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

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.

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:
• thanks for the answer. Is the curve $C$ mentioned in the answer irreducible? Are the intersection numbers $C.C$ and $C.H$ immediate from the discussion? (apology in advance, if this is trivial to compute). Finally, Is theorem $(II)3.1$ mentioned there relevant in this context? – UPOKROMONIKA 2 days ago
• also as the general surface of degree $d \geq4$ is not linear determinantal, but linear pfaffian if $d \leq 15$, so in this case are we talking about smooth quintic surfaces which is not general? – UPOKROMONIKA 2 days ago
• The curve $C$ is cut out by the maximal minors of a $d \times (d-1)$-submatrix of the matrix of linear forms cutting out $X,$ so by general results on determinantal varieties we can ensure $C$ is smooth and connected, and therefore irreducible. The degree $C \cdot H$ and genus can of course be read off the Hilbert polynomial, which in turn can be read off the Eagon-Northcott resolution of the associated determinantal ideal. You can then determine $C \cdot C$ from the adjunction formula on $X.$ I can't access the reference at the moment, so I can't speak to the relevance of Theorem (II)3. – Yusuf Mustopa 2 days ago
• As to your second comment, you are correct that the smooth surfaces in under discussion are not general when $d \geq 4.$ – Yusuf Mustopa 2 days ago