The Full Flag Hilbert Scheme of Nodal Curves and the Punctual Hilbert Scheme of Points of the Cusp Curve

TL;DR

This paper investigates the singularities of the full flag Hilbert scheme of points on nodal and cusp curves, revealing complete intersection singularities and an isomorphism to projective space.

Contribution

It characterizes the singularities of the Hilbert scheme of points on nodal and cusp curves, showing they are complete intersections and identifying the structure of the punctual Hilbert scheme.

Findings

Full flag Hilbert scheme of nodal curves has only complete intersection singularities.

Punctual Hilbert scheme of the cusp curve is isomorphic to ^1.

Hilbert scheme has a unique singularity along the punctual scheme.

Abstract

In this paper, we show the full flag Hilbert scheme of points of nodal curves has only complete intersection singularities. We study the Hilbert scheme of points of the cusp curve and show the punctual Hilbert scheme is isomorphic to \mathbb{P}^1. Furthermore Hilbert scheme has only one singularty along the punctual one.