Hilbert schemes of points and Fulton-MacPherson compactifications
Denis Nesterov
TL;DR
The paper develops a stability-based interpolation between Hilbert schemes of points and Fulton–MacPherson compactifications via ε-weighted degenerations, enabling universal wall-crossing formulas for tautological invariants in dimensions up to three. Central to the approach is the I-function, a universal object encoding how tautological data on Hilb_n(X) relate to those on FM spaces, and its role in expressing Hilbert-scheme integrals in terms of FM data and the universal $ ext{Hilb}_n(oldsymbol C^d)$ invariants. By constructing a master space and invoking Zhou’s entanglement technique, the authors derive explicit wall-crossing formulas across walls at $oldsymbol extepsilon_0=1/n_0$, as well as generalized results for $n$-fold products and δ-weighted stability, with concrete computations for Euler characteristics in dimensions 1–3. The framework yields universality statements for tautological integrals, connections to classical generating functions (Macdonald, Göttsche, MNOP), and effective methods to compute higher-dimensional invariants via reductions to affine models and fixed loci. Overall, the work provides a robust bridge between singular Hilbert schemes and smooth FM configurations, with broad implications for enumerative geometry and related moduli problems.
Abstract
We relate Hilbert schemes of points and Fulton-MacPherson compactifications by an interpolating stability condition. We then derive wall-crossings formulas and some applications for the enumerative geometry of Hilbert schemes.