The Riemann Hurwitz formula.
The canonical divisor is a fundamental invariant in algebraic geometry, so it is essential to have means of computing it. In this video we look at the Riemann-Hurwitz formula which relates the degrees of canonical divisors of two projective smooth curves, given a regular morphism between them. The formula is in terms of the degree of the morphism and the ramification divisor. We explain the degree in a number of ways, geometrically it says how finite to one a morphism is. We also explain the ramification divisor which gives a measure of deviation that a morphism is from being a local diffeomorphism. We finish by giving an application of the Riemann-Hurwitz formula to computing the degree of the canonical divisor of a plane curve. It is thus a powerful tool for computing canonical divisors.