Let $X \subset \mathbb{P}^3$ be a non-singular surface defined over $\mathbb{Q}$ of degree $d \geq 3$. It is a theorem of Colliot-Thelene (see the appendix to this paper: <http://www.jstor.org/stable/3062125?seq=1#page_scan_tab_contents>) that there are finitely many curves of degree $\delta \leq d-2$ contained in $X$. Let $\mathcal{C}(X)$ denote this finite set of curves. Since each curve is defined over some number field whose degree over $\mathbb{Q}$ is finite, it follows that there is a unique largest finite extension of $\mathbb{Q}$, which we denote by $K = K\_{\mathcal{C}(X)}$, for which all curves in $\mathcal{C}(X)$ are defined over $K$.


Do we know anything about this field $K$? For example, can we bound $[K:\mathbb{Q}]$ in terms of $d$, or maybe the coefficients of the polynomial defining $X$? A quick literature review turned up nothing. Any help would be appreciated.



A certain invariant of non-singular algebraic surfaces
I will explain what happens in the case of cubic surfaces. As is well known, any smooth cubic surface over an algebraically closed field contains $27$ lines. For the generic cubic surface over $\mathbb{Q}$, its configuration space of lines has Galois group $W(E\_6)$ (Weyl group of the $E\_6$ root system). Hence an application of Hilbert's irreducibility theorem shows that for "most" cubic surfaces over $\mathbb{Q}$, the Galois group of its splitting field is $W(E\_6)$, hence has degree $\#W(E\_6)=51840$.


An explicit example of such a surface is
$$11w^3 + 56w^2x + 33x^3 + 28x^2y + 11y^3 + 14y^2z + 28wz^2 + 11z^3 = 0,$$
as found by Ekedahl in his paper "An effective version of Hilbert's irreducibility theorem".


Of course the case of cubic surfaces is very special. In general I don't see how one can do any better than what Jason Starr suggests.


