Tuesday, 9 October 2007

ag.algebraic geometry - Is there any rational curve on an Abelian variety?

Over mathbbCmathbbC you can argue as follows.
Suppose you have a morphism mathbbP1(mathbbC)toAmathbbP1(mathbbC)toA (AA= abelian variety ). Since mathbbP1(mathbbC)mathbbP1(mathbbC) is simply connected , the morphism lifts to the universal cover of AA, affine space mathbbCnmathbbCn. But since mathbbP1(mathbbC)mathbbP1(mathbbC) is complete and connected, the lift to affine space must be constant and hence the original morphism is constant too.



The answers by Charles, Felipe and jvp are better because they work over arbitrary fields, but since the argument just given is so ridiculously elementary (introductory topology), I thought it might still be of some interest ( also it works in the holomorphic category if AA is a complex torus, maybe not algebraic).

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