Paper detail

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

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.