We study the Hilbert scheme of smooth, irreducible, non-degenerate and
linearly normal curves of degree d and genus g in Pr (rβ₯3)
whose complete and very ample hyperplane linear series D have
relatively small index of speciality i(D)=gβd+r. In particular we
determine the existence as well as the non-existence of Hilbert schemes of
linearly normal curves Hd,g,rLβ for every possible
triples (d,g,r) with i(D)=5 and rβ₯3. We also determine the
irreducibility of the Hilbert scheme Hg+rβ5,g,rLβ
when the genus g is near to the minimal possible value with respect to the
dimension of the projective space Pr for which
Hg+rβ5,g,rLβξ =β , say r+9β€gβ€r+11.
In the course of proofs of key results, we show the existence of linearly
normal curves of degree dβ₯g+1 with arbitrarily given index of speciality
with some mild restriction on the genus g.Comment: 60 pages, added two figures illustrating main results and corrected
typo