Let $X$ be a projective smooth surface. I read two different definitions of $X$ to be a ruled surface, namely:
- $X$ is birational equivalent to some $B\times \mathbb P^1$, where $B$ is projective smooth curve.
- Through every point on $X$ there exists a projective line. (I am a little confused about this: this means $X$ is already embedded into some $\mathbb P^n$? or means through every point there exists some curve which $\cong \mathbb P^1$?)
I want to know are these two definitions equivalent?