definitions:

A non-singular complex projective surface $S$ is a ruled surface if it is birationally equivalent to $C\times_{\text{Spec} \mathbb C}\mathbb P^1_{\mathbb C}$ where $C$ is a non-singular complex projective curve.

A non-singular complex projective surface $S$ is a geometrically ruled surface if there exists a surjective morphism $f:S\longrightarrow C$ over a non-singular complex projective curve $C$ such that every fiber $S_x$ ($x\in C$) is isomorphic to $\mathbb P^1_{\mathbb C}$.

Questions:

1. Suppose that $S\cong^{\text{bir}} C \times\mathbb P^1_{\mathbb C}$ is a ruled surface defined over $\mathbb {\overline Q}$, namely $S\cong S_{\overline{\mathbb Q}}\times_{\text{Spec }{\overline{\mathbb Q}}}\text{Spec }{\mathbb C}$, then can I conclude that $C$ is defined over $\overline{\mathbb Q}$?
2. Suppose that $f:S\longrightarrow C$ is a geometrically ruled surface such that the curve $C$ is defined over $\overline{\mathbb Q}$, then can I conclude that $S$ is defined over $\overline{\mathbb Q}$?

Many thanks in advance