Components of the Hilbert scheme of smooth projective curves using ruled surfaces II: existence of non-reduced components

Abstract

Let \(\mathcal {I}_{d,g,r}\) be the union of irreducible components of the Hilbert scheme whose general points represent smooth, irreducible, non-degenerate curves of degree d and genus g in \(\mathbb {P}^r\). Using a family of curves found on ruled surfaces over smooth curves of genus \(\gamma \), we show that for \(\gamma \ge 7\) and \(g \ge 6 \gamma + 5\), the scheme \(\mathcal {I}_{2g-4\gamma + 1, g, g - 3\gamma + 1}\) acquires a non-reduced component \(\mathcal {D}^{\prime }\) such that \({\text {dim}}T_{[X^{\prime }]} \mathcal {D}^{\prime } = {\text {dim}}\mathcal {D}^{\prime } + 1\) for a general point \([X^{\prime }] \in \mathcal {D}^{\prime }\).