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 con-jecture of Colliot-The ' le`ne 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 hy-persurfaces of fixed degree and dimension by height, we prove that almost every such hypersurface satisfies the Hasse principle provided that the di-mension is at least 3. This proves a conjecture of Poonen and Voloch in every case except for cubic surfaces.