Torsion orders of Fano hypersurfaces

TL;DR

This paper establishes new lower bounds on torsion orders of general Fano hypersurfaces, impacting their unirational parametrizations and rationality, especially in characteristic two.

Contribution

It provides the first bounds applicable in arbitrary characteristic, including characteristic two, extending previous results and addressing the rationality problem.

Findings

Lower bounds on torsion orders for Fano hypersurfaces

Implications for the degree of unirational parametrizations

Resolution of the rationality problem in characteristic two

Abstract

We find new lower bounds on the torsion orders of very general Fano hypersurfaces over (uncountable) fields of arbitrary characteristic. Our results imply that unirational parametrizations of most Fano hypersurfaces need to have enormously large degree. Our results also hold in characteristic two, where they solve the rationality problem for hypersurfaces under a logarithmic degree bound, thereby extending a previous result of the author from characteristic different from two to arbitrary characteristic.