On the intersection numbers of the generators of $\text{Pic}(X)$ of a smooth quintic surface

Question

There exists the following result in the literature: There exists a polarized $K3$ surface $(X, H)$ of genus $3$ and a smooth irreducible curve $C$ on $X$ satisfying $C^2 =4$, $C.H=6$ such that $\text{Pic}(X) \cong \mathbb Z[H] \oplus Z[C]$. The theorem follows from [https://arxiv.org/pdf/math/9805140.pdf] theorem $1.1(iv)$.

Now Let's consider $X$ to be a smooth quintic hypersurface in $\mathbb P^3$ with Picard number $2$. Then I think it can be shown that $\text{Pic}(X) \cong \mathbb Z[H] \oplus Z[H']$, where $H$ is the hyperplane class and $H'$ is some divisor. Now in order to locate the ample line bundles in $\text{Pic}(X)$ using the Nakai-Moishezon criterion, we must know the intersection numbers $H.H'$ and $H'^2$.

In this context my question is the following: Does there exist in the literature an analogous existence result as the first-mentioned theorem for smooth quintic hypersurface with Picard number $2$?

To be more precise: Does there exist a polarized smooth quintic hypersurface $(X, H)$ in $\mathbb P^3$ and a smooth irreducible curve $C$ for which the intersection numbers $C^2$ and $C.H$ are known and $\text{Pic}(X) \cong \mathbb Z[H] \oplus Z[C]$?

Can someone give me any reference which could be even remotely useful in the context of finding out such $(X,H)$ and $C$

Any help from anyone is welcome.

Answer 1

A natural source of surfaces in $\mathbb{P}^{3}$ with Picard number $> 1$ is given by linear determinantal surfaces, i.e. zero sets of square matrices of linear forms. In what follows, let $X \subseteq \mathbb{P}^{3}$ be a smooth linear determinantal surface of degree $d \geq 2$ (smoothness can arranged by taking the $d \times d$ matrix to be sufficiently general) and let $H \in {\rm Pic}(X)$ be the class of a hyperplane section.

It can be checked that $X$ contains a curve $C$ which is the degeneracy locus of a $d \times (d-1)$ matrix of linear forms. For degree/genus reasons, this $C$ cannot be a complete intersection of $X$ and another surface in $\mathbb{P}^{3}$.

There is a Noether-Lefschetz-type result which says that if $X$ is a generic linear determinantal surface of degree $d,$ the Picard group ${\rm Pic}(X)$ is isomorphic to $\mathbb{Z}H \oplus \mathbb{Z}C.$ Here is a reference for this result, as well as others which go further than determinantal surfaces:

Lopez, Angelo Felice; Noether-Lefschetz theory and the Picard group of projective surfaces, Mem. Amer. Math. Soc. 89 (1991), no. 438, x+100 pp.

EDIT: The aforementioned result on generic linear determinantal varieties is a corollary of the following result of Lopez; if $C \subseteq \mathbb{P}^{3}$ is a smooth connected curve whose homogeneous ideal is generated in degree at most $d-1,$ then the general surface of $S$ of degree $d$ containing $C$ is smooth and ${\rm Pic}(S) \cong \mathbb{Z}H \oplus \mathbb{Z}C.$ This holds in particular when $d \geq 2$ and $C$ is a line in $\mathbb{P}^{3}.$