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Mathmos6
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Homework Statement
How would I go about showing that if [itex] X = \{(x,\,y) \in \mathbb{C}^2 | x^2 = y^3\}[/itex] then X is birational but not isomorphic to the affine space [itex] \mathbb{A}^1[/tex]? I have found the obvious birational map, sending [itex](x,\,y) \to \frac{x}{y}[/itex], so I have shown the spaces are birational, but how do I show they are not isomorphic? (Isomorphic here means there exists a morphism in the algebraic geometry sense with a two-sided inverse.) It is obvious that X is a sort of line with a 'cusp' at the origin, but I can't see how to show explicitly that they are non-isomorphic: presumably the behaviour at the origin is the reason why this occurs, but I don't think I can get away with saying "it's not smooth around 0 in one case and it's smooth around 0 in the other".
I am only an undergraduate so please do keep the algebraic geometry as simple as possible! Many thanks.
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