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Algorithm for motivic Hilbert zeta function of some curve singularities

Wenhao Zhu, Yizi Chen, Hussein Mourtada

Abstract

We develop an algorithm for computing the motivic Hilbert zeta function for curve singularities with a monomial valuation group or for singular curves defined by $y^{k}=x^{n}$, where $gcd(k,n)=1$. It is well known that the Hilbert scheme of points on a smooth curve is isomorphic to the symmetric product of the curve. However, the geometrical structure of Hilbert scheme of points on singular curves remains less understood. The algorithm we propose computes the motivic Hilbert zeta function, $Z_{(C,O)}^{Hilb}(q)\in K_{0}(Var_{\mathbb{C}})[[q]]$, for such curve singularities. This function is represented as a series with coefficients in the Grothendieck ring of varieties over $\mathbb{C}$. The main computational challenge arises from the infinity of $Γ$. To address this, we approximate $Γ$ by truncating it to a finite subset to allow effective algorithm operation. We also analyze the time complexity and estimate the range of the effective finite length of $Γ$ necessary for reliable results. The Python implementation of our algorithm is available at https://github.com/whaozhu/motivic_hilbert.

Algorithm for motivic Hilbert zeta function of some curve singularities

Abstract

We develop an algorithm for computing the motivic Hilbert zeta function for curve singularities with a monomial valuation group or for singular curves defined by , where . It is well known that the Hilbert scheme of points on a smooth curve is isomorphic to the symmetric product of the curve. However, the geometrical structure of Hilbert scheme of points on singular curves remains less understood. The algorithm we propose computes the motivic Hilbert zeta function, , for such curve singularities. This function is represented as a series with coefficients in the Grothendieck ring of varieties over . The main computational challenge arises from the infinity of . To address this, we approximate by truncating it to a finite subset to allow effective algorithm operation. We also analyze the time complexity and estimate the range of the effective finite length of necessary for reliable results. The Python implementation of our algorithm is available at https://github.com/whaozhu/motivic_hilbert.

Paper Structure

This paper contains 7 sections, 15 theorems, 18 equations, 1 figure, 4 algorithms.

Table of Contents

  1. Introduction
  2. Punctual Hilbert scheme and monomial semigroup
  3. Tree structure associated with valuation semigroup
  4. Piecewise fibration
  5. Algorithm
  6. Effective finite length of Lg
  7. Experimentation

Key Result

Proposition 1

pfister1992reduced. For $\ell\geq 0$, there exists a closed embedding $\phi_{\ell}: C^{[\ell]}\rightarrow \mathscr{M}$ and the map are bijective when $\ell\geq c$.

Figures (1)

  • Figure 1: Tree for the case of $E_{6}$ type singularity

Theorems & Definitions (29)

  • Proposition 1
  • Lemma 1
  • Remark 1
  • Definition 1
  • Definition 2
  • Proposition 2
  • Lemma 2
  • Corollary 1
  • Lemma 3
  • Lemma 4
  • ...and 19 more