Classification of sextic curves in the Fano 3-fold ${\mathcal{V}}_5$ with rational Galois covers in ${\mathbb{P}}^3$

In this paper, we classify sextic curves in the Fano $3$-fold ${\mathcal{V}}_{\mathbf{5}}$ (the smooth quintic del Pezzo $3$-fold) that admit rational Galois covers in the complex ${\mathbb{P}}^3$. We show that the moduli space of such sextic curves is of complex dimension $2$ through the invariants of the engaged Galois groups for the explicit constructions. This raises the intriguing question of understanding the moduli space of sextic curves in ${\mathcal{V}}_5$ through their Galois covers in ${\mathbb{P}}^3$.