Basic Algebraic Geometry 2 by Igor R. Shafarevich

Author:Igor R. Shafarevich
Language: eng
Format: epub
Publisher: Springer Berlin Heidelberg, Berlin, Heidelberg

Theorem 7.5

If X is a nonsingular projective curve then the space can be triangulated, and is a combinatorial surface.

Proof

We prove first that has a triangulation. We indicate a triangulation which, although not the most economic, is useful for subsequent applications.

The decomposition of the surface of an octahedron into its faces, edges and vertexes gives a triangulation. Suppose that the octahedron is inscribed in the 2-sphere . Projecting this triangulation outwards from an interior point of the octahedron provides a triangulation of S 2. Since is homeomorphic to S 2, we get a triangulation of .

We identify with the plane of one complex variable together with a point at infinity. The triangulation constructed in terms of the octahedron is the decomposition realised by the real and imaginary axes and the circle |z|=1 (see Figure 30). It has

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