Let $\Bbbk$ be a field and $X$ a smooth scheme over $\Bbbk$.
If $X$ were a variety complex, smooth, projective variety, then I'd know that for fixed $n\in\Bbb N$, all rational maps $X\dashrightarrow\Bbb P^n_\Bbbk$ are all given by some choice of a line bundle $\mathcal L$ on $X$ and a $\Bbbk$-vector subspace $V\subseteq \mathcal L(X)$ of dimension $n+1$ such that the base locus of this linear system is of codimension at least two in $X$. This is, for example, in Griffiths & Harris, Chapter 4.
Is (something like) this still true if $X$ is any scheme over $\Bbbk$? Can anyone point me to a reference for this statement? I cannot seem to find it in any textbook on algebraic geometry that I checked.
PS: In my case, $X$ is actually just the spectrum of a regular local ring over $\Bbbk=\Bbb C$, in which case I am rather confident it will be true. I would still like to know how general this fact is.