The Hasse principle for random Fano hypersurfaces
Number Theory
2020-06-04 v1
Abstract
It is known that the Brauer--Manin obstruction to the Hasse principle is vacuous for smooth Fano hypersurfaces of dimension at least over any number field. Moreover, for such varieties it follows from a general conjecture of Colliot-Th\'el\`ene 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 . This proves a conjecture of Poonen and Voloch in every case except for cubic surfaces.
Keywords
Cite
@article{arxiv.2006.02356,
title = {The Hasse principle for random Fano hypersurfaces},
author = {Tim Browning and Pierre Le Boudec and Will Sawin},
journal= {arXiv preprint arXiv:2006.02356},
year = {2020}
}