Dual family of torsion-free rank-1 sheaves on Gorenstein curves 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 $Hom(\_, \omega_X)$?
 A: 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.
