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

Abstract

For $\gamma \ge 7$ and $g\ge 6\gamma +5$, we construct a family ${\mathcal{F}}^{\prime }$ of curves lying on cones in ${\mathbb{P}}^{g-3\gamma +1}$ over smooth non-degenerate curves of genus $\gamma$ and degree $g-2\gamma$ in ${\mathbb{P}}^{g-3\gamma +1}$. We show that $\dim {\mathcal{F}}^{\prime }=2g-\gamma -1+(g-3\gamma +1{)}^2$. For a general curve $X^{\prime }$ from the family ${\mathcal{F}}^{\prime }$, we compute the dimension of the space of its first-order deformations. We prove that the family ${\mathcal{F}}^{\prime }$ gives rise to an irreducible, non-reduced component ${\mathcal{D}}^{\prime }$ of the Hilbert scheme ${\mathcal{I}}_{2g-4\gamma +1,g,g-3\gamma +1}$, which parametrizes smooth, irreducible, non-degenerate curves of degree $2g-4\gamma +1$ and genus $g$ in ${\mathbb{P}}^{g-3\gamma +1}$. We obtain $\dim T_{[X^{\prime }]}{\mathcal{D}}^{\prime }=\dim {\mathcal{D}}^{\prime }+1=\dim {\mathcal{F}}^{\prime }+1$.