Hilbert scheme of smooth curves of degree sixteen 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 $\mathcal{H}_{16,g,5}$ for almost every possible genus $g$ and chasing after its irreducibility. We also study the natural moduli map $\mathcal{H}_{d,g,5}\stackrel{\mu}{\to}\mathcal{M}_g$ and several key properties such as gonality of a general element as well as characterizing smooth elements in each component.