Hilbert scheme of points on non-reduced nodal curves

TL;DR

This paper constructs a new stratification of the punctual Hilbert scheme of points on non-reduced nodal plane curves, introducing combinatorial objects called weak diagonal partitions, and computes related topological invariants.

Contribution

It introduces a novel stratification based on weak diagonal partitions and analyzes the structure of each stratum, including affine and torus times affine cases, for non-reduced nodal curves.

Findings

Each stratum is affine when u=1,2.

Each stratum is isomorphic to a torus times an affine space for specific v values.

Computed Poincaré polynomials for certain cases of the Hilbert scheme.

Abstract

We construct a stratification of the punctual Hilbert scheme of points on a non-reduced and nodal plane curve, $x^uy^v=0$. Each stratum is indexed by a new combinatorial object we define: a weak diagonal partition. The approach is based on introducing filtrations on ideals, together with a valuation adapted to the non-reduced structure, which allows us to analyze generators and their degrees of freedom in a systematic way. In particular, each stratum is affine when $u=1,2$; and each stratum is isomorphic to an algebraic torus times an affine space, $(\mathbb{C}^*)^{m_1} \times \mathbb{C}^{m_2}$, when $u=v,v-1,v-2$. We consequently compute the Poincar\'e polynomials of the punctual Hilbert scheme of points on curves $x^uy^v=0$ when $u=1,2,v-2,v-1,v$. As an application, we prove the colored Oblomkov-Rasmussen-Shende conjecture for the Hopf link for $u=1, v$ arbitrary, showing that the Poincar\'e polynomial is the row-colored link homology up to change of variables.