  We take a new look at the curvilinear Hilbert scheme of points on a smooth
projective variety $X$ as a projective completion of the non-reductive quotient
of holomorphic map germs from the complex line into $X$ by polynomial
reparametrisations. Using an algebraic model of this quotient coming from
global singularity theory we develop an iterated residue formula for
tautological integrals over curvilinear Hilbert schemes.
