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 $Hom(\\_, \omega\_X)$?



Dual family of torsion-free rank-1 sheaves on Gorenstein curves
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**](http://dx.doi.org/10.1215/kjm/1250520873), J. Math. Kyoto Univ. 26, 375-386 (1986). [ZBL0613.14008](https://zbmath.org/?q=an:0613.14008).


*Altman, Allen B.; Kleiman, Steven L.*, [**Compactifying the Picard scheme**](http://dx.doi.org/10.1016/0001-8708(80)90043-2), Adv. Math. 35, 50-112 (1980). [ZBL0427.14015](https://zbmath.org/?q=an:0427.14015).


