The Riemann-Roch Theorem for Surfaces
The original Riemann-Roch theorem was for curves and was an indispensable tool for controlling the space of global sections of a line bundle on a curve. It gives the Euler characteristic of the line bundle as a simple linear function of the degree of the line bundle. In this video, we look at the Riemann-Roch theorem for surfaces which gives the Euler characteristic of the line bundle O(D) as a quadratic function of D. We give the proof by reducing to the curve case. As for the Riemann-Roch theorem for curves, its utility lies in ensuring there are non-zero global sections. We give a criterion for this to occur which is a surface analogue of the fact that a sufficiently positive divisor on a curve is linearly equivalent to an effective divisor.