2
$\begingroup$

Let $f:X\rightarrow S$ be a family of smooth projective varieties. For $s\in S$ set $X_s := f^{-1}(s)$, and let $\rho(X_{s})$ be the Picard number of the fiber over $s\in S$. Fix a point $s_0\in S$.

Does there exist a Zariski open subset $U\subset S$ such that $s_0\in U$ and $\rho(X_s)\leq\rho(X_{s_0})$ for all $s\in U$?

Improve this question
$\endgroup$
2
  • 4
    $\begingroup$ No, in general the locus where the Picard rank jumps is a countable union of Zariski closed sets and if we are over the complex numbers, this union can even be dense in the classical topology. This happens for example for K3 surfaces where e.g. the locus of Picard rank 2 K3s inside the moduli space of degree d K3s is a countable union over all possible rank 2 lattice polarizations. $\endgroup$
    – Dori Bejleri
    Commented Nov 5, 2022 at 23:06
  • $\begingroup$ Dear @DoriBejleri, thanks a lot for your answer. Do you have a reference for the claim "the locus where the Picard rank jumps is a countable union of Zariski closed sets"? Thank you. $\endgroup$
    – Srinivasa Granujan
    Commented 2 days ago

0

Reset to default

You must log in to answer this question.