Curves contained in a quartic determinantal surface containing a line

Let $X\subseteq \mathbb{P}^{3}$ be a very general element of the Noether-Lefschetz divisor that parametrizing smooth quartic surfaces containing a line.

Let $L\subseteq X$ denote the corresponding line.

We study the curves contained in $X$ and analyze their behavior in the Hilbert scheme.

We first determine which linear systems contain smooth irreducible curves.

For most classes, we verify that the general member is a smooth point of the expected Hilbert scheme.

Finally we compute the Rao function of any curve on $X$.