- #1

Diophantus

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I have the definition that a rational map phi (from some affine variety V to an affine variety W) is dominant if the image of V is not contained in a proper subvariety of W.

In my lecture notes there is some remark about how if we compose phi with another such map psi going in the reverse direction W to V (to get a map V to V) then it may not make sense unless phi is dominant.

Whilst reading around the subject I have noticed that the notion of dominance crops up in a lot of theorem hypotheses so I really feel I ought to get a grip on it.

My problem is this: Rational maps are not generally functions anyway since they may have undefined points. OK. Now, in the remark when it talks about the composition 'not making sense', one would usually assume by the phrase 'not making sense' that it has undefined points - but why are we suddenly bothered about this now when we weren't before?

Any help would be greatly appreciated.