Let $f:X \to Y$ be a dominant morphism between integral algebraic surfaces (so a $2$-dimensional, proper $k$-scheme).
I know that there I exist a open set $V \subset Y$ such that the restriction of $f$ to $f \vert _{f^{-1}(V)} \to V$ is a isomorphism.
Futhermore, if $Y$ is also normal $V$ can be choosen as $V := Y \backslash \{x_1, ..., x_n\}$ where $\{x_1, ..., x_n\}$ is a finite set of closed points.
$E_i := f^{-1}(x_i)$ are called exceptional curves. If $Y= Spec(f_*\mathcal{O}_X)$ then it can be showed that $E_i$ are connected.
But why they are curves (= $1$-dimensional, proper $k$-scheme) and not points?