Technical Summary: Algorithm for Motivic Hilbert Zeta Function of Some Curve Singularities

1. Problem Statement

The paper addresses the computational challenge of determining the motivic Hilbert zeta function for singularities of complex curves. While the geometry of the Hilbert scheme of points on a smooth curve is well-understood (isomorphic to the symmetric product), the geometry of the punctual Hilbert scheme $C[\ell]$ on a singular curve $(C, O)$ remains complex and less understood.

Specifically, the authors aim to compute two generating series with coefficients in the Grothendieck ring of complex varieties, $K_0(\text{Var}_{\mathbb{C}})$:

1. The Classical Motivic Hilbert Zeta Function:
   $$Z_{\text{Hilb}}^{(C,O)}(q) = \sum_{\ell=0}^{\infty} [C[\ell]] q^\ell$$
   where $C[\ell]$ parametrizes 0-dimensional subschemes of length $\ell$ supported at the singularity.
2. The Refined Motivic Hilbert Zeta Function:
   $$Z_{\text{mHilb}}^{(C,O)}(a^2, q^2) = \sum_{\ell \ge 0} \sum_{m \ge 1} q^{2\ell}(1-a^2)^{m-1} [C[\ell], m]$$
   where $C[\ell], m$ is the subvariety of ideals with exactly $m$ minimal generators.

The primary difficulty lies in the fact that the value semigroup $\Gamma$ associated with the singularity is infinite, making direct computation of the stratification of the Hilbert scheme impossible without approximation.

2. Methodology

The authors develop a recursive algorithmic framework based on the combinatorial structure of the value semigroup $\Gamma$ and the stratification of the Hilbert scheme.

2.1. Theoretical Foundations

• Stratification by Semimodules: Following Pfister and Steenbrink, the punctual Hilbert scheme $C[\ell]$ is stratified by $\Gamma$-subsemimodules $\Delta \subset \Gamma$. A stratum $C[\Delta]$ consists of ideals whose valuation set is exactly $\Delta$. The complement size $\#(\Gamma \setminus \Delta)$ corresponds to the length $\ell$.
• Tree Structure: Building on work by Chen, Mourtada, Zhu, and Hajli, the set of $\Gamma$-subsemimodules $D_\ell$ (those with complement size $\ell$) forms a tree structure. There is a directed edge from $\Delta \in D_\ell$ to $\Delta' \in D_{\ell-1}$ if $\Delta' = \Delta \cup \{\gamma_\Delta\}$, where $\gamma_\Delta = \max(\Gamma \setminus \Delta)$ is the Frobenius element.
• Geometric Morphisms: For specific singularities (monomial semigroups or $y^k = x^n$ with $\gcd(k,n)=1$), an edge in this tree corresponds to a piecewise trivial fibration between strata $C[\Delta]$ and $C[m(\Delta)]$. The fiber is an affine space whose dimension depends on the number of minimal generators of $\Delta$ and its syzygy module below $\gamma_\Delta$. This allows the class $[C[\Delta]]$ to be computed recursively from $[C[m(\Delta)]]$.

2.2. Algorithmic Approach

The authors propose a suite of algorithms to compute these classes:

• Truncation Strategy: Since $\Gamma$ is infinite, the algorithms operate on a truncated set $\Gamma_{\le n^*} = \Gamma \cap [0, n^*]$. The paper provides a theoretical bound for the minimal effective truncation length $N$ required to ensure correctness.
• Algorithm 1 (Classical Zeta Function):
  • Input: A curve singularity with a monomial valuation semigroup or defined by $y^k = x^n$ ($\gcd(k,n)=1$), and its conductor $c$.
  • Process:
    1. Generate the tree of subsemimodules $D_1, \dots, D_c$ using Algorithm 4 (deletion maps).
    2. For each $\Delta \in D_\ell$, compute the Frobenius element $\gamma_\Delta$.
    3. Identify minimal generators of $\Delta$ (Algorithm 2) and the syzygy module Syz($\Delta$) (Algorithm 3).
    4. Apply the recursive formula from Corollary 4.2:
       $$[C[\Delta]] = [C[m(\Delta)]] \cdot \mathbb{L}^{|\{\gamma_i < \gamma_\Delta\}| - |\{\sigma_i < \gamma_\Delta\}|}$$
       where $\mathbb{L} = [\mathbb{A}^1]$.
    5. Sum the classes for each $\ell$ to obtain the coefficients of the zeta function.
• Algorithms 5–7 (Refined Zeta Function):
  • Targeted specifically at singularities $y^p = x^q$ ($\gcd(p,q)=1$).
  • Utilizes inclusion-exclusion principles on subsets of minimal generators to compute the classes $[C[\Delta], m]$ (ideals with exactly $m$ generators).
  • Relies on the decomposition of strata into locally closed sets $Y_{ij}$ defined by syzygy conditions, computing their classes as products of $(\mathbb{C}^*)^k$ and affine spaces.

2.3. Complexity and Bounds

• Time Complexity: The complexity is analyzed in terms of the conductor $c$, the number of subsemimodules $\chi_\ell = \#D_\ell$, and the truncation bound $n^*$. The total complexity for Algorithm 1 is roughly $O(\sum c \cdot \chi_\ell \cdot n^* \cdot \alpha_1 \cdot (n^* - \ell))$.
• Effective Truncation Bound: Theorem 7.3 establishes that a truncation bound $N \le (c-1)(\alpha_1 + 2)$ is sufficient for the algorithms to produce correct results, where $\alpha_1$ is the smallest generator of $\Gamma$.

3. Key Contributions

1. Algorithmic Implementation: The paper provides the first explicit algorithms (implemented in Python) to compute the motivic Hilbert zeta functions for a broad class of curve singularities, moving beyond theoretical existence to effective computation.
2. Refined Zeta Function: It extends the computation to the refined zeta function $Z_{\text{mHilb}}$, which tracks the number of minimal generators, a finer invariant than the classical zeta function.
3. Theoretical Justification of Truncation: The authors prove a concrete bound for the truncation of the infinite semigroup $\Gamma$, ensuring that the finite approximations used in the code are mathematically rigorous for the range of $\ell$ up to the conductor.
4. Verification on Specific Types: The algorithms are applied to specific families, including monomial curves and the $W_8$ singularity type ($\Gamma = \langle 4, 5, 6 \rangle$), reproducing and verifying known results for ADE singularities (types $A_{2k}, E_6, E_8$) previously studied by Soma and Watari.

4. Results

The paper demonstrates the efficacy of the algorithms through several examples:

• Example 8.1: Computes the motivic Hilbert zeta function for the singularity with local ring $\mathbb{C}[[t^4, t^5, t^6]]$ (Type $W_8$). The authors explicitly list the classes $[C[\ell]]$ for $\ell=1$ to $8$ and construct the generating series.
• Example 8.2: Computes the refined motivic Hilbert zeta function $Z_{\text{mHilb}}$ for the singularity $y^3 = x^7$. The output is a detailed expansion in terms of $q$, $a$, and the Lefschetz motive $\mathbb{L}$, showing the distribution of ideals by length and number of generators.

The results confirm that the recursive tree structure allows for the systematic calculation of these invariants, which were previously difficult to obtain due to the complexity of the defining equations in the Grassmannian.

5. Significance and Claims

The authors position this work as a tool to overcome the computational barriers in studying the geometry of Hilbert schemes on singular curves.

• Accessibility: By providing a Python implementation (available on GitHub), the authors make these advanced motivic invariants accessible for further experimentation and verification.
• Scope: The methods are currently limited to singularities with monomial valuation semigroups or those defined by $y^k = x^n$ with coprime exponents. The paper does not claim to solve the problem for arbitrary curve singularities.
• Modesty: The paper frames its contribution as an algorithmic realization of existing theoretical frameworks (specifically the tree structure and fibration results from [1] and [2]) rather than a discovery of new geometric phenomena. The significance lies in the effective computation of these invariants, enabling the study of specific families (like ADE singularities) where manual calculation is infeasible.

The work serves as a bridge between the abstract theory of motivic integration and concrete computational algebraic geometry, providing a verified method to compute the classes of punctual Hilbert schemes for a significant class of curve singularities.