To give a rational function $f$ in the function field $K(X)$ of a smooth projective curve $X$ is equivalent to giving a finite morphism $f:X\to \mathbb P^1$.
What is the precise analogue of this statement for higher-dimensional varieties?
I would guess that a rational function on a smooth projective variety gives rise to a rational morphism $X\dashrightarrow\mathbb P^1$ and that this is a morphism up to blowing-up $X$. Is that correct?
If $X$ is an integral scheme of finite type over a field $k$ ($k$-variety), there is a bijection $$K(X)\simeq \{\textrm{Rational functions }X\dashrightarrow\mathbb P^1\}.$$ The functor $K(-)$ establishes a duality between the category of $$ k\textrm{-varieties, with dominant rational maps,} $$ and the category of $$ \textrm{function fields over }k. $$ In particular, to a dominant rational map $f:X\dashrightarrow \mathbb P^1$ there corresponds a monomorphism $f^\ast: k(t)\to K(X)$. The image of $t$ under $f^\ast$ "is" the rational function $f$.