How do you realize that the complex curve $x^4+y^4=1$ looks topologically like three tori glued together with four points at infinity?

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    $\begingroup$ By plotting it! :D $\endgroup$ – vrugtehagel Feb 11 '16 at 19:36
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    $\begingroup$ @vrugtehagel I think that the OP works in $\Bbb{C}^2$, so there is no way to plot it. $\endgroup$ – Crostul Feb 11 '16 at 19:43
  • $\begingroup$ You actually probably could find a way to plot in 4 dimensions (using time or color?), but I'm not sure that would help here. $\endgroup$ – user2469 Feb 11 '16 at 21:30
  • $\begingroup$ Is it a plane curve? $\endgroup$ – Alan Muniz Feb 12 '16 at 0:59
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    $\begingroup$ Is your question about visualizing the curve or understanding why the curve is topological a genus three surface? The Genus-Degree Formula implies that the geometric genus of $C$ is $\frac12(4-1)(4-2) = 3 $. A classical theorem says that nonsingular smooth projective algebraic curves are in fact compact Riemann surfaces (endowed with the standard complex topology instead of the Zariski) and vice versa. The notions of genus coincide, so $C$ is a closed, orientable genus 3 surface. Finally, the classification of surfaces implies that $C$ is homeomorphic to the connected sum of three tori. $\endgroup$ – Charlie Feb 13 '16 at 18:01

By the genus-degree formula (or Riemann-Hurwitz), the projective closure of this curve has genus $3$, so looks like a three-holed torus. Next you need to figure out how many points taking the projective closure added, and as in the comments it's not hard to see that there are four.


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