A note on Hilbert schemes of nodal curves

We study the Hilbert scheme and punctual Hilbert scheme of a nodal curve, and the relative Hilbert scheme of a family of curves acquiring a node. The results are then extended to flag Hilbert schemes, parametrizing chains of subschemes. We find, notably, that if the total space X of a family X/B is smooth (over an algebraically closed field k), then the relative Hilbert scheme Hilbm(X/B) is smooth over k and the flag Hilbert schemes are normal and locally complete intersection, but generally singular.