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Fix a smooth sextic curve curve $C = \{f_6(x,y,z) = 0\}$ in $\mathbb{P}^2$, and consider the double cover $X_{f_6}$ defined by $z^2 = f_6$ in the appropriate weighted projective space. This is known to be a K3 surface, and the rank of $Pic(X)$ is known to be generically $1$ (with respect to the choice of $f_6$).

It is known that the rank of $Pic(X)$ may jump for particular choices of $f_6$. For example, there might be a tangent conic or a tritangent line to the curve $C$. The existence of such algebraic cycles follows from the Lefschetz (1,1) theorem, though the beautiful underlying geometry seems specific to the situation.

Now for my question, and I apologize for the lack of specificity, but I would dearly love to see similar examples to this phenomenon. Any and all references would be appreciated. Examples where one has family of polynomials ${f}$ and a family of varieties $X_{f}$, such that one can explicitly (if partially) predict the rank jumping of the transcendental invariants of $X_{f}$ by means of imposing reasonably geometric conditions on the polynomials $f$.

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    $\begingroup$ Smooth hypersurfaces in $\mathbb{P}^3$ of degree $d\geq 4$ have "extra" integral $(1,1)$-classes whenver the surface contains a specified curve in projective space of degree not divisible by $d$, e.g., when it contains a line. $\endgroup$
    – Jason Starr
    Commented Dec 1, 2017 at 20:04
  • $\begingroup$ I would also like to give the example of cubic fourfolds in $\mathbf P^5$, which generically have $h^{2,2}=1$ but acquire extra integral classes when the fourfold contains a surface which is not a complete intersection. Whether such containment is a "reasonably geometric" condition on the polynomial defining the variety is open to debate. $\endgroup$
    – Pooter
    Commented Dec 4, 2017 at 14:37

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