A note on the Hilbert scheme of points on a cusp curve (Q2888842)

The author investigates the Hilbert scheme of points on a cusp curve \(X\). Specifically, let \(X\) be a projective cusp curve with a cusp locally defined by \(x^2=y^3\) and let \(H\) denote the Hilbert scheme parameterizing zero-dimensional length \(m\) subschemes \(X\), for a fixed \(m\). Also let \(H^0\) denote the subscheme of \(H\) parameterizing those subschemes supported only at the cusp. The author establishes the two nice theorems:NEWLINENEWLINE1. The reduced scheme \((H^0)^{\mathrm{red}}\) is isomorphic to \(\mathbb{P}^1\) for \(m \geq 2\).NEWLINENEWLINE2. \(H\) has one singular point along \(H^0\).NEWLINENEWLINEThe proofs are explicit and thorough and will be of interest to those studying Hilbert schemes and in particular punctual Hilbert schemes.