The Genus of a Curve

The genus of a curve is a birational invariant which plays an important role in the parametrization of algebraic curves (and in the geometry of algebraic curves in general). In fact, only curves of genus 0 can be rationally parametrized. So in the process of parametrization we will first compute the genus of the curve C. This will involve an analysis of the singularities of C, and we will determine the genus as the deficiency between a bound on the number of singularities and the actual number of singularities of C. But in order to arrive at a definition of the genus, we first need to consider divisors on C and their associated linear spaces.We do not want to repeat this classical development here (for further details see for instance [Ful89] or [Wal50]), but we give a kind of road map for getting to the definition of the genus and from there to a method for computing it.