Wikipedia (https://en.m.wikipedia.org/wiki/Smooth_morphism) defines a smooth morphism of schemes as one that is locally finitely presented (lft) and flat with regular (=smooth) geometric fibers. But the article goes on to say that this is equivalent to the geometric fibers being smooth varieties, which I don't agree with: varieties are separated but nothing in the first definition rules out the fibers being non-separated. Indeed the affine line with two origins is locally the usual affine line, so I want it to be smooth.
This issue comes up in Vakil's notes too: in theorem 25.2.2, the following definitions are claimed to be equivalent:
(ii) a smooth morphism is lft, flat of relative dimension n, and the sheaf of relative differentials is locally free of rank n
(iii/iv) a smooth morphism is lft and flat with (geometric) fibers smooth (in the sense of the Jacobian criterion) varieties of dimension n.
But again the first definition allows affine line with two origins over a field, whereas the second rules it out.
My guess is that everyone really means k-scheme instead of variety, but am I missing something?
Best Answer
You are correct that there is no reason for the fibers of a smooth morphism to be separated, so calling them "smooth varieties" is probably not a great idea, and ``$k$-schemes" is indeed better, since most definitions of varieties do include a separatedness hypothesis. Moreover, the use of the term "smooth varieties" (or "smooth $k$-schemes") to describe the fibers as part of the definition of a smooth morphism seems slightly objectionable to me (slightly circular). One could say that a morphism is smooth if it is flat, locally finitely presented, and the geometric fibers are regular. This is sometimes taken as the definition, and varieties are sometimes called smooth if they are (geometrically) regular. If this is your definition of smooth, then it's immediate that a geometrically regular variety is smooth (whatever your definition of variety, which presumably includes at least a (locally of) finite type hypothesis).
I personally prefer (if I were setting up the theory) to have this characterization of smoothness as a theorem, rather than the definition. For example this is the approach taken in the Stacks Project, where an obviously local, but admittedly complicated notion of smoothness derived from a corresponding notion for ring maps is used. Then one has to prove (which is non-trivial) that a $k$-scheme locally of finite type ($=$ locally of finite presentation) is smooth over $k$ if and only if it is geometrically regular. But the openness of the smooth locus is immediate from this approach. On yet another hand, proving flatness of a smooth morphism using the Stacks definition is a fair bit of work.
Anyway, to reiterate after my digression, you are correct about the separatedness. There is no separatedness (of the fibers or otherwise) built into the standard notion of a smooth morphism.