Complex Curves and Real Surfaces
A complex projective algebraic curve resides in n-dimensional complex projective space CPn. This has complex dimension n, but topological dimension, as a real manifold, 2n, and is compact, connected, and orientable. An algebraic curve likewise has topological dimension two; in other words, it is a surface. A nonsingular complex projective algebraic curve will then be a smooth orientable surface as a real manifold, embedded in a compact real manifold of dimension 2n which is CPn regarded as a real manifold.
The topological genus of this surface, that is the number of handles or donut holes, is equal to the genus of the algebraic curve that may be computed by algebraic means. In short, if one consider a plane projection of a non singular curve, that has degree d and only ordinary singularities (singularities of multiplicity two with distinct tangents), then the genus is (d - 1)(d - 2)/2 - k, where k is the number of these singularities.
Read more about this topic: Algebraic Curve
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