GENUS OF AN ALGEBRAIC CURVE

 Synonym: deficiency.

The genus of a plane algebraic curve of degree n is the integer minus the orders of multiplicity of singular points of the curve in the complex projective plane.

It is the genus of this curve considered as a surface (i.e. a manifold of dimension 2) of the complex projective plane, which itself is a manifold of dimension 4 (see genus of a surface).

A real curve of genus g is therefore the section of a torus with g holes by a plane, and, as a result, has, in general and at most, g + 1 connected components in the real projective plane.

By the Harnack theorem (1878), we can find curves of any degree having g + 1 real connected components (and the 16th problem of Hilbert consisted in classifying these curves).
See examples at quartic.

The curves of genus zero are the rational curves, that can be parametrized by rational functions, and the curves of genus 1, the elliptic curves, that can be parametrized by elliptic functions; the curves of maximal genus are the smooth curves.