Unirationality of hypersurfaces via highly tangent lines

Abstract

This article describes a unirationality construction for general low degree complete intersections in projective space which is based on a variety of highly tangent lines. Applied to hypersurfaces, this implies that a general hypersurface of degree $d \geq 6$ in projective $n$-space is unirational as soon as $n \geq 2^{(d-1)2^{d-5}}$, significantly improving classical bounds.