Surface bundles and the section conjecture
Surface bundles and the section conjecture
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DOI：
10.1007/s00208-022-02421-9
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发表时间：
2020-10
影响因子：
1.4
通讯作者：
Wanlin Li;Daniel Litt;Nick Salter;P. Srinivasan
中科院分区：
文献类型：
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作者：
Wanlin Li;Daniel Litt;Nick Salter;P. Srinivasan
我们制定了一个热带类似的Grothendieck的部分猜想：对每一个稳定的图的属，和每一个字段k，一般曲线与减少类型超过k满足部分猜想。我们证明了这个猜想的许多情况。在这样做，我们显示了许多例子的曲线没有合理的点满足部分猜想领域的几何兴趣，然后通过Chebotarev参数在p-进领域和数域的存在。我们构造了两个Galois上同调类和，它们阻碍了-截面的存在，从而阻碍了有理点的存在。首先是一个阿贝尔障碍，密切相关的曲线的周期和上同调类的moduli space的curvesstudied森田。第二个是2-幂零阻塞，似乎是新的。我们研究退化这些类通过拓扑技术，我们产生的例子，这些类阻碍部分的表面上的表面束。然后，我们使用这些结构，以显示存在的曲线上的p-adic领域和数域，每个类的障碍部分，因此合理的点。在我们的几何结果是一个新的证明部分猜想的一般曲线的属，并证明部分猜想的一般曲线的甚至属与一个合理的除数类的程度1（其中的障碍，以存在一个部分是真正的非阿贝尔）。
We formulate a tropical analogue of Grothendieck’s section conjecture: that for every stable graphof genus, and every fieldk, the generic curve with reduction typeoverksatisfies the section conjecture. We prove many cases of this conjecture. In so doing we show the existence of many examples of curves with no rational points satisfying the section conjecture over fields of geometric interest, and then overp-adic fields and number fields via a Chebotarev argument. We construct two Galois cohomology classesand, which obstruct the existence of-sections and hence of rational points. The first is an abelian obstruction, closely related to the period of a curve and to a cohomology class on the moduli space of curvesstudied by Morita. The second is a 2-nilpotent obstruction and appears to be new. We study the degeneration of these classes via topological techniques, and we produce examples of surface bundles over surfaces where these classes obstruct sections. We then use these constructions to show the existence of curves overp-adic fields and number fields where each class obstructs-sections and hence rational points. Among our geometric results are a new proof of the section conjecture for the generic curve of genus, and a proof of the section conjecture for the generic curve of even genus with a rational divisor class of degree one (where the obstruction to the existence of a section is genuinely non-abelian).