When topologists speak of an "immersion", they are quite deliberately describing something that is not necessarily an "embedding." But I cannot think of any use of the word "embedding" in algebraic geometry, except sometimes as a word for an immersion of varieties. And the notion of an "immersion" of schemes, especially an "open immersion," seems much more similar to the topologists' "embedding" than their "immersion." [Closed immersions at least have the somewhat flimsy rationale that the scheme structure does not depend solely on the choice of subset.]

Does anyone know of a good reason, other than cultural momentum, to use the word "immersion" rather than "embedding"?

[Note: this has come up in Ravi Vakil's blog on his Algebraic Geometry notes.]

algebraicgeometry, at least for "closed immersions"!) $\endgroup$3more comments