Solvable points on intersections of quadrics, cubics, and quartics

TL;DR

This paper establishes lower bounds on the dimension needed for intersections of quadrics, cubics, and quartics to have many solvable points over a solvable closure, using polar hypersurfaces and Fano varieties.

Contribution

It introduces a novel approach linking polar hypersurfaces to Fano varieties to analyze solvable points on complex intersections.

Findings

Derived polynomial lower bounds on ambient dimension for solvable points

Connected classical polar hypersurfaces to Fano varieties for better arithmetic control

Improved understanding of the distribution of solvable points on intersections

Abstract

Let $k$ be a field of characteristic not 2 or 3. We establish polynomial lower bounds on the ambient dimension $N$ for an intersection $X\subset\mathbb{P}^N$ of quadrics, cubics and quartics to have a dense collection of solvable points, i.e. points in $X(k^{\mathsf{Sol}})$ where $k^{\mathsf{Sol}}/k$ is a solvable closure. Our method connects the classical theory of polar hypersurfaces, as redeveloped by Sutherland, to Fano varieties $\mathcal{F}(j,X)$ of $j$-dimensional linear subspaces on $X$, and we use this to obtain improved control on the arithmetic of $\mathcal{F}(j,X)$.