We denote by $\mathcal{ H}_{ d,g,r}$ 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 $\mathbb{ P}^r.$ In this article, we study $\mathcal{ H}_{ 15,g,5}$ for every possible genus $g$ and determine when it is irreducible. We also study the moduli map $\mathcal{ H}_{ 15,g,5}\rightarrow\mathcal{ M}_g$ and several key properties such as gonality of a general element as well as characterizing smooth elements of each component.
Categories: math.AG