Given a smooth proper variety $V/K$ of dimension $n$ and an element $f \in K(V)$, does this define a map $V \to \mathbf{P}^n$? Probably only a rational map outside a set of codimension $> 1$?



map defined by element of function field of a variety
The question was answered in the comments, so I'm adding this to knock the question off the "unanswered" list. The literal answer to your first question is "Yes if and only if $n \leq 1$".


