Riemann–Hurwitz formula

Remarks

Triangulate $Y$ so that every branch value is a vertex, and lift the triangulation to $X$ along the branched covering $f$. Each edge and face has exactly $n$ preimages, and a vertex $y$ has $n - \sum_{f(P) = y} (e_P - 1)$ preimages. Comparing Euler characteristics with $\chi = 2 - 2g$ gives the formula. The same formula holds for a finite separable morphism of smooth projective curves over an algebraically closed field when every $e_P$ is prime to the characteristic. With wild ramification the term $e_P - 1$ must be replaced by a larger number. Example: the map $z \mapsto z^n$ on the Riemann sphere has degree $n$ and ramifies only at $0$ and $\infty$ with $e = n$; the formula reads $-2 = n(-2) + 2(n - 1)$, which is true. Example: a double cover $X \to \mathbb{P}^1$ branched at $2g + 2$ points has $2 g_X - 2 = -4 + 2g + 2$, so $g_X = g$; this is how the genus of a hyperelliptic curve is found.

Going deeper

For a student

The formula is the main tool to compute genera of curves given as covers, such as hyperelliptic curves, Fermat curves and modular curves. It shows at once that there is no nonconstant map from a curve of genus $g$ to a curve of larger genus. It also shows that an unramified cover of a genus-$1$ curve has genus $1$. Applied to the quotient map by a group of automorphisms, it gives Hurwitz's automorphisms theorem. It is the one-dimensional case of the comparison of canonical divisors $K_X = f^* K_Y + R$, where $R$ is the ramification divisor.