The Riemann—Roch Theorem

Abstract

Fix a compact Riemann surface X, and let g = g _X be its genus, M its field of meromorphic functions, and Ω the space of meromorphic 1-forms on X. A divisor D = Σm _P P on X is just another word for a 0-chain. That is, it assigns an integer m _P to each point P in X, with only finitely many being nonzero. We say that the order of D at P is m _P , and write ord _P (D) = m _P . The divisors on X form an abelian group. As for 0-chains, the degree of a divisor is the sum of the coefficients: deg(D) = Σm _P . If E = Σn _P P is another divisor, we write E ≥ D to mean that n _p ≥ m _P for all P in X. A divisor D is called effective if each coefficient m _P is nonnegative, i.e., D ≥ 0.