geometry on nodal curves

Abstract

given a family x/b of nodal curves we construct canonically and compatibly with base-change, via an explicit blow-up of the cartesian product x^r/b, a family w^r(x/b) that we show is isomorphic to the relative flag hilbert scheme parametrizing flags of subschemes of fibres of x/b with colengths $1,\dotsc,r$. although w^r(x/b) is singular, the important sheaves on it are locally free, which allows us to study some intersection theory on it and deduce enumerative applications, including some relative multiple point formulae enumerating the length-r schemes contained simultaneously in some fibre of x/b and some fibre of a given map from x to a smooth variety.