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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!

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  • $\begingroup$ 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). $\endgroup$
    – Mohan
    May 31 '20 at 15:26
  • $\begingroup$ @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. $\endgroup$
    – Mike
    May 31 '20 at 18:52
  • $\begingroup$ 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. $\endgroup$
    – Mohan
    May 31 '20 at 18:56
  • $\begingroup$ @Mohan I see, thank you very much. By the way what is exactly Picard number of $x^4+y^4+z^4+w^4$? $\endgroup$
    – Mike
    May 31 '20 at 20:29
  • $\begingroup$ I have found the answer, the Picard number is 20. $\endgroup$
    – Mike
    May 31 '20 at 20:37
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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 $\rho(S_{0})\leq 1, \rho(S_{F})=20$.

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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.

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