Applications of Riemann-Roch to Classifying Low genus curves
The Riemann-Roch theorem is one of the most powerful tools in algebraic geometry. In this video we give applications of this theorem to studying smooth projective curves of genus zero and one. For curves of genus zero, it can be used to quickly show these curves are isomorphic to the projective line. Genus one curves, also called elliptic curves (when a distinguished point is provided) are studied next. We see how curves which have an algebraic group structure must have genus one. We then show conversely, using Riemann-Roch, that any genus one curve is a cubic curve and also has a natural algebraic group structure. Hence the Riemann-Roch theorem gives essentially complete information about low genus curves. The proofs given illustrate many key methods in algebraic geometry, use the Riemann-Roch theorem to find global sections of line bundles which can be used to map curves to projective space. It is most effective when Serre duality is used to provide effective vanishing of the first cohomology group. The Riemann-Roch theorem can further be used to study relations between the various global sections of line bundles.