the Riemann-Hurwitz formula
/ REE-mahn HOOR-vits /
If you wrap one surface over another as a branched cover, how do their numbers of holes relate? Naively, covering a sphere d times d-fold should give a surface with d times the Euler characteristic — but pinching the sheets together at branch points costs you. The Riemann-Hurwitz formula is the exact accounting: it says the genus of the cover is determined by the genus of the base, the degree, and a precise penalty paid at every ramification point.
Precisely, let f: X -> Y be a degree-d branched cover of compact Riemann surfaces with genera g_X and g_Y. For each point p in X let e_p be its ramification index (e_p = 1 at unramified points). Then 2 g_X - 2 = d (2 g_Y - 2) + sum over p of (e_p - 1). In Euler-characteristic form this is chi(X) = d chi(Y) - sum over p of (e_p - 1): a d-fold cover would multiply chi by d, and each ramification point of index e_p subtracts (e_p - 1) because that many sheets got glued into one. The sum is finite since only finitely many points ramify. The number R = sum over p of (e_p - 1) is the total ramification.
Why it matters: this is the workhorse for computing genus. Given a curve presented as a branched cover — say y^2 = f(x) over the sphere — you read off the branch data and the genus drops out instantly, no triangulation needed. It also forces structural facts: a nonconstant map can only LOWER or keep the genus (you cannot map a sphere onto a torus), and it constrains how many branch points are possible. Honesty notes: the formula needs X and Y compact; the ramification term counts (e_p - 1), NOT e_p; and over each branch point you must include EVERY ramified preimage, since several sheets can ramify above one value.
For y^2 = (x - a_1)...(x - a_6) the map to the x-sphere has degree 2, ramified over the six roots a_i and over infinity (seven branch points, e = 2 each), so R = 7 - ... actually the parity forces an even count: ramify over the 6 roots only when the degree is even at infinity. Taking R = 6, Riemann-Hurwitz gives 2 g_X - 2 = 2(2*0 - 2) + 6 = 2, so g_X = 2 — a genus-2 surface.
A 2-cover of the sphere branched over 6 points has genus 2: 2g - 2 = 2(-2) + 6.
The ramification term is sum of (e_p - 1), not sum of e_p, and X, Y must be compact. Be careful at infinity: for y^2 = (degree-n polynomial) the point at infinity ramifies exactly when n is odd, which changes R and hence the genus.