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Motivic Hilbert zeta functions for curve singularities and related invariants

Masahiro Watari

Abstract

In the present paper, we show that the motivic Hilbert zeta function for a curve singularity yields the generating functions for Euler numbers of punctual Hilbert schemes when any punctual Hilbert scheme admits an affine cell decomposition. This fact allows us to derive the relations among the motivic Hilbert zeta function and other invariants such as the generating function for semi-modules of the semi-group of the singularity, the HOMFLY polynomial and the degrees of Severi strata of the miniversal deformation of the singularity. As an application of the fact above, we also generalize Kawai's result regarding the generating function for the Euler numbers of a singular curve.

Motivic Hilbert zeta functions for curve singularities and related invariants

Abstract

In the present paper, we show that the motivic Hilbert zeta function for a curve singularity yields the generating functions for Euler numbers of punctual Hilbert schemes when any punctual Hilbert scheme admits an affine cell decomposition. This fact allows us to derive the relations among the motivic Hilbert zeta function and other invariants such as the generating function for semi-modules of the semi-group of the singularity, the HOMFLY polynomial and the degrees of Severi strata of the miniversal deformation of the singularity. As an application of the fact above, we also generalize Kawai's result regarding the generating function for the Euler numbers of a singular curve.
Paper Structure (11 sections, 8 theorems, 43 equations)

This paper contains 11 sections, 8 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Proof of main theorems
  3. Proof of Theorem \ref{['main 2']}
  4. Preliminaries for the proof of Theorem \ref{['main 4']}
  5. Proof of Theorem \ref{['main 4']}
  6. Examples of motivic Hilbert zeta functions
  7. The $A_1$ and $A_{2d}$-singularities
  8. The $E_6$ and $E_{8}$-singularities
  9. An application
  10. Generalization of Kawai's result
  11. Proof of Theorem \ref{["Generalization of Kawai's Thm"]}

Key Result

Lemma 2

Let $(C,p)$ be a reduced curve singularity that satisfies the condition $(*)$. The class $[C^{[l]}_p]$ of $C^{[l]}_p$ in $K_0(\mathrm{Var}_{\mathbb{C}})$ is a polynomial in $\mathbb{C}[\mathbb{L}]$. Furthermore, the Euler number $\chi (C^{[l]}_p)$ is equal to the number of affine cells of $C^{[l]}$.

Theorems & Definitions (17)

  • Remark 1
  • Lemma 2
  • proof
  • Proposition 3
  • Proposition 4: PS, Theorem 3
  • Corollary 5
  • Proposition 6: SW1, Proposition 6
  • Remark 7
  • Remark 8
  • Corollary 9
  • ...and 7 more