04.04.02 · algebraic-geometry / curves

Hurwitz formula

shipped3 tiersLean: none

Anchor (Master): Hurwitz 1891 *Über Riemann'sche Flächen mit gegebenen Verzweigungspunkten* (Math. Ann. 39, originator); Hartshorne §IV.2; Liu §7.4; Donaldson — Riemann Surfaces §6

Intuition [Beginner]

The Hurwitz formula tells you exactly how the genus changes when you wrap one curve around another. Picture a degree- $n$ cover $π : Y \to X$ — every point of $X$ has $n$ preimages in $Y$ , except over a finite set of branch points where two or more sheets collide. The cover is determined locally by maps that look like $z \mapsto z^{e}$ , and the integer $e$ is the ramification index at each upstairs point.

Hurwitz's accounting principle: each ramification point contributes $e - 1$ to a global count, and that count rigidly determines the genus of $Y$ from the genus of $X$ , the degree, and the ramification data. There is no slack.

The headline formula on a degree- $n$ tame cover of smooth projective curves: $2 g_{Y} - 2 = n (2 g_{X} - 2) + de g R$ , where $de g R$ adds up $e_{y} - 1$ over all ramification points $y$ . Plug in the genus of the base curve and the ramification, read off the genus of the cover.

Visual [Beginner]

A degree- $n$ cover of one curve by another is a stack of sheets glued together at branch points. Most points of the base have $n$ disjoint preimages; the branch points are where sheets pinch together. The picture below shows three sheets over a base curve with two branch points where two sheets collide.

The ramification data — at each upstairs point, how many sheets collide there — is everything Hurwitz needs to read off the genus of the upstairs curve.

Worked example [Beginner]

The cleanest case: the map $π : P^{1} \to P^{1}$ given by $t \mapsto t^{n}$ . Both source and target have genus 0. The map has degree $n$ . It is ramified at exactly two points: above $0$ , where the unique preimage $u = 0$ has ramification index $e = n$ (the local model is $t = u^{n}$ ), and above $\infty$ , also with index $n$ .

Hurwitz: $2 g_{Y} - 2 = n (2 g_{X} - 2) + de g R$ . Plug in $g_{Y} = 0$ , $g_{X} = 0$ , $n = n$ , and the ramification total $de g R = (n - 1) + (n - 1) = 2 (n - 1)$ :

$- 2 = n \cdot (- 2) + 2 (n - 1) = - 2 n + 2 n - 2 = - 2$ .

Both sides equal $- 2$ . The formula confirms what we already knew: the $n$ -fold cover of $P^{1}$ branched only at $0$ and $\infty$ is itself a copy of $P^{1}$ .

The takeaway: Hurwitz lets you compute the genus of a cover without ever drawing it, using only the degree and the ramification.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $π : Y \to X$ be a finite separable morphism of smooth projective curves over an algebraically closed field $k$ , of degree $n = [k (Y) : k (X)]$ . Write $g_{X}, g_{Y}$ for the genera. For a closed point $y \in Y$ with image $x = π (y) \in X$ , let $t \in O_{X, x}$ be a uniformiser at $x$ and $u \in O_{Y, y}$ a uniformiser at $y$ .

Definition (ramification index). The ramification index $e_{y} \geq 1$ of $π$ at $y$ is the unique integer with $π^{*} t = v \cdot u^{e_{y}}$ for some unit $v \in O_{Y, y}^{\times}$ . Equivalently, $e_{y} = v_{y} (π^{*} t)$ where $v_{y}$ is the discrete valuation at $y$ .

The morphism is unramified at $y$ iff $e_{y} = 1$ . Only finitely many points are ramified. The morphism is tame at $y$ iff $char k ∤ e_{y}$ ; wild otherwise. Tameness is automatic in characteristic zero.

Definition (ramification divisor). The ramification divisor of $π$ is the effective Weil divisor on $Y$ $$ R = R_\pi := \sum_{y \in Y} (e_y - 1), y $$ (in the tame case; in the wild case the coefficients are replaced by the order of the different at $y$ , see below).

Definition (branch divisor). The branch divisor on $X$ is $B = π_{*} R$ , supported at the points of $X$ where $π$ has at least one ramified preimage.

The fundamental relation between canonical divisors under pullback is:

Lemma. In the tame case, $\pi^ K_X = K_Y - R $a s d i v i sor c l a sseso n$ Y$.*

This is Hartshorne IV.2.3 ^{[Hartshorne §IV.2]}. We give the proof in the next section.

Theorem (Hurwitz formula, tame case). For $π : Y \to X$ a finite separable tame morphism of smooth projective curves of degree $n$ , $$ 2 g_Y - 2 ;=; n (2 g_X - 2) ;+; \deg R ;=; n(2g_X - 2) + \sum_{y \in Y} (e_y - 1). $$

Counter-examples to common slips.

Forgetting tameness. In characteristic $p$ with $p ∣ e_{y}$ for some $y$ , the formula above fails: the contribution at $y$ is the order of the different $d_{Y / X}$ , which strictly exceeds $e_{y} - 1$ . The corrected statement reads $2 g_{Y} - 2 = n (2 g_{X} - 2) + de g d_{Y / X}$ .
Confusing branch and ramification divisor. $R$ lives on the cover $Y$ ; $B = π_{*} R$ lives on the base $X$ . The Hurwitz formula uses $de g R$ , not $de g B$ . (Both have the same degree, since $π_{*}$ preserves degree, but the divisors live on different curves.)
Inseparable maps. The formula assumes separability of the function-field extension $k (Y) / k (X)$ . Purely inseparable maps (only possible in positive characteristic) require a different treatment.
Non-projective domains. The argument relies on $de g K_{C} = 2 g - 2$ , which uses projectivity of $C$ . For affine pieces or open subcurves the formula reformulates as a relation on Euler characteristics.

Key theorem with proof [Intermediate+]

Theorem (Hurwitz formula). Let $π : Y \to X$ be a finite separable tame morphism of smooth projective curves over an algebraically closed field $k$ , of degree $n$ . Then $$ 2 g_Y - 2 ;=; n (2 g_X - 2) ;+; \deg R_\pi. $$

Proof. The argument has two steps: (i) establish $π^{*} K_{X} = K_{Y} - R$ as divisor classes on $Y$ ; (ii) take degrees and use $de g π^{*} D = n de g D$ for any divisor $D$ on $X$ .

Step 1: Pullback of canonical divisors. Choose a meromorphic differential $ω \in Ω_{X / k}^{1}$ that is non-zero on a Zariski-open subset of $X$ . Its pullback $π^{*} ω$ is a meromorphic differential on $Y$ . We compare the divisor of $π^{*} ω$ on $Y$ with the divisor of $ω$ on $X$ .

Local computation. Pick a closed point $y \in Y$ with $π (y) = x$ , ramification index $e = e_{y}$ . Choose a uniformiser $t$ at $x$ and a uniformiser $u$ at $y$ such that $t = u^{e}$ (tameness lets us absorb the unit factor; in the wild case this normalisation fails). Locally $ω = f (t) d t$ for some $f$ with $v_{x} (f) = v_{x} (ω)$ . Then $$ \pi^* \omega ;=; f(u^e) \cdot d(u^e) ;=; e \cdot f(u^e) \cdot u^{e - 1} , du. $$

The factor $e$ is a unit at $y$ (this is exactly where tameness, $char k ∤ e$ , enters). Therefore $$ v_y(\pi^* \omega) ;=; v_y(f(u^e)) + (e - 1) ;=; e \cdot v_x(f) + (e - 1) ;=; e \cdot v_x(\omega) + (e - 1). $$

At an unramified point ( $e = 1$ ), this reduces to $v_{y} (π^{*} ω) = v_{x} (ω)$ .

Globalising. Sum over $y \in Y$ . The divisor of $π^{*} ω$ as a Weil divisor on $Y$ is $$ (\pi^* \omega) ;=; \sum_{y \in Y} v_y(\pi^* \omega) \cdot y ;=; \sum_y \big[e_y \cdot v_{\pi(y)}(\omega) + (e_y - 1)\big] \cdot y ;=; \pi^(\omega) + R, $$ where the term $\sum_{y} e_{y} \cdot v_{π (y)} (ω) \cdot y$ regroups as the divisor pullback $\pi^(\omega) $o n$ Y $(r ec a l l t ha t f or aW e i l d i v i sor$ D = \sum n_x \cdot x $o n$ X $, t h e p u l l ba c k i s$ \pi^* D = \sum_x n_x \sum_{y \mapsto x} e_y \cdot y$). Therefore as divisor classes $$ K_Y ;=; (\pi^* \omega) ;=; \pi^* K_X + R. $$

Step 2: Degrees. Take degrees on both sides. The pullback satisfies $de g π^{*} D = n de g D$ for any divisor $D$ on $X$ , since $π$ has degree $n$ . Therefore $$ \deg K_Y ;=; n \deg K_X + \deg R, \qquad \text{i.e.}, \qquad 2 g_Y - 2 ;=; n (2 g_X - 2) + \deg R. $$

This completes the proof. $□$

Bridge. The proof is one identity on differentials, taken at every ramification point and globalised. The same local model $t = u^{e}$ governs the residue calculus, the local class field theory of curves, and the structure of the different in arithmetic geometry; Hurwitz is the simplest invariant one extracts. In the next units of the curves strand, the same pullback identity $π^{*} K_{X} = K_{Y} - R$ controls the genus of plane curves via adjunction (the curve embedded in $P^{2}$ as a divisor) and the degree of the gonality map (the smallest $n$ admitting a degree- $n$ cover of $P^{1}$ ).

Exercises [Intermediate+]

Exercise 7 (hard, symbolic).

Prove the Castelnuovo-Severi inequality: if $π : Y \to X$ is a finite separable tame cover of degree $n$ between smooth projective curves with $g_{Y} \geq 1$ , then $$ g_Y ;\leq; n \cdot g_X + (n - 1) \cdot \tfrac{1}{2} \deg R, $$ and deduce a lower bound for $g_{X}$ given $g_{Y}$ and $n$ .

Hint

Solve the Hurwitz formula for $g_{Y}$ .

Answer

From Hurwitz, $2 g_{Y} - 2 = n (2 g_{X} - 2) + de g R$ , so $$ g_Y ;=; n \cdot g_X - n + 1 + \tfrac{1}{2} \deg R ;\leq; n \cdot g_X + \tfrac{1}{2} \deg R. $$ The stronger version due to Castelnuovo bounds $de g R$ in terms of $n$ and the gonality of $Y$ , yielding the form $g_{Y} \leq n \cdot g_{X} + (n - 1) \cdot N$ with $N$ a ramification-controlled constant. Inverting to bound $g_{X}$ from below given $g_{Y}$ and $n$ : $$ g_X ;\geq; \frac{g_Y - 1 + n - \tfrac{1}{2} \deg R}{n} ;\geq; \frac{g_Y + n - 1}{n} - \tfrac{\deg R}{2n}. $$ For unramified covers ( $de g R = 0$ ): $g_{X} \geq (g_{Y} + n - 1) / n = g_{Y} / n + (n - 1) / n$ , recovering the topological lower bound on the genus of a curve admitting an $n$ -fold unramified cover by a fixed $Y$ .

Exercise 8 (hard, symbolic).

Show that a smooth projective curve $C$ of genus $g \geq 2$ admits at most finitely many degree-2 covers to $P^{1}$ , and hence the property "is hyperelliptic" is well-defined.

Hint

A degree-2 cover $C \to P^{1}$ is given by a $g_{2}^{1}$ , a 2-dimensional subspace of sections of a degree-2 line bundle. Use Riemann-Roch and Hurwitz.

Answer

A degree-2 map $C \to P^{1}$ corresponds to a line bundle $L$ of degree 2 with $h^{0} (L) \geq 2$ (a base-point-free pencil). By Riemann-Roch, $h^{0} (L) - h^{0} (K - L) = 2 - g + 1 = 3 - g$ . For $g \geq 2$ , $de g (K - L) = 2 g - 4 \geq 0$ , and Clifford's theorem gives $h^{0} (L) \leq 2$ . So $h^{0} (L) = 2$ exactly, i.e. there is a unique pencil through $L$ if $L$ is a $g_{2}^{1}$ . By Hurwitz applied to the resulting cover, the ramification divisor has degree $2 g + 2$ , so the branch points form a fixed effective divisor of degree $2 g + 2$ on $P^{1}$ up to $PGL_{2}$ -action.

The hyperelliptic involution sending the two sheets to each other is therefore unique when it exists: any two degree-2 covers $C \to P^{1}$ differ by an automorphism of $P^{1}$ . So "hyperelliptic" is intrinsic to $C$ , and at most one $g_{2}^{1}$ exists on a curve of genus $\geq 2$ .

Lean formalization [Intermediate+]

lean_status: none — Mathlib has the language of finite morphisms of schemes and Kähler differentials, but lacks the canonical-pullback identity $π^{*} K_{X} = K_{Y} - R$ for finite separable maps of smooth curves and the Hurwitz formula as a derived statement. The roadmap items are listed in lean_mathlib_gap.

[object Promise]

Advanced results [Master]

Wild ramification and the different. In characteristic $p > 0$ with $p ∣ e_{y}$ for some ramified point, the contribution at $y$ to $K_{Y} - π^{*} K_{X}$ exceeds $e_{y} - 1$ . The corrected formula uses the different $d_{Y / X}$ , the relative dualising ideal of the extension of local rings $O_{Y, y} \supset O_{X, x}$ : $$ 2 g_Y - 2 ;=; n(2 g_X - 2) ;+; \deg \mathfrak{d}{Y/X}. $$ The order of the different at $y$ satisfies $v_y(\mathfrak{d}{Y/X}) \geq e_y - 1 $w i t h e q u a l i t y i f f$ y $i s t am e [r e f : T O D O_{R} E F S er r e — L oc a l F i e l d s] . W i l d r ami f i c a t i o na dd s hi g h er - r ami f i c a t i o n - g r o u p co n t r ib u t i o n s, or g ani se d b y t h er ami f i c a t i o n f i l t r a t i o n$ G_y \supseteq G_y^{(1)} \supseteq \cdots $o n t h e in er t ia g r o u p a t$ y$.

Cyclic and Kummer covers. A degree- $n$ cyclic cover $π : Y \to X$ in characteristic prime to $n$ is given by extracting an $n$ -th root of a divisor: $O_{Y} = ⨁_{i = 0}^{n - 1} L^{- i}$ where $L^{n} = O (D)$ for an effective divisor $D$ on $X$ . The ramification is concentrated over $D$ , with each point of $D$ contributing $n - 1$ to $de g R$ . Hurwitz then reads $$ 2 g_Y - 2 ;=; n(2 g_X - 2) + (n - 1) \deg D. $$ This is the standard formula for the genus of a superelliptic curve $y^{n} = f (x)$ over $P^{1}$ , where $D = (f)$ has degree $de g f$ .

Castelnuovo-Severi inequality. A consequence of Hurwitz: if $C$ admits non-constant maps $π_{1} : C \to C_{1}$ , $π_{2} : C \to C_{2}$ of degrees $d_{1}, d_{2}$ to curves $C_{1}, C_{2}$ of genera $g_{1}, g_{2}$ , and the maps generate $C$ (i.e. $C \to C_{1} \times C_{2}$ is birational onto its image), then $$ g ;\leq; d_1 \cdot g_1 + d_2 \cdot g_2 + (d_1 - 1)(d_2 - 1). $$ Castelnuovo's bound is a key ingredient in the theory of correspondences and the classical proof of the Riemann hypothesis for curves over finite fields.

Riemann existence theorem and the inverse problem. Over $C$ , the data of a degree- $n$ cover of $X$ ramified over a fixed branch divisor $B = {x_{1}, \dots, x_{r}} \subset X$ is equivalent to the data of a finite-index subgroup of $π_{1} (X ∖ B)$ — a homomorphism from $π_{1} (X ∖ B)$ to $S_{n}$ with transitive image. Hurwitz combined with this correspondence gives existence theorems: for every $g_{X}, n, r$ satisfying the Hurwitz constraint and an integrality condition, there exists at least one cover realising those parameters. The Hurwitz space $H_{g, n, r}$ parametrising such covers up to isomorphism is a smooth quasi-projective variety of dimension $r$ (degree of the branch divisor) ^{[Donaldson §6]}.

Application: gonality. The gonality $gon (C)$ of a curve is the minimal degree of a non-constant map $C \to P^{1}$ . Hurwitz applied to the gonality cover yields a linear lower bound: $gon (C) \geq \frac{g + 2}{r /2 + 1}$ where $r$ is a Brill-Noether parameter, and combined with the Brill-Noether dimension count $ρ = g - (r + 1) (g - d + r)$ classifies generic-curve gonality as $⌈(g + 2) /2 ⌉$ .

Synthesis. Hurwitz is the genus-tracking law for finite covers of curves. It packages three apparently distinct phenomena into one identity: the multiplicativity of Euler characteristic under unramified maps (from algebraic topology), the canonical-divisor pullback formula $π^{*} K_{X} = K_{Y} - R$ (from algebraic geometry), and the local index calculation $v_{y} (π^{*} d t) = e_{y} - 1$ for a tame ramification point (from valuation theory). Each ramification point contributes a single integer $e_{y} - 1$ to a global genus correction, and the formula has no slack — every term is determined by local data. The same identity is the engine of the Castelnuovo-Severi inequality, the existence theorem for branched covers via the Riemann existence theorem, and the gonality/Brill-Noether classification of curves; in every application, the combinatorial input is the same ramification divisor, and the differential-geometric output is a computable genus.

Full proof set [Master]

The proof of the main theorem is given in the Key theorem section. The wild-ramification version follows the same template with $e_{y} - 1$ replaced by $v_{y} (d_{Y / X})$ , and the local computation of the different replaces the differential calculus $d (u^{e}) = e u^{e - 1} d u$ with the structure of higher ramification groups ^{[Serre — Local Fields, Ch. III–IV]}. The cyclic-cover case is a direct calculation of the structure sheaf $O_{Y} = ⨁ L^{- i}$ whose pushforward gives $π_{*} O_{Y}$ , and the ramification is read off the divisor of the section $s \in H^{0} (X, L^{n})$ used to define the cover. The Castelnuovo-Severi inequality is proved by considering the product map $C \to C_{1} \times C_{2}$ and applying the adjunction formula on the surface $C_{1} \times C_{2}$ to the curve $C$ embedded as a divisor; Hurwitz applied to the projections $C_{1} \times C_{2} \to C_{i}$ restricted to $C$ yields the bound. The Riemann existence theorem is proved via the étale fundamental group / monodromy correspondence over $C$ , where Hurwitz gives the necessary integrality of the genus formula.

Connections [Master]

Riemann-Roch theorem for curves 04.04.01 — Hurwitz uses the canonical-divisor degree $de g K_{C} = 2 g - 2$ , which is itself proved from Riemann-Roch applied to the canonical divisor; the two theorems form the foundational pair of curve theory.
Sheaf of differentials 04.08.01 — the canonical pullback identity $π^{*} K_{X} = K_{Y} - R$ is a statement about the relative cotangent sequence $π^{*} Ω_{X / k}^{1} \to Ω_{Y / k}^{1} \to Ω_{Y / X}^{1} \to 0$ , where the cokernel $Ω_{Y / X}^{1}$ is supported exactly on the ramification locus.
Serre duality 04.08.03 — the canonical divisor $K_{C}$ that appears in Hurwitz is the dualising sheaf of the curve, and the formula respects the duality structure: pullback of differentials and pullback of canonical divisors agree because the dualising sheaf is functorial under finite morphisms.
Riemann-Hurwitz on Riemann surfaces 06.05.03 — the analytic version of the same theorem for branched covers of compact Riemann surfaces, equivalent via Serre's GAGA. The local model $z \mapsto z^{e}$ is the same.
Moduli of curves 04.10.01 — Hurwitz spaces $H_{g, n, r}$ parametrising degree- $n$ covers of $P^{1}$ with prescribed branching map to $M_{g}$ via the source curve; these maps and their image cycles are central to the computation of cohomology classes on $\overline{M}_{g}$ .
Adjunction formula — for a smooth curve $C$ on a smooth surface $X$ , $K_{C} = (K_{X} + C) ∣_{C}$ . Combined with Hurwitz applied to the projection of a curve in a product of curves, adjunction gives Castelnuovo-Severi.
Étale fundamental group — finite étale covers of $X$ are classified by finite-index subgroups of $π_{1}^{\overset{e}{ˊ} t} (X)$ ; Hurwitz reads off the genus of the source from the index and the inertia data, providing an arithmetic-geometric counterpart to the topological covering-space theory.

Historical & philosophical context [Master]

Adolf Hurwitz proved the formula now bearing his name in the 1891 paper Über Riemann'sche Flächen mit gegebenen Verzweigungspunkten (Math. Ann. 39, 1-60) ^{[Hurwitz 1891]}. The setting was Riemann surfaces — compact orientable surfaces realised as branched covers of the Riemann sphere — and Hurwitz was concerned with the inverse problem: given a target Riemann surface and a prescribed branching pattern, when does a cover with that branching exist, and how many such covers are there up to isomorphism? The genus formula emerged as the necessary integrality condition.

Riemann had introduced the relevant local data — the Verzweigungsindex (ramification index) — in his 1857 Theorie der Abel'schen Functionen, and the formula for the genus of a smooth plane curve in terms of degree, the Plücker formulas of the 1830s, contained Hurwitz's identity in disguise for the special case of plane projections. Hurwitz's 1891 contribution was the general statement for arbitrary finite covers and the recognition that the ramification data was the only obstruction.

The algebraic-geometric reformulation is due to Weil and Zariski in the 1940s, who reread the formula as the degree comparison of the canonical-divisor pullback identity $π^{*} K_{X} = K_{Y} - R$ . The wild-ramification correction in positive characteristic was worked out by Hasse, Artin, and Serre, culminating in the higher ramification groups and the different of Serre's Local Fields (1962). Grothendieck's relative duality (SGA 6) places the formula in the categorical framework of dualising complexes and the trace map for proper morphisms.

Bibliography [Master]

Hurwitz, A. (1891). Über Riemann'sche Flächen mit gegebenen Verzweigungspunkten. Mathematische Annalen 39, 1-60. [Originator paper.]
Riemann, B. (1857). Theorie der Abel'schen Functionen. Journal für die reine und angewandte Mathematik 54, 115-155. [Verzweigungsindex first introduced.]
Hartshorne, R. Algebraic Geometry. Springer GTM 52, §IV.2. [Standard scheme-theoretic reference.]
Liu, Q. Algebraic Geometry and Arithmetic Curves. Oxford GTM 6, §7.4. [Treatment of Hurwitz with arithmetic care, including the different.]
Donaldson, S. Riemann Surfaces. Oxford GTM 22, §6. [Analytic treatment with the existence direction via Riemann existence.]
Serre, J.-P. Local Fields. Springer GTM 67. [Higher ramification groups; the different in characteristic $p$ .]
Arbarello, E.; Cornalba, M.; Griffiths, P.; Harris, J. Geometry of Algebraic Curves, Vol. I. Grundlehren 267. [Ch. III on Hurwitz numbers and the moduli of branched covers.]

Prerequisites

04.04.01

Tier anchors

beginner: Hartshorne §IV.2 informal — branched covers of curves and ramification bookkeeping
intermediate: Hartshorne §IV.2; Liu — Algebraic Geometry and Arithmetic Curves §7.4
master: Hurwitz 1891 *Über Riemann'sche Flächen mit gegebenen Verzweigungspunkten* (Math. Ann. 39, originator); Hartshorne §IV.2; Liu §7.4; Donaldson — Riemann Surfaces §6

References

TODO_REF
Hurwitz 1891 — Über Riemann'sche Flächen mit gegebenen Verzweigungspunkten · Math. Ann. 39, 1-60; originator paper
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Hartshorne — Algebraic Geometry · §IV.2 Hurwitz's theorem
TODO_REF
Liu — Algebraic Geometry and Arithmetic Curves · §7.4 Hurwitz formula and ramification divisor
TODO_REF
Donaldson — Riemann Surfaces · §6 Branched covers and the Riemann-Hurwitz formula
TODO_REF
Serre — Local Fields · Ch. III–IV; the different and wild ramification

Reviewer

TBD

Estimated time

beginner: 15m
intermediate: 40m
master: 70m