Suppose given a flat, projective morphism of finite type noetherian $\mathbb{C}$-schemes $X \rightarrow T$ and a coherent sheaf $M$ on $X$. Define a contravariant functor $F:Sch/T \rightarrow Grp$ taking each $Y \rightarrow T$ to the set of sections of $M|_{X_Y}$. Is $F$ representable?
If $M$ is locally free, then the answer is yes, by Lemma 3.1.3 of
http://arxiv.org/pdf/1104.4828.pdf .
Unfortunately in my case, $M$ is not locally free, although $M|_{X_t}$ is locally free for all $t$ in $T$.
(To be specific, my case is $T$ a smooth affine curve, $X = \mathbb{P}^1_T$, and $M$ is a sheaf that is trivial of rank 2 except in one fiber).