Singularities of generic projection hypersurfaces in arbitrary characteristic

Algebraic and Tropical Geometry
Event time: 
Thursday, October 24, 2019 - 3:00pm
Rosenfeld Hall (109 Grove St)
Takumi Murayama
Speaker affiliation: 
Princeton University
Event description: 

Classically, it is known that every algebraic variety is birational to a hypersurface in some projective or affine space. Using generic linear projections, Doherty proved that over the complex numbers, this hypersurface can be taken to have at worst semi-log canonical singularities in dimensions up to five. This extends classically known results for curves and surfaces. We present a positive-characteristic analogue of Doherty's theorem, by showing that in the same dimensions, generic projection hypersurfaces in positive characteristic are F-pure. This proves cases of a conjecture of Bombieri, Andreotti, and Holm. To show our result, we study F-injective singularities, which are the analogue of Du Bois singularities in positive characteristic, and their behavior under flat morphisms. This work is joint with Rankeya Datta.