the Riemann-Roch theorem
/ REE-mahn ROKK /
Here is the central counting problem of the whole subject: given a budget for poles — 'you may have a pole of order at most 2 here, a simple pole there' — how many meromorphic functions fit inside that budget? You expect the answer to grow as you allow more poles, but on a surface with holes there are mysterious obstructions. The Riemann-Roch theorem is the exact formula that counts the dimension of this space, with a correction term that precisely measures the obstruction.
Precisely, let X be a compact Riemann surface of genus g, and D a divisor on it. Let L(D) be the vector space of meromorphic functions f with div(f) + D >= 0 (poles no worse than D allows, zeros allowed), and write l(D) = dim L(D). Let K be the canonical divisor. Then l(D) - l(K - D) = deg(D) - g + 1. The term l(D) is what you want; the correction l(K - D) (the dimension of holomorphic differentials vanishing to order D, the 'index of speciality') is the obstruction. When deg(D) > 2g - 2 the correction l(K - D) vanishes, giving the clean Riemann inequality become equality: l(D) = deg(D) - g + 1. Two sanity checks fall right out: D = 0 gives l(0) = 1 (only constants) and l(K) = g (there are g holomorphic differentials); and D = K gives deg K = 2g - 2.
Why it matters: Riemann-Roch is the computational heart that turns 'abstract surface' into 'concrete projective curve.' It tells you when a divisor has enough functions to define a map to projective space, hence proves curves embed; it forces the genus to equal the dimension of holomorphic 1-forms; it yields the Weierstrass gap structure; and it generalizes upward to Hirzebruch-Riemann-Roch and Atiyah-Singer. Honesty points: there are MANY 'Riemann-Roch theorems' (curves, surfaces, Hirzebruch, Grothendieck, the index theorem); this is the original CURVE version. The clean form l(D) = deg D - g + 1 holds only when the special term vanishes (e.g. deg D > 2g - 2 or deg D < 0); in the middle range you genuinely need both terms. And l(D) depends only on the linear-equivalence class of D, not the particular representative.
On the sphere (g = 0, K has degree -2) take D = n[infinity] with n >= 0. Then deg D = n > -2 = 2g - 2, so l(K - D) = 0 and l(D) = n - 0 + 1 = n + 1. Indeed L(n[infinity]) is spanned by 1, z, z^2, ..., z^n — the polynomials of degree at most n, exactly n + 1 of them. Riemann-Roch reproduces this count with no algebra.
On the sphere l(n[infinity]) = n + 1 counts polynomials of degree <= n — Riemann-Roch with the special term zero.
There are many Riemann-Roch theorems; this is the curve version. The simple formula l(D) = deg D - g + 1 holds only when the special term l(K - D) vanishes (e.g. deg D > 2g - 2); in the intermediate range you must keep both terms, and l(D) depends only on the linear-equivalence class of D.