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 \([X] \in {\mathcal {H}}\).