nLab
image of a rational map

As rational map $f:X⤏Y$ among projective algebraic varieties is not defined everywhere there is also a nontrivial definition of an image of a rational map: it is defined by help of a graph of a rational map.

For this embed $Y$ in a projective space $P^n=P_k^n$ of dimension $n$; now the image of $f$ is the image of the composition still denoted $f:X⤏P^n$. It is defined as a regular map on some open dense subset $f_U:U\to P^n$. Let the set theoretical graph be $\mathrm{setgraph}{f}_U\subset U\times P^n$; then define the graph of the rational map $f$ as the closure of the $\mathrm{setgraph}{f}_U$ within $X\times P^n$. This does not depend on the choice of the dense open subset $U$ and agrees with the usual graph when $f$ is regular.

Notice that not every $k$-point in the image of a rational map is actually set theoretically in image of the underlying map of sets of $k$-points.

A rational map $f:X⤏Y$ is dominant if its image as a rational map is the whole of $Y$.