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

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).