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

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