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

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 $\gamma \ge 3$ and $e\ge 4\gamma +5$ 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 $[X]\in \mathcal{H}$.