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.

To appear in a special issue on Bollettino UMI under the title A Projective Geometry Landscape