Dominant classical morphisms and rational maps

A morphism $\varphi :U\to Y$ from a nonempty open in an affine variety to an affine variety is dominant when $\overline{\varphi (U)}=Y$. A rational map is dominant when one representative is dominant. This does not depend on the representative. Indeed, if $\varphi$ is dominant and $W\subseteq U$ is nonempty open, then for each nonempty open $T\subseteq Y$, ${\varphi }^{-1}(T)$ is a nonempty open in $U$ by A classical morphism pulls Zariski closed sets back to closed sets and dominance. It meets $W$ by Every nonempty open of a classical affine variety is dense, so $\varphi (W)$ meets every such $T$, proving $\varphi {\mathrm{\mid }}_W$ dominant. If two representatives agree on a nonempty common open, restricting a dominant one to that open gives a dense image contained in the image of the other. Thus every representative is dominant. Conversely, if any restriction is dominant, the original image contains a dense subset and is dominant.

Source comparison: Milne, Algebraic Geometry, v6.10, Proposition 3.34 p. 72 and §§5k–l pp. 116–117. Conventions here distinguish arbitrary affine algebraic sets from nonempty irreducible varieties.