In CP^2, any homogeneous polynomial in the coordinates, like P(X,Y,Z) = X^3 + Y^3 + Z^3, defines a compact Riemann surface (or what algebraic geometers like to call a "projective curve"). (P isn't well-defined on CP^2, but the locus P^(-1)(0) is.) If d := deg(P), then the genus g of this surface is given by the standard formula g = (d-1)(d-2)/2. (Assuming a technical condition that's satisfied for this P(X,Y,Z).) So, (3-1)*(3-2)/2 = 1 and hence P^(-1)(0) is a torus. Projective curves defined by X^n + Y^n + Z^n = 0 for some n >= 1 are called Fermat curves, after FLT. --Dan << On May 25, 2011, at 4:17 PM, "Cordwell, William R" <wrcordw@sandia.gov> wrote:

So, starting with the set CxCxC minus the origin, define the equivalence class of a point x to be all nonzero (complex) multiples of x, and "mod out" by that equivalence relation. This forms the complex projective space CP^2. I'm not sure how the topology is inherited or defined (topologists, please jump in)--maybe it's the largest one that keeps the mapping continuous?

For this problem, one also wants initially to restrict to the subset of the original points in C^3 where x^3 + y^3 + z^3 = 0. The claim is that the final subset of CP^2 is homeomorphic to the torus.

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Sometimes the brain has a mind of its own.