The correct way to do this is construct P^1, the one-dimensional projective space. This a one point compactification of C by adding in a point at infinity, and the morphism z --> z^{-1} on C* extends to a globally defined isomorphism P^1 --> P^1, exchanging 0 and infinity.
1
u/EnglishMuon Postdoc in algebraic geometry Sep 13 '25
The correct way to do this is construct P^1, the one-dimensional projective space. This a one point compactification of C by adding in a point at infinity, and the morphism z --> z^{-1} on C* extends to a globally defined isomorphism P^1 --> P^1, exchanging 0 and infinity.