Unirationality of hypersurfaces via highly tangent lines

TL;DR

This paper presents a new method for proving the unirationality of general low degree hypersurfaces using highly tangent lines, significantly improving classical bounds for when such hypersurfaces are unirational.

Contribution

It introduces a novel unirationality construction based on highly tangent lines, extending results to higher degrees and dimensions with improved bounds.

Findings

Unirationality holds for hypersurfaces of degree d ≥ 6 in projective n-space when n ≥ 2^{(d-1)2^{d-5}}.

The method improves classical bounds for the unirationality of hypersurfaces.

The approach applies to general low degree complete intersections in projective space.

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.