We denote by Hd,g,rβ the Hilbert scheme of smooth curves, which
is the union of components whose general point corresponds to a smooth
irreducible and non-degenerate curve of degree d and genus g in
Pr. In this article, we show that H15,14,5β is non
empty and reducible with two components of the expected dimension hence
generically reduced. We also study the birationality of the moduli map up to
projective motion and several key properties such as gonality of a general
element as well as specifying smooth elements of each components.Comment: Final version; corrected many typos in the exposition. To appear in
Bollettino dell'Unione Matematica Italian