Degrees of closed points on hypersurfaces

Abstract

Let $k$ be any field. Let $X \subset \mathbb{P}_k^N$ be a degree $d \geq 2$ hypersurface. Under some conditions, we prove that if $X(K) \neq \emptyset$ for some extension $K/k$ with $n:=[K:k] \geq 2$ and $\gcd(n,d)=1$, then $X(L) \neq \emptyset$ for some extension $L/k$ with $\gcd([L:k], d)=1$, $n \nmid [L:k]$, and $[L:k] \leq nd-n-d$. 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 $X \subset \mathbb{P}^3_k$ by showing that if $X(K) \neq \emptyset$ for some extension $K/k$ with $\gcd([K:k], 3)=1$, then $X(L) \neq \emptyset$ for some $L/k$ with $[L:k] \in \{1, 10\}$.