Assume that $C$ is a projective curve and $X$ is an elliptic fibration over $C$.
What is the picard group of $X$? can we say something about it?
I think it should be generated by multisections and (components of ) fibres. How can I see it formally?
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3$\begingroup$ Every integral divisor on $X$ is either horizontal (multisection) or vertical, so indeed, the Picard group is generated by components of the fibres and multisections. But, you can't say much more in full generality. Just consider the examples: $C\times E$ over $C$ with $E$ an elliptic curve, or an elliptic K3 surface $X\to \mathbb{P}^1$ (for which the Picard rank could be essentially anything between 11 and 20), Enriques surfaces and Kodaira dimension one surfaces... $\endgroup$– Ariyan JavanpeykarOct 27, 2022 at 13:29
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2$\begingroup$ You can replace multi sections with sections as long as there is at least one section, using the group law. $\endgroup$– Will SawinOct 27, 2022 at 13:37
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1 Answer
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Do you know about the Shioda-Tate formula. (I'm surprised there isn't a Wikipedia article about it.) Anyway, it says that if the elliptic fibration does not split as a product and if you have at least one section, then the Neron-Severi group is generated by the following divisors:
- One irreducible fiber.
- For each reducible fiber, take all but one of the components.
- The zero-section.
- Generators for the group of sections.
answered Oct 27, 2022 at 15:11