Let $X$ be a smooth, complex projective algebraic variety defined over a number field $K$.
Let $D$ be a divisor of $X$ defined over $K$ with the following property:

> For any curve $C$ defined over $K$, we have $\operatorname{deg (D_{|C})=0}$

Is it then true that $c_1(D)=0$?


In general, in order to have $c_1(D)=0$ ,I should check that $\operatorname{deg (D_{|C})=0}$
 for *any curve* (not just the ones defined over $K$). I'm asking if in this particular setting, the curves defined over $K$ are enough.