The Hasse principle for lines on diagonal surfaces

Jörg Jahnel; Daniel Loughran

doi:10.1017/S0305004115000596

The Hasse principle for lines on diagonal surfaces

Number Theory 2015-12-16 v2 Algebraic Geometry

Authors: Jörg Jahnel , Daniel Loughran

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Abstract

Given a number field $k$ and a positive integer $d$ , in this paper we consider the following question: does there exist a smooth diagonal surface of degree $d$ in $\mathbb{P}^3$ over $k$ which contains a line over every completion of $k$ , yet no line over $k$ ? We answer the problem using Galois cohomology, and count the number of counter-examples using a result of Erd\H{o}s.

Keywords

algebraic surfaces del pezzo surfaces incidence geometry of points and lines

Cite

@article{arxiv.1503.05021,
  title  = {The Hasse principle for lines on diagonal surfaces},
  author = {Jörg Jahnel and Daniel Loughran},
  journal= {arXiv preprint arXiv:1503.05021},
  year   = {2015}
}