The Hasse principle for random Fano hypersurfaces


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.


Tim Browning

Institute of Science and Technology Austria, Klosterneuburg, Austria

Pierre Le Boudec

Departement Mathematik und Informatik, Fachbereich Mathematik, Universität Basel, Basel, Switzerland

Will Sawin

Columbia University, New York, NY