A transfer principle for unirationality

TL;DR

This paper introduces a transfer principle linking polynomial strength to unirationality of hypersurfaces, providing new cases and an alternative proof for existing results, advancing understanding in algebraic geometry.

Contribution

It establishes that high collective strength in polynomials ensures unirationality and introduces a transfer principle connecting polynomial strength to properties of Fano schemes.

Findings

High strength polynomials produce unirational hypersurfaces.

Unirationality is preserved under substitution of high collective strength.

Provides an alternative proof of a result by Xi Chen and advances the de Jong-Debarre Conjecture.

Abstract

We apply ideas related to the strength of polynomials to provide new cases of unirational hypersurfaces. It is famously known that hypersurfaces that are smooth in very high codimension are unirational, and a simple corollary then implies that any polynomial of sufficiently high strength will give rise to a unirational hypersurface. Our main result shows that unirationality is preserved under a substitution of high collective strength. In particular, we prove that polynomials of sufficiently high secondary strength are unirational. Along the way, we introduce a ``transfer principle,'' showing that polynomials of high collective strength have Fano schemes defined by polynomials of high collective strength. This gives an alternate proof of a result of Xi Chen on unirationality of Fano schemes, and proves a weakened form of the de Jong-Debarre Conjecture. Combined with some ideas of Starr, this implies a version of Kazhdan and Ziegler's result about the universality of complete intersections of polynomials.