Riemann Roch: Proof, part 1

This talk is the first of two talks that give a proof of the Riemann Roch theorem, in the spacial case of nonsingular complex plane curves.

We divide the Riemann-Roch theorem into 3 pieces: Riemann's theorem, a topological theorem identifying the three definitions of the genus, and Roch's duality theorem. The we prove Riemann's theorem that l(D)=deg D+1-p(0)+p(D) where p(D) is the index of speciality of D, the dimension of the space of obstructions to the Mittag-Leffler problem. To do this we compare the theorem for D and D+P where P is a point. We check that i(D) is finite by calculating i(0) explicitly and finding that it is (d-1)(d-2)/2.

Finally we comment on what happens if C has n double points: the main effect of this is to reduce the arithmetic genus (of its normalization) by n.

• #education
• #riemann_roch
• #index_of_speciality
• #mittag_leffler_problem
• #arithmetic_genus