Unirationality and geometric unirationality for hypersurfaces in positive characteristics

Abstract

Building on work of Segre and Koll'ar on cubic hypersurfaces, we construct over imperfect fields of characteristic p\geq 3 particular hypersurfaces of degree p, which show that geometrically rational schemes that are regular and whose rational points are Zariski dense are not necessarily unirational. A likewise behaviour holds for certain cubic surfaces in characteristic p=2.