Mori dream 纤维与几何一般纤维
Mori dream fibers and the geometric generic fiber
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- School of Mathematics, Korea Institute for Advanced Study（韩国高等科学研究院数学学院）
- Gakushuin University（学习院大学）
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中文总结 AI 辅助
本文构造了有理曲面族，证明其纤维的 Mori dream 性质由法丛挠性决定，并给出纤维化成为 Mori dream 态射的充分条件。
中文摘要 AI 辅助
我们在 $\mathbb{G}_{m,\mathbb{Z}}$ 上构造了一个光滑的射影有理曲面族。纤维的 Mori dream 性质由反典范循环的法丛的挠性决定。在 $\mathbb{C}$ 上，Mori dream 纤维的轨迹是 Zariski 稠密的。对于每个素数 $p$，模 $p$ 约化闭点上的每个几何纤维都是 Mori dream 曲面，而几何一般纤维不是 Mori dream 空间。在这两种情形下，限制到任何非空开子集都不是 Mori dream 态射。我们还证明了，在任意代数闭域上，只要具有 Mori dream 纤维的点集不包含在可数个真闭子集的并集中，射影纤维化在收缩基后就会成为 Mori dream 态射。
英文摘要
We construct a smooth projective family of rational surfaces over $\mathbb{G}_{m,\mathbb{Z}}$. The Mori dream property of a fiber is determined by the torsion of the normal bundle of an anticanonical cycle. Over $\mathbb{C}$, the locus of Mori dream fibers is Zariski dense. For every prime $p$, every geometric fiber over a closed point of the reduction modulo $p$ is a Mori dream surface, whereas the geometric generic fiber is not a Mori dream space. In either setting, no restriction to a nonempty open subset is a Mori dream morphism. We also prove that, over any algebraically closed field, a projective fibration becomes a Mori dream morphism after shrinking the base whenever the set of points with Mori dream fibers is not contained in a countable union of proper closed subsets.
