An inclusion-exclusion principle for tautological sheaves on Hilbert schemes of points

TL;DR

This paper introduces an inclusion-exclusion principle for calculating Euler characteristics of tautological sheaves on Hilbert schemes of points, establishing universal polynomials and addressing a conjecture through degeneration techniques.

Contribution

It develops a new combinatorial inclusion-exclusion approach for Euler characteristics and proves the existence of universal polynomials for tautological sheaves.

Findings

Derived an equation for Euler characteristics on Hilbert schemes

Established universal polynomials for these Euler characteristics

Reduced a conjecture to cases involving products of projective spaces

Abstract

We show an equation of Euler characteristics of tautological sheaves on Hilbert schemes of points on the fibers of a double point degeneration. This equation resembles a computation of such Euler characteristics via a combinatorial inclusion-exclusion principle. As a consequence, we show the existence of universal polynomials for the Euler characteristics of tautological sheaves on the Hilbert scheme of points on smooth proper algebraic spaces. We apply this result to a conjecture of Zhou on tautological sheaves on Hilbert schemes of points, and reduce the conjecture to the cases of products of projective spaces. Our main tools are good degenerations and algebraic cobordism.