Example of Picard number in family of smooth variety jumping

For a scheme or formal scheme $$X$$, let $$\mathrm{Pic} X$$ be its Picard group. If $$X$$ is a smooth proper variety over an algebraically closed field, let $$\mathrm{Pic}^{0}(X)$$ be the subgroup consisting of isomorphism classes of line bundles algebraically equivalent to $$\mathscr{O}_{X}$$, and define the Neron-Severi group $$\mathrm{NS}X :=\mathrm{Pic} X/\mathrm{Pic}^{0}(X)$$. By Neron-Severi theorem, $$\mathrm{NS}X$$ is finitely generated and its rank call the Picard number of X, denote as $$\rho(X)$$

Now suppose $$X$$ and $$S$$ are two varieties, $$f: X\to S$$ is a smooth proper morphism, let $$X_{b}$$ be the fiber of $$X$$ over $$b$$. My question is following:

Could you show me one example that the function $$h: S\to \mathbb{Z}: h(b)=\rho({X_{b}})$$ is not locally constant?

I use google to search this question. But I can only find that this conclusion is true and can not find a specific example. So could you give me one specific example? Thank you very much!

• Take $S$ to be the open set of a projective space parametrizing degree $d\geq 4$ smooth hypersurfaces in $\mathbb{P}^3$, with $X$ the universal hypersurface over $S$. Then $\rho$ is not locally constant. (Try googling Noether-Lefschetz theorem). May 31 '20 at 15:26
• @Mohan I googled it and I could only find the statement of Noether-Lefschetz theorem, that is the general hypersurface of degree $d\geq 4$ has the Picard group $Z$. But I could not find why $\rho$ is not locally constant? Could you recommend me some more specific reference about it? Thanks a lot.
– Mike
May 31 '20 at 18:52
• Connect a general hypersurface with Picard group $\mathbb{Z}$ and the Fermat hypersurface of degree $d$. The Fermat has Picard group large and show that taking $S$ as this line (open set of a line containing only smooth ones), show that $\rho$ is not locally constant. May 31 '20 at 18:56
• @Mohan I see, thank you very much. By the way what is exactly Picard number of $x^4+y^4+z^4+w^4$?
– Mike
May 31 '20 at 20:29
• I have found the answer, the Picard number is 20.
– Mike
May 31 '20 at 20:37

To give such an example, we need to use two facts.

Noether-Lefschetz theorem: If $$S_{d}\subseteq \mathbb{P}^{3}$$ is a very general surface, then $$\mathrm{Pic}(S_{d})\cong \mathbb{Z}$$.

Hence we could choose a quartic surface $$S_{0}$$ in $$\mathbb{P}^{3}$$, such that $$\rho(S_{0})\leq 1$$.

The Picard number of Fermat's surface $$S_{F}: x^4+y^4+z^4+w^4=0$$ is 20.

Construction: Let $$Y$$ be the open set of a projective space parametrizing degree $$d = 4$$ smooth hypersurfaces in $$\mathbb{P}^{3}$$ , $$X$$ be the universal hypersurface over $$Y$$. Then $$\rho$$ is not locally constant, since $$\rho(S_{0})\leq 1, \rho(S_{F})=20$$.

For an elliptic curve $$E$$ over $$\mathbb{Z}$$, the Picard number of $$E \times E$$ is $$3$$ if $$E$$ has no complex multiplication (generated by horizontal, vertical, diagonal classes) and 4 if it does (also take the class of the graph of a non-integer endomorphism).

So you can take any family of elliptic curves which includes all isomorphism classes, e.g. the Legendre family $$y^2 = x(x-1)(x-\lambda),$$ and consider the fiber product of this family with itself over the base to get a family of elliptic surfaces where the Picard number jumps at special points.