We show that any smooth projective cubic hypersurface of dimension at least 29 over the rationals contains a rational line. A variation of our methods provides a similar result over p-adic fields. In both cases, we improve on previous results due to the second author and Wooley.We include an appendix in which we highlight some slight modifications to a recent result of Papanikolopoulos and Siksek. It follows that the set of rational points on smooth projective cubic hypersurfaces of dimension at least 29 is generated via secant and tangent constructions from just a single point.

中文翻译：

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三次超曲面上的有理线

我们表明，任何在有理数上至少为 29 维的光滑射影三次超曲面都包含一条有理线。我们方法的一种变体提供了类似的结果p-adic 领域。在这两种情况下，由于第二作者和 Wooley，我们改进了之前的结果。我们包括一个附录，其中我们强调了对 Papanikolopoulos 和 Siksek 最近结果的一些细微修改。由此可见，尺寸至少为 29 的光滑射影三次超曲面上的有理点集是通过仅从单个点的割线和切线构造生成的。