Complex Algebraic Curves are not Compact

Algebraic Geometry
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Tsakanikas Nickos
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Complex Algebraic Curves are not Compact

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Post by Tsakanikas Nickos »

Recall that, given a non-constant polynomial $ P(x,y) \in \mathbb{C}[x,y] $ without repeated factors, the complex algebraic curve $ C $ in $ \mathbb{C}^2 $ determined by $ P $ is defined as
\[ C = \left\{ (x,y) \in \mathbb{C}^2 \ | \ P(x,y) = 0 \right\} \hskip -2pt . \] Show that any complex algebraic curve in $ \mathbb{C}^2 $ is not compact.
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