Unirationality of hypersurfaces via highly tangent lines
Raymond Cheng
TL;DR
This work introduces a novel unirationality construction for general low-degree complete intersections in projective space, exploiting a family of highly tangent lines (penultimate tangents) and a residual point map to transfer unirationality from fibres to the base. The authors develop an inductive framework on multi-degrees, make precise the notions of pointed lines and penultimate tangents, and prove dominance of the residual map under suitable dimension bounds. In the hypersurface case, they obtain a substantially improved bound: a general hypersurface of degree $d\ge6$ in $P^n$ is unirational whenever $n\ge 2^{(d-1)2^{d-5}}$, with explicit computations and a detailed combinatorial analysis via multiplicity sequences. The approach connects to classical parameterizations and offers a new angle on longstanding questions about rationality and unirationality, providing both concrete bounds and a versatile framework for further refinement and generalization.
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.