On the Hilbert scheme of smooth curves of degree $d=15$ in ${\mathbb{P}}^5$

Abstract

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 ${\mathcal{H}}_{15,g,5}\to {\mathcal{M}}_g$ and several key properties such as gonality of a general element as well as characterizing smooth elements of each component.

Major stylistic changes. Improved readerbility. Shorten the size of the manuscript. Many typos were corrected. 36 pages