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Is it true that any family of K3 surfaces over $\mathbb{C}$ whose Picard number is constant is isotrivial? Here isotrivial means locally analytically trivial.

Speculation: Let $\mathcal{M}$ be the moduli space of K3 surfaces over $\mathbb{C}$. It is known that for any point $[S]\in \mathcal{M}$ its open neighborhood $U$ contains a K3 surface $[T]$ such that $Pic(S)>Pic(T)$ and such points are dense. So basically I am asking whether I can take a family avoiding these points or not.

Thank you in advance.

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You can take arbitrary family and remove the points where the Picard number jumps. –  Sasha Nov 22 '12 at 10:24

2 Answers 2

The answer to your question is yes. This is proven in a paper by Oguiso.

http://arxiv.org/abs/math/0011258

There is a slightly more general criterion for the density of Hodge loci which appears for instance in this survey of Voisin (section 3.2)

http://www.math.polytechnique.fr/~voisin/Articlesweb/hodgeloci.pdf

and is attributed to Mark Green.

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I think the answer to your question is yes:

http://arxiv.org/abs/alg-geom/9701013

Edit: As Sasha points out, this is only for compact families.

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Isn't it about compact families only? –  Sasha Nov 22 '12 at 10:23
    
Sorry, yes I was assuming that was what was wanted! –  Jonny Evans Nov 22 '12 at 12:18
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The result holds more generally even if the base isn't compact; by a result of green, the jumping locus is analytically dense so you can't just remove the points. –  PRL Nov 22 '12 at 13:01
    
Sorry - edited to reflect comments. –  Jonny Evans Nov 22 '12 at 16:21

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