Let $X$ be a 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?

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?