Hilbert scheme of linearly normal curves in ${\mathbb{P}}^r$ with index of speciality five and beyond

We study the Hilbert scheme of smooth, irreducible, non-degenerate and linearly normal curves of degree $d$ and genus $g$ in ${\mathbb{P}}^r$ ($r\ge 3$) whose complete and very ample hyperplane linear series $\mathcal{D}$ have relatively small index of speciality $i(\mathcal{D})=g-d+r$. In particular we determine the existence as well as the non-existence of Hilbert schemes of linearly normal curves ${\mathcal{H}}_{d,g,r}^{\mathcal{L}}$ for every possible triples $(d,g,r)$ with $i(\mathcal{D})=5$ and $r\ge 3$. We also determine the irreducibility of the Hilbert scheme ${\mathcal{H}}_{g+r-5,g,r}^{\mathcal{L}}$ when the genus $g$ is near to the minimal possible value with respect to the dimension of the projective space ${\mathbb{P}}^r$ for which ${\mathcal{H}}_{g+r-5,g,r}^{\mathcal{L}}\mathrm{\ne }\mathrm{\varnothing }$, say $r+9\le g\le r+11$. In the course of proofs of key results, we show the existence of linearly normal curves of degree $d\ge g+1$ with arbitrarily given index of speciality with some mild restriction on the genus $g$.