On the geometry of punctual Hilbert schemes on singular curves and their motivic zeta functions

TL;DR

This work develops a combinatorial–geometric framework for punctual Hilbert schemes on irreducible curve singularities by introducing the $\Gamma$-subsemimodule tree $G_{\Gamma}$. It proves that for plane curve singularities defined by $x^{p}-y^{q}=0$ with $\gcd(p,q)=1$ or for monomial semigroups, each edge of $G_{\Gamma}$ induces a piecewise trivial fibration between strata $C^{[\Delta]}$ and $C^{[m(\Delta)]}$, with fiber an affine space whose dimension is dictated by the semimodule data. This enables an algorithmic computation of the motivic Hilbert zeta function and yields explicit formulas for simple singularities ($A_{2d}$, $E_6$, $E_8$, $W_8$, $Z_{10}$), as well as a refined stratification by the minimal number of generators, captured by a generalized motivic zeta function $Zm^{\mathrm{Hilb}}_{(C,O)}$. The results unify and extend previous analyses (including Oblomkov–Rasmussen–Shende and Watari) and illuminate the connections between semigroup combinatorics, affine fibrations of Hilbert scheme strata, and motivic invariants, with monomial semigroups providing particularly tractable, explicit descriptions.

Abstract

Inspired by the work of Soma and Watari, we define a tree structure on certain subsemimodules of the semigroup $Γ$ associated with an irreducible plane curve singularity $(C,O)$. Building on results of Oblomkov, Rasmussen, and Shende, we show that for specific classes of singularities, this tree encodes key aspects of the geometry of the punctual Hilbert schemes of $(C,O)$. As an application, we compute the motivic Hilbert zeta function for a family of singular curves. \vskip 0.1cm A point in the Hilbert scheme corresponds to an ideal in the local ring $\mathcal{O}_{C,O}$ of the singularity. We study the stratification of these Hilbert schemes induced by constraints on the minimal number of generators of the defining ideals, and we describe geometric properties of these strata, including their dimension and closure relations.\vskip 0.1cm More importantly, we study their motivic zeta functions, particularly the motivic Hilbert zeta function, which encodes the classes of all punctual Hilbert schemes in the Grothendieck ring of varieties.

Figures (1)

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

Theorems & Definitions (78)

• Theorem 1.1: Main Theorem
• Theorem 1.2
• Theorem 1.3
• Proposition 2.1: pfister1992reduced, Theorem 3
• Lemma 2.1
• Lemma 3.1
• proof
• Definition 3.1
• Lemma 3.2
• proof