On the Hilbert scheme of smooth curves of degree $15$ and genus $14$ in $\mathbb{P}^5$

TL;DR

This paper investigates the Hilbert scheme of smooth degree 15, genus 14 curves in projective 5-space, demonstrating its non-emptiness, reducibility with two components, and analyzing properties like gonality and moduli map birationality.

Contribution

It proves the non-emptiness and reducibility of al{H}_{15,14,5} with two expected-dimensional components and studies their geometric properties and moduli map behavior.

Findings

al{H}_{15,14,5} is non-empty and reducible with two components

Each component has the expected dimension and is generically reduced

The paper analyzes gonality and the birationality of the moduli map

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 show that $\mathcal{H}_{15,14,5}$ is non empty and reducible with two components of the expected dimension hence generically reduced. We also study the birationality of the moduli map up to projective motion and several key properties such as gonality of a general element as well as specifying smooth elements of each components.