We develop a new approach to the study of the multipoint loci of holomorphic maps between complex manifolds. We relate the k-fold locus to the curvilinear component of the Hilbert scheme of k points on the source space of the map, and using equivariant localisation, we derive a closed iterated multipoint residue formula. Our work is motivated by ideas and conjectures of M. Kazarian and R. Rimányi on residual polynomials.
