Hypersurfaces passing through the Galois orbit of a point

TL;DR

This paper extends a result about points not lying on certain hypersurfaces over fields of size greater than two, confirming the result for size two and exploring related geometric properties.

Contribution

It proves that the previous hypersurface avoidance result holds for fields of size two and generalizes to points with prescribed vanishing properties in linear systems.

Findings

Confirmed the existence of points outside hypersurfaces over fields of size two.

Generalized the result to points with specific vanishing conditions in linear systems.

Discussed applications to special properties of hypersurfaces and linear systems.

Abstract

Asgarli, Ghioca, and Reichstein proved that if $K$ is a field with $|K|>2$, then for any positive integers $d$ and $n$, and separable field extension $L/K$ with degree $m=\binom{n+d}{d}$, there exists a point $P\in \mathbb{P}^n(L)$ which does not lie on any degree $d$ hypersurface defined over $K$. They asked whether the result holds when $|K| = 2$. We answer their question in the affirmative by combining various ideas from arithmetic geometry. More generally, we show that for each positive integer $r$ and separable field extension $L/K$ with degree $r$, there exists a point $P \in \mathbb{P}^n(L)$ such that the vector space of degree $d$ forms over $K$ that vanish at $P$ has the expected dimension. We also discuss applications to linear systems of hypersurfaces with special properties.