Degrees of closed points on hypersurfaces

Francesca Balestrieri

Mathematische Nachrichten, 2024, vol. 297, issue 5, 1831-1837

Abstract: Let k$k$ be any field. Let X⊂PkN$X \subset \mathbb {P}_k^N$ be a degree d≥2$d \ge 2$ hypersurface. Under some conditions, we prove that if X(K)≠∅$X(K) \ne \emptyset$ for some extension K/k$K/k$ with n:=[K:k]≥2$n:=[K:k] \ge 2$ and gcd(n,d)=1$\gcd (n,d)=1$, then X(L)≠∅$X(L) \ne \emptyset$ for some extension L/k$L/k$ with gcd([L:k],d)=1$\gcd ([L:k], d)=1$, n∤[L:k]$n \nmid [L:k]$, and [L:k]≤nd−n−d$[L:k] \le nd-n-d$. Moreover, if a K$K$‐solution is known explicitly, then we can compute L/k$L/k$ explicitly as well. As an application, we improve upon a result by Coray on smooth cubic surfaces X⊂Pk3$X \subset \mathbb {P}^3_k$ by showing that if X(K)≠∅$X(K) \ne \emptyset$ for some extension K/k$K/k$ with gcd([K:k],3)=1$\gcd ([K:k], 3)=1$, then X(L)≠∅$X(L) \ne \emptyset$ for some L/k$L/k$ with [L:k]∈{1,10}$[L:k] \in \lbrace 1, 10\rbrace$.

Date: 2024
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