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.