein curve over a field an consider the compactified Jacobian parametrizing torsion-free, rank-1 sheaves on $X$.

Is there a chance that the dual functor $Hom(\_, \mathcal O_X)$ is well defined (geometric, i.e. preserve flatness of families)?

Or maybe is it the Gorenstein-dual functor $Hom(\_, \omega_X)$?

  • 1
    $\begingroup$ Just a comment that since $X$ is Gorenstein, $\omega_X$ is a line bundle hence local Hom to $\mathcal{O}_X$ and to $\omega_X$ are essentially the same. $\endgroup$ Jun 24 at 12:07

Yes. This follows from Theorem 1.10(ii) of the paper of Altman-Kleiman cited below.

More precisely, let $S$ be a scheme and let $\mathcal{F}$ be a locally finitely presented $\mathcal{O}_{X_S}$-module on $X_S$, flat over $S$, with the property that $\mathcal{F}$ is torsion-free rank $1$ in every geometric fibre of $S$. Such a sheaf $\mathcal{F}$ determines an $S$-point of the generalized Jacobian of $X$.

Let $\mathcal{G} = \mathcal{Hom}(\mathcal{F},\mathcal{O}_{X_S})$. I interpret your question as asking whether $\mathcal{G}$ defines an $S$-point of the generalized Jacobian as well. By Lemma 1.1(a) of Hartshorne (see below), we have $\mathcal{Ext}^1(\mathcal{F}_s,\mathcal{O}_{X_s}) =0$ for every geometric point $s$ of $S$. By theorem 1.10(ii) of Altman-Kleiman (applied to $I = \mathcal{F}, F = \mathcal{O}_{X_S}$ and $p=0$), this implies that:

  • $\mathcal{G}$ is locally finitely presented and flat over $S$.
  • The formation of $\mathcal{G}$ commutes with base-change: for every $T \rightarrow S$ we have $\mathcal{G}_T \simeq \mathcal{Hom}(\mathcal{F}_T,\mathcal{O}_{X_T})$.

Therefore $\mathcal{G}_s \simeq \mathcal{Hom}(\mathcal{F}_s,\mathcal{O}_{X_s})$ for every geometric point $s$ of $S$. Since $\mathcal{Hom}(\mathcal{F}_s,\mathcal{O}_{X_s})$ is obviously torsion-free rank $1$, we see that $\mathcal{G}$ indeed defines an $S$-point of the generalized Jacobian.

Hartshorne, Robin, Generalized divisors on Gorenstein curves and a theorem of Noether, J. Math. Kyoto Univ. 26, 375-386 (1986). ZBL0613.14008.

Altman, Allen B.; Kleiman, Steven L., Compactifying the Picard scheme, Adv. Math. 35, 50-112 (1980). ZBL0427.14015.

  • $\begingroup$ Thanks! Lemma1.1(a) of Hart87 requires the curve to be integral. But we can use Proposition 1.6 in Hartshorne's 1994 paper (Generalized divisors on Gorenstein Schemes) and I think everything works as well $\endgroup$
    – Raffaele C
    Jun 23 at 10:01
  • $\begingroup$ Thanks for the correction, I guess I assumed the curve was integral because that's often assumed in references (but is not strictly necessary). If your happy with the answer, you can accept it so that this question is closed. $\endgroup$
    – Jef
    Jun 23 at 13:14
  • $\begingroup$ You're right! Btw if you are interested, I give a definition of the compactified Jacobian over a (possibly reducible, non reduced) projective curve in my PhD thesis arxiv.org/abs/2006.13034 Best! $\endgroup$
    – Raffaele C
    Jun 23 at 14:35

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