Non-reduced components of the Hilbert scheme of curves using triple covers

TL;DR

This paper demonstrates the existence of a non-reduced component in the Hilbert scheme of certain smooth curves on cones, using triple covers and deformation techniques, revealing new geometric properties of these curves.

Contribution

It introduces a novel approach to studying non-reduced components of the Hilbert scheme via triple covers and deformation theory, extending understanding of curve components on cones.

Findings

Existence of a non-reduced component in the Hilbert scheme for specified genus and degree.

Dimension of the tangent space at a general point exceeds the component dimension by one.

Curves deform within cones over deformations of the base curve.

Abstract

In this paper we consider curves on a cone that pass through the vertex and are also triple covers of the base of the cone, which is a general smooth curve of genus $\gamma$ and degree $e$ in $\mathbb{P}^{e-\gamma}$. Using the free resolution of the ideal of such a curve found by Catalisano and Gimigliano, and a technique concerning deformations of curves introduced by Ciliberto, we show that the deformations of such curves remain on cones over a deformation of the base curve. This allows us to prove that for $\gamma \geq 3$ and $e \geq 4\gamma + 5$ there exists a non-reduced component $\mathcal{H}$ of the Hilbert scheme of smooth curves of genus $3e + 3\gamma$ and degree $3e+1$ in $\mathbb{P}^{e-\gamma+1}$. We show that $\dim T_{[X]} \mathcal{H} = \dim \mathcal{H} + 1 = (e - \gamma + 1)^2 + 7e + 5$ for a general point $[X] \in \mathcal{H}$.