Hilbert scheme and Hilbert functions of smooth curves of degrees at most $15$ in $\mathbb{P}^5$

TL;DR

This paper investigates the Hilbert functions of smooth curves with degree up to 15 in projective 5-space, analyzes the irreducibility of their Hilbert schemes, and explores the properties of the associated moduli map.

Contribution

It provides new results on the Hilbert functions, irreducibility, and projective normality of smooth curves in 5, especially for low degrees, and examines the fibers of the moduli map.

Findings

Determined irreducibility of 5 Hilbert schemes for degrees 4.

Computed Hilbert functions for general curves of degree 5 to 15.

Established conditions for projective normality and ACM property.

Abstract

Denoting $\mathcal{H}_{d,g,5}$ by the Hilbert scheme of smooth curves of degree $d$ and genus $g$ in $\mathbb{P}^5$, let $\mathcal{H}$ be an irreducible component of $\mathcal{H}_{d,g,5}$. We study the Hilbert function $h_X:\mathbb{N}\longrightarrow\mathbb{N}$, $h_X(t):= h^0(\mathcal{I}_X(t))$ of a general member $X\in\mathcal{H}$, especially when the degree of the curve is low; $d\le 15$. We also determine the irreducibility of $\mathcal{H}_{d,g,5}$ for $d\le 14$ and study the natural functorial map $\mu :$\mathcal{H}_{d,g,5}$ \longrightarrow \mathcal{M}_g$ in some detail. We describe the fibre $\mu^{-1}\mu(X)$ for a general $X\in\mathcal{H} $ as well as determining the projective normality (or being ACM).