Degrees of closed points on hypersurfaces

Abstract

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

arXiv commentThe structure of the paper has been changed and the main theorem is now the one for the more general case; suggestions and comments from the referees have been incorporated