AI 中文总结
本文研究有理椭圆曲面纤维积的 crepant 消解与 Kummer 商，给出局部模型、Hodge 数公式及特征 5、7 下的约化结果。
AI 中文摘要
设 $S_1$ 和 $S_2$ 是特征零代数闭域上带截面的有理椭圆曲面，并设 $X=S_1\times_{\mathbb{P}^1}S_2$。我们确定由 Kodaira 纤维对（包括非约化纤维）产生的局部方程，并为可容许的局部模型构造 crepant 消解。对于每个构造，我们记录其是否为射影的，并计算消解后中心纤维的不可约分支数和 Euler 特征。仅在由 terminal factorial 障碍覆盖的正规 Gorenstein 情形中，我们断言不存在性；其余条目对于本文的方法而言是开放的。在显式的全局相容性假设下，局部数据给出光滑射影消解纤维积的 Hodge 数的公式。我们还研究由纤维方向对合得到的 Kummer 商。它们的固定曲线通过多截面的约化纤维积的 monodromy 轨道描述来分析。最后一节记录了相同构造在特征 $5$ 和 $7$ 约化后所蕴含的结果，但不将特征 $p$ 的 Euler 和 Picard 数据解释为复 Hodge 数。
英文摘要
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.
Comments34 pages
