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$\begingroup$ Thank you! Two points would like to clarify: You mentioned that for the Kodaira dimension we have always $\kappa(X) \geq \kappa(J)$ and is even an equality if $X \to B$ has no multiple fibres. Then cases $X$ rational and $X$ Enriques (esp. $X \to B$ admits multiple fibres, so we have a drop) are treated in Cossec and Dolgachev in 5.6, resp. 5.7. Could you elaborate the ideas or give a reference treating general scenario? Can this be extracted from $P_n(J) = n(2g_B-2+\chi(\mathcal{O}_X))+1-g_B$? $\endgroup$

user267839
– user267839

2024-11-27 01:52:36 +00:00
Commented Nov 27, 2024 at 1:52
$\begingroup$ Also, could you maybe elaborate the part with why $J$ neccessarily happens to be an abelian surface only if it is a product in the case above? $\endgroup$

user267839
– user267839

2024-11-27 01:52:53 +00:00
Commented Nov 27, 2024 at 1:52

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