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

For $\gamma \ge 7$ and $g\ge 6\gamma +5$, we construct a family ${\mathcal{F}}^{\mathrm{\prime }}$ 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 $\dim {\mathcal{F}}^{\mathrm{\prime }}=2g-\gamma -1+(g-3\gamma +1{)}^2$. For a general curve $X^{\mathrm{\prime }}$ from the family ${\mathcal{F}}^{\mathrm{\prime }}$, we compute the dimension of the space of its first-order deformations. We prove that the family ${\mathcal{F}}^{\mathrm{\prime }}$ gives rise to an irreducible, non-reduced component ${\mathcal{D}}^{\mathrm{\prime }}$ 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 $\dim T_{[X^{\mathrm{\prime }}]}{\mathcal{D}}^{\mathrm{\prime }}=\dim {\mathcal{D}}^{\mathrm{\prime }}+1=\dim {\mathcal{F}}^{\mathrm{\prime }}+1$.