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

Abstract

For $γ\geq 7$ and $g \geq 6γ+ 5$, we construct a family $\mathcal{F}^{\prime}$ of curves lying on cones in $\mathbb{P}^{g-3γ+1}$ over smooth non-degenerate curves of genus $γ$ and degree $g-2γ$ in $\mathbb{P}^{g-3γ+1}$. We show that $\dim \mathcal{F}^{\prime} = 2g-γ-1 + (g-3γ+1)^2$. 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 $\mathcal{I}_{2g-4γ+ 1, g, g - 3γ+ 1}$, which parametrizes smooth, irreducible, non-degenerate curves of degree $2g-4γ+ 1$ and genus $g$ in $\mathbb{P}^{g-3γ+1}$. We obtain $\dim T_{[X^{\prime}]} \mathcal{D}^{\prime} = \dim \mathcal{D}^{\prime} + 1 = \dim \mathcal{F}^{\prime} + 1$.