Degrees of closed points on hypersurfaces

2023/04/07 by Balestrieri, Francesca
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2304.04562

Abstract

Let k be any field. Let X ⊂ ℙ_k^N be a degree d ≥ 2 hypersurface. Under some conditions, we prove that if X(K) ≠ ∅ for some extension K/k with n:=[K:k] ≥ 2 and gcd(n,d)=1, then X(L) ≠ ∅ for some extension L/k with gcd([L:k], d)=1, n \nmid [L:k], and [L:k] ≤ nd-n-d. Moreover, if a K-solution is known explicitly, then we can compute L/k explicitly as well. As an application, we improve upon a result by Coray on smooth cubic surfaces X ⊂ ℙ³_k by showing that if X(K) ≠ ∅ for some extension K/k with gcd([K:k], 3)=1, then X(L) ≠ ∅ for some L/k with [L:k] ∈ \1, 10\.

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