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Let $f: X \rightarrow Y$ be a morphism of schemes. When the image $f(X)$ is dense in $Y$, we say that $f$ is dominant.

No doubt, dominant morphisms pop up all over the place and have an important role in birational geometry. But a lot of the time it seems to me like the property we are really interested in is that $f(X)$ contains all the generic points of irreducible components of $Y$. See, for example, the stacks project entry on Dominant Morphisms Stacks 01RI, which is entirely devoted to the question of when these properties coincide. In geometric situations, like when our schemes have finitely many irreducible components, or when our morphisms are quasi-compact, the concepts coincide. But in general, dominance is a much weaker property.

Let's for now denote by $(*)$ the property of a morphism that its image contains all the generic points of the irreducible components of the target (aka codimension $0$ points aka maximal points).

A couple questions:

(1) Does $(*)$ have an agreed upon name in the literature? Is it interesting in its own right? More/less useful than dominance in general scheme theory?

(2) Is there a historical reason for the prevailing definition of dominance? Some thoughts: in EGA dominance is introduced along with other purely topological properties of morphisms, like being open, closed, surjective. Meanwhile $(*)$ can be expressed in purely topological terms, but if we even want compositions of $(*)$ morphisms to be stable under composition then we need to restrict to spectral spaces. So in some sense dominance makes sense as a purely topological property but $(*)$ heavily depends on respecting structure sheaves.

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