Irreducible components of Hilbert scheme of points on non-reduced curves

TL;DR

This paper classifies the irreducible components of the Hilbert scheme of points on non-reduced plane curves, providing a formula for their multiplicities and revealing their structure indexed by partitions.

Contribution

It introduces a complete classification of irreducible components and multiplicities for the Hilbert scheme of points on non-reduced algebraic curves.

Findings

Irreducible components are indexed by partitions of n.

All components have dimension n.

Multiplicities are given by a polynomial in the parts of partitions.

Abstract

We classify the irreducible components of the Hilbert scheme of $n$ points on non-reduced algebraic plane curves, and give a formula for the multiplicities of the irreducible components. The irreducible components are indexed by partitions of $n$; all have dimension $n$; and their multiplicities are given as a polynomial of the parts of the corresponding partitions.