Linear system of hypersurfaces passing through a Galois orbit

2023/10/16 by Asgarli, Shamil, Ghioca, Dragos, Reichstein, Zinovy
#14G15 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT) #Primary 14N05 #Secondary 14J70

paper · doi:10.48550/arxiv.2310.10361

Abstract

Let d and n be positive integers, and E/F be a separable field extension of degree m=\binomn+dn. We show that if |F| > 2, then there exists a point P∈ ℙⁿ(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 x₀, x₁, …, x_n. As an application, we determine the maximal dimensions of linear systems L₁ and L₂ of hypersurfaces in \mathbb Pⁿ over a finite field F, where every F-member of L₁ is reducible and every F-member of L₂ is irreducible.

Related

• Hypersurfaces passing through the Galois orbit of a point
• Linear families of smooth hypersurfaces over finitely generated fields
• Linear automorphisms of smooth hypersurfaces giving Galois points
• Low degree subvarieties of universal hypersurfaces
• Degrees of closed points on hypersurfaces