It is known that every smooth cubic surface with coefficients in $\mathbb{Q}$ has $27$ lines defined over a number field extension of $\mathbb{Q}$ of degree at most $51840$ as the group $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ acting on the lines is a subgroup of the Weyl group $W(E_6)$ which has order $51840$.

I was wondering if a similar argument can prove the existence of an absolute constant $c$ such that whenever a smooth cubic surface has coefficients in a number field $K$ then all $27$ lines are defined in a number field extension $L$ of $K$ of degree at most $[L:K]\leq c.$

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    $\begingroup$ The same argument works over any field. If $X$ is the moduli space of rational cubic surfaces, then there is a Galois cover $Y \rightarrow X$ of degree $27$ corresponding to the lines, and the Galois group is $W(E_6)$. So, for any field $K$, one has the same bound. Conversely, for a number field, Hilbert irreducibility implies that for a generic cubic surface (outside a thin set in the sense of Serre) over $K$ will have lines defined over a field $L$ with the largest possible Galois group $W(E_6)$. Of course, over some fields (like $\mathbf{C}$ or $\mathbf{F}_p$) the bound will be smaller. $\endgroup$ Jan 13, 2017 at 4:38


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