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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 varietyFor the purposes of algebraic geometry over the complex numbers, an abelian variety is a complex torus (a torus of real dimension 2''n that is a complex manifold) that is also a projective algebraic variety of dimension n i. can be defined in projective s of dimension g