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

Xiaowen Hu

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.

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

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.

Paper Structure

This paper contains 13 sections, 37 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Good degeneration of Hilbert schemes of points revisited
  3. Good degeneration of Hilbert schemes of points on smooth projective schemes
  4. Good degeneration of Hilbert schemes of points on smooth proper algebraic spaces
  5. Flatness of the good degenerations of Hilbert spaces of points
  6. Flatness of Hilbert spaces of points
  7. Flatness of good degenerations
  8. Decomposition of the central fiber of a good degeneration
  9. Base change property of higher direct images of sheaves on stacks
  10. Algebraic cobordism of a list of bundles
  11. Tautological sheaves and the degeneration formula
  12. Moishezon's theorem for algebraic spaces
  13. An identity of Chern classes in double point degenerations

Key Result

Theorem 1.1

Let $\Bbbk$ be a field of characteristic zero. Let $X$, $Y_1$, $Y_2$ and $D$ be smooth proper algebraic spaces over $\Bbbk$, and $C$ a smooth curve over $\Bbbk$. Let $X\overset{\mathfrak{X}}{\rightsquigarrow}Y_1\cup_D Y_2$ be a double point degeneration with total space $\mathfrak{X}\rightarrow C$. as formal series of $u,v$ and $Q$. Here $E|_{\mathbb{P}_D}$ (resp. $F|_{\mathbb{P}_D}$) is the pull

Theorems & Definitions (86)

  • Theorem 1.1
  • Lemma 1.2
  • Theorem 1.3
  • Conjecture 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4: Expanded degeneration
  • ...and 76 more