Linear system of hypersurfaces passing through a Galois orbit

Author:

Shamil Asgarli, Dragos Ghioca, Zinovy Reichstein

Keyword:

Mathematics, Algebraic Geometry, Algebraic Geometry (math.AG), Number Theory (math.NT)

journal:

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date:

2023-10-15 16:00:00

Abstract

Let $d$ and $n$ be positive integers, and $E/F$ be a separable field extension of degree $m=\binom{n+d}{n}$. We show that if $|F| > 2$, then there exists a point $P\in \mathbb{P}^n(E)$ which does not lie on any degree $d$ hypersurface defined over $F$. In other words, the $m$ Galois conjugates of $P$ impose independent conditions on the $m$-dimensional $F$-vector space of degree $d$ forms in $x_0, x_1, \ldots, x_n$. As an application, we determine the maximal dimension of an $F$-linear system $\mathcal{L}$ of hypersurfaces such that every $F$-member of $\mathcal{L}$ is irreducible over $F$.