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

TL;DR

This paper investigates the structure and properties of the Hilbert scheme of smooth degree 15 curves in projective 5-space, focusing on irreducibility, moduli mapping, gonality, and component characterization across all possible genera.

Contribution

It provides a comprehensive analysis of the irreducibility and geometric properties of $\

Findings

Determines when $\

Analyzes the moduli map $\

Characterizes smooth elements of each component

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