Algebraic curve
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In algebraic geometry, an algebraic curve is an algebraic variety of dimension equal to 1. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.
Using the intrinsic concept of tangent space, points P on an algebraic curve C are classified as non-singular or singular. Singular points include crossings over itself, and also types of cusp, for example that shown by the curve with equation X3 = Y2 at (0,0).
A curve C has at most a finite number of singular points. If it has none, it can be called non-singular. For this definition to be correct, we must use an algebraically closed field and a curve C in projective space (i.e. complete in the sense of algebraic geometry). If for example we simply look at a curve in the real affine plane there might be singular points 'at infinity', or that needed complex number co-ordinates for their expression.
The theory of non-singular algebraic curves over the complex numbers coincides with that of the compact Riemann surfaces. Every algebraic curve has a genus defined. In the Riemann surface case that is the same as the topologist's idea of genus of a 2-manifold. The genus enters into the statement of the Riemann-Roch theorem and can be characterized as the only integer that makes this theorem correct. This can serve as a definition of the genus for curves over other fields.
The case of genus 1 - elliptic curves - has in itself a large number of deep and interesting features. For higher genus g some of those carry over to the Jacobian variety, an abelian variety of dimension g