Using the intrinsic concept of tangent space, points P on an algebraic curve C are classified as nonsingular or singular. Singular points include crossings over itself, and also types of cusp, for example that shown by the curve with equation X^{3} = Y^{2} at (0,0).
A curve C has at most a finite number of singular points. If it has none, it can be called nonsingular. 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 coordinates for their expression.
The theory of nonsingular algebraic curves over the complex numbers coincides with that of the compact Riemann surfaces[?]. Every algebraic curve has a genus (mathematics) genus defined. In the Riemann surface case that is the same as the topologist's idea of genus of a 2manifold. The genus enters into the statement of the RiemannRoch theorem.
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
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