2018/09/30 by Julia Brandes, Rainer Dietmann
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Cubic surface #Dimension (graph theory) #Geometry #Hypersurface #Line (geometry) #Mathematical analysis #Mathematics #Meromorphic and Entire Functions #Point (geometry) #Projective line #Projective space #Projective test #Pure mathematics #Rational number #Tangent #math.NT #msc:11D72 #msc:11D88 #msc:11E76
paper · pdf · doi:10.1017/s0305004120000079
published as Math. Proc. Camb. Phil. Soc. 171 (2021) 99-112 · An oversight in Lemma 3.1 as well as a few typos have been corrected
arxiv created 2020/04/05 · openalex publication_date 2020/04/24 · arxiv updated 2021/07/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
Abstract 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.