The Hasse principle for random Fano hypersurfaces

Tim Browning, Pierre Le Boudec, Will Sawin

Research output: Contribution to journalArticlepeer-review

Abstract

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.

Original languageEnglish (US)
Pages (from-to)1115-1203
Number of pages89
JournalAnnals of Mathematics
Volume197
Issue number3
DOIs
StatePublished - 2023
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Mathematics (miscellaneous)

Keywords

  • Fano hypersurfaces
  • Hasse principle
  • heights
  • rational points

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