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I apologize if this question is sophomoric as my knowledge of projective geometry is rather elementary. But I'm curious if there exists a good intuitive geometric explanation for why the curve $y-x^3=0$ in $\mathbb{P}^2(\mathbb{R})$ has a singularity at infinity.

Thanks

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    @Nils Matthes: Yes looking at it in the $y=1$ chart was how I discovered its singularity at infinity. My understanding was that looking at it in a different affine chart was just a tool to gain a 'fuller' understanding of the geometry of $y-x^3$. But it appears what you're implying is that the its projective version is in some sense the 'true' curve and it just so happens that its $z=1$ portion happens to miss the singularity. Nevertheless, I can't help but feel that there's probably something interesting to say which relates the way in which $y-x^3$ diverges to its singularity at infinity.2012-12-02

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