Existence and the reducibility of the Hilbert scheme of linearly normal curves in ${\mathbb{P}}^r$ of relatively high degrees

Let ${\mathcal{H}}_{d,g,r}$ be the Hilbert scheme parametrizing smooth irreducible and non-degenerate curves of degree $d$ and genus $g$ in ${\mathbb{P}}^r\mathrm{.}$ We denote by ${\mathcal{H}}_{d,g,r}^{\mathcal{L}}$ the union of those components of ${\mathcal{H}}_{d,g,r}$ whose general element is linearly normal. In this article we show that ${\mathcal{H}}_{d,g,r}^{\mathcal{L}}$ ($d\ge g+r-3$)isnon-emptyinacertainoptimalrangeoftriples$(d,g,r)$ and is empty outside the range. This settles the existence (or non-emptiness if one prefers) of the Hilbert scheme ${\mathcal{H}}_{d,g,r}^{\mathcal{L}}$ of linearly normal curves of degree $d$ and genus $g$ in ${\mathbb{P}}^r$ for $g+r-3\le d\le g+r$,$r\ge 3$\mathrm{.}Wealsodetermineallthetriples$(d,g,r)$with$g+r-3\le d\le g+r$forwhich$$\mathcal{H}$^$\mathcal{L}$_{d,g,r}$ is reducible (or irreducible).