Degrees of closed points on diagonal-full hypersurfaces

Abstract

Let $k$ be any field. Let $X \subset \mathbb{P}_k^N$ be a diagonal-full degree $d$ hypersurface, where $d$ is an odd prime. We prove that if $X(K) \neq \emptyset$ for some extension $K/k$ with $n:=[K:k]$ prime and $gcd(n,d)=1$, then $X(L) \neq \emptyset$ for some extension $L/k$ with $gcd([L:k], nd)=1$ and $[L:k] \leq nd-n-d$. Moreover, if a $K$-solution is known explicitly, then we can compute $L/k$ explicitly as well. When $n$ or $d$ is not prime, we can still say something about the possible values of $[L:k]$. As an example, we improve upon a theorem by Coray on smooth cubic surfaces $X \subset \mathbb{P}^3_k$, in the case when $X$ is diagonal-full, 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 $[L:k] \in \{1, 10\}$.Comment: Comments welcome