Abstract Niels Lindner
Density of hypersurfaces with defect
Over a field of characteristic zero, a general hypersurface in projective space is smooth by Bertini's theorem. For finite fields, the situation is somewhat different: By a result of Bjorn Poonen, the asymptotic fraction of singular hypersurfaces among all hypersurfaces defined over the ground field turns out to be very close to zero, but yet positive. This small failure of Bertini's theorem seems to stem from singular hypersurfaces without defect, which are known as singular Q-factorial threefold hypersurfaces in the special case of projective 4-space. The crucial step is to link certain aspects of the cohomology of a hypersurface with the number of its singularities. This connection can be made rigorous in characteristic zero, and this method is expected to generalize to fields of positive characteristic.