This may be a question with a trivial answer. Given a morphism of schemes $X \to \mathbf{P}_k^1$, where $k$ is a field and $\mathbf{P}_k^1$ the projective line over $k$, I am trying to understand what the generic fibre of this morphism is.
By definition - e.g. in Liu's book - the generic fibre of $X_{\eta}$ is just the fibre over the generic point. But what is the generic point of the projective line?
Is it correct to just choose an affine part $\text{Spec }k[t]$ of $\mathbf{P}_k^1$ and then the zero ideal of $k[t]$ corresponds to the generic point of $\mathbf{P}_k^1$?