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

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 $h_X:\mathbb{N}⟶\mathbb{N}$, $h_X(t):=h^0({\mathcal{I}}_X(t))$ 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}$_{d,g,5}$⟶{\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).