Quartic curves in the quintic del Pezzo threefold

In this paper, we prove that the Hilbert scheme $\mathbf{H}_{4}(X_{5})$ of rational quartic curves on the quintic del Pezzo threefold $X_{5}$ is isomorphic to a Grassmannian bundle over the Hilbert scheme of lines on $X_{5}$. In particular, $\mathbf{H}_{4}(X_{5})$ is smooth and irreducible. Our approach builds upon the geometry of rational quartic curves on $X_{5}$ studied by Fanelli–Gruson–Perrin in their work on the moduli space of stable maps to $X_{5}$.