Let $X$ be a complete nonsingular curve and $S$ a scheme over $k$ algebraically closed, and $\cal{F}$ a coherent sheaf on $X \times S$, generated by finitely many global sections and flat over $S$ (via the projection). So my question is the following: is the locus of points $s\in S$ such that $\cal{F}\_s$ is locally free on $X$ open? And if so, is there an easy way to see this?



Is local freeness open for curves?
To elaborate on Jason's comment: consider a morphism of schemes $f:\mathrm{X}\rightarrow\mathrm{S}$, and an $f$-flat coherent sheaf $\mathscr{F}$ on $\mathrm{X}$. Then


(1) the singular locus $\mathrm{Sing}(\mathscr{F})$ of $\mathscr{F}$ (given by the points $x\in\mathrm{X}$ such that $\mathscr{F}$ is not locally free at $x$) is closed - if $r$ is the rank of $\mathscr{F}$, it is cut out by the $r$-th Fitting ideal sheaf of $\mathscr{F}$;


(2) $\mathscr{F}$ is locally free at $x\in\mathrm{X}$ if and only if $\mathscr{F}\_{f(x)}$ is locally free at $x$, where $\mathscr{F}\_{f(x)}$ denotes the restriction of $\mathscr{F}$ to $f^{-1}(f(x))$ - this follows from flatness of $\mathscr{F}$ and the Nakayama lemma.


By (1) and (2), $f(\mathrm{Sing}(\mathscr{F}))$ consists of all $s\in\mathrm{S}$ such that $\mathscr{F}\_{s}$ is not locally free. If $f$ is proper, then of course the latter image is closed in $\mathrm{S}$.


Lemma 2.1.8 in *The Geometry of Moduli Spaces of Sheaves* by Huybrechts & Lehn is a good reference.


