deformations of vector bundles on curves

Question

Let $X$ be a smooth algebraic curve. Suppose I have a flat family $V_y\to X$ of vector bundles on $X$ over an affine scheme $S$. Let $p=Spec(k)$ be one geometric point of $S$. If the determinant of $V_y$ is trivial for all $y\in S-p$, does $V_p$ have trivial determinant? If not, under what hypotheses would it have?

Answer 1

Yes if $X$ is proper. Note that $\det(V)$ is itself a line bundle on $X\times S$, so the question is: given a line bundle $L$ on $X\times S$, with $X$ proper, is the locus of $s\in S$ such that $L_s$ is trivial a closed subset of $S$? In fact, there is a natural maximal closed subscheme of $S$ over which $L$ is fiberwise trivial. This is beautifully explained in Chapter 2 of Mumford's "Abelian Varieties" book.

For non-proper $X$, I think this should still be true, but I don't see how to show it.