Tautological integrals on curvilinear Hilbert schemes

TL;DR

This paper introduces a new algebraic approach to compute tautological integrals on curvilinear Hilbert schemes of points on smooth projective varieties, using iterated residue formulas derived from singularity theory.

Contribution

It develops an algebraic model for the quotient of holomorphic map germs and derives an iterated residue formula for integrals on curvilinear Hilbert schemes, advancing the computational tools in this area.

Findings

Derived an explicit iterated residue formula for tautological integrals.

Provided a new algebraic framework connecting singularity theory and Hilbert schemes.

Enhanced computational methods for geometric invariants of Hilbert schemes.

Abstract

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.