A relative GAGA principle for families of curves

A classical result, due to Riemann, is that every Riemann surface has a non-constant meromorphic function. Equivalently, every Riemann surface is a projective algebraic variety. When considering this problem in families, the answer is more subtle. I will, however, describe a simply stated criterion that permits a family of possibly singular Riemann surfaces to be algebraized. This turns out to be an easy consequence of some analytic comparison results, or GAGA, for Deligne--Mumford stacks.