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
Famous quotes containing the words complex, curves, real and/or surfaces:
“All of life and human relations have become so incomprehensibly complex that, when you think about it, it becomes terrifying and your heart stands still.”
—Anton Pavlovich Chekhov (1860–1904)
“For a hundred and fifty years, in the pasture of dead horses,
roots of pine trees pushed through the pale curves of your ribs,
yellow blossoms flourished above you in autumn, and in winter
frost heaved your bones in the ground—old toilers, soil makers:
O Roger, Mackerel, Riley, Ned, Nellie, Chester, Lady Ghost.”
—Donald Hall (b. 1928)
“I’m a Yankee Doodle Dandy,
A Yankee Doodle do or die;
A real live nephew of my Uncle Sam’s,
Born on the fourth of July.”
—George M. Cohan (1878–1942)
“But ice-crunching and loud gum-chewing, together with drumming on tables, and whistling the same tune seventy times in succession, because they indicate an indifference on the part of the perpetrator to the rest of the world in general, are not only registered on the delicate surfaces of the brain but eat little holes in it until it finally collapses or blows up.”
—Robert Benchley (1889–1945)