Rational Algebraic Curves pp 67-86 | Cite as

# The Genus of a Curve

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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.

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