2025/01/03 by Asgarli, Shamil, Love, Jonathan, Yip, Chi Hoi
#11G25 #14J70 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT) #Primary: 14G15 #Secondary: 14N05
paper · doi:10.48550/arxiv.2501.01906
Asgarli, Ghioca, and Reichstein recently 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=\binomn+dd, there exists a point P∈ ℙn(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 ∈ ℙn(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.