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 study H15,g,5β for every
possible genus g and determine their irreducibility. We also study the
birationality of the moduli map up to projective equivalence and several key
properties such as gonality of a general element as well as characterizing
smooth elements of each component.Comment: Incorrect statement in Remark 3.8 has been corrected. Several typos
has been fixe