论文标题
随机Fano Hypersurfaces的Hasse原理
The Hasse principle for random Fano hypersurfaces
论文作者
论文摘要
众所周知，对哈斯原理的brauer the骨障碍物对于平滑的尺寸的流畅性高度弯曲至少$ 3 $，在任何数字字段上都是空虚的。此外，对于此类品种，它遵循的是Colliot-Thélène的一般猜想，即Brauer-Manin tossion to to Hasse原理应该是唯一的一种，因此预计Hasse原则将持有。在理性数字领域工作，并按照高度订购固定学位和尺寸的Fano Hypersurfaces，我们证明，几乎所有这样的超表面都满足了Hasse原则，前提是尺寸至少为$ 3 $。在每种情况下，这都是Poonen和Voloch的猜想，除了立方表面。
It is known that the Brauer--Manin obstruction to the Hasse principle is vacuous for smooth Fano hypersurfaces of dimension at least $3$ over any number field. Moreover, for such varieties it follows from a general conjecture of Colliot-Thélène that the Brauer--Manin obstruction to the Hasse principle should be the only one, so that the Hasse principle is expected to hold. Working over the field of rational numbers and ordering Fano hypersurfaces of fixed degree and dimension by height, we prove that almost every such hypersurface satisfies the Hasse principle provided that the dimension is at least $3$. This proves a conjecture of Poonen and Voloch in every case except for cubic surfaces.
