Crepant resolutions of fiber products and Kummer fibrations of rational elliptic surfaces with non-reduced fibers

RESUMO

Let $S_1$ and $S_2$ be rational elliptic surfaces with section over an algebraically closed field of characteristic zero, and let $X=S_1\times_{\mathbb{P}^1}S_2$. We determine the local equations produced by pairs of Kodaira fibers, including non-reduced fibers, and construct crepant resolutions for the admissible local models. For every construction we record whether it is projective and compute the number of irreducible components and the Euler characteristic of the resolved central fiber. Non-existence is asserted only in the normal Gorenstein cases covered by the terminal factorial obstruction; the remaining entries are stated as open for the methods of this paper. Under an explicit global compatibility hypothesis, the local data give formulas for the Hodge numbers of a smooth projective resolved fiber product. We also study Kummer quotients by fiberwise involutions. Their fixed curves are analyzed through the monodromy-orbit description of normalized fiber products of multisections. The final section records what the same constructions imply after reduction in characteristics $5$ and $7$, without interpreting characteristic-$p$ Euler and Picard data as complex Hodge numbers.