Differential Forms, Riemann-Roch, and Hurwitz's Formula

Based on Shafarevich §III.3–4, Fulton Algebraic Curves Ch. 7–8, Hartshorne §II.8 and §IV.1

Relations

Builds on: Divisors and the Picard Group, Local Properties of Varieties Extended by: (Elliptic curves note — no note yet) Concepts used: Classical Algebraic Geometry

Table of Contents

• 1. Introduction
• 2. Kähler Differentials
  • 2.1 The Module of Kähler Differentials
  • 2.2 Kähler Differentials for Varieties
  • 2.3 Examples
• 3. The Canonical Divisor
  • 3.1 Divisor of a Differential Form
  • 3.2 Degree of the Canonical Class
  • 3.3 Geometric Genus
• 4. The Riemann-Roch Theorem
  • 4.1 Serre Duality
  • 4.2 Statement and Proof Sketch
  • 4.3 Key Corollaries
• 5. Applications of Riemann-Roch
  • 5.1 Genus 0: Classification
  • 5.2 Genus 1: Weierstrass Form from Riemann-Roch
  • 5.3 Genus Formula for Smooth Plane Curves
• 6. Hurwitz’s Formula
  • 6.1 Finite Morphisms and Ramification
  • 6.2 Ramification Divisor and Hurwitz
  • 6.3 Applications
• References

Throughout, \(k\) is an algebraically closed field and \(X\) is a smooth projective curve over \(k\) unless stated otherwise. We write \(\ell(D) = \dim_k H^0(X, \mathcal{O}(D))\) and \(K_X\) for the canonical class. For divisors, \(\mathrm{Pic}(X)\), and \(\mathcal{O}(D)\), see Divisors and the Picard Group; for DVRs and smooth points, see Local Properties.

1. Introduction 📐

The study of rational functions on a curve — their zeros and poles, their linear systems — reaches its apex in the Riemann-Roch theorem. But to state it requires a new ingredient: the canonical class \(K_X\), the divisor class carried by the differential forms on \(X\).

The three main developments of this note are:

1. Kähler differentials. For any \(k\)-algebra \(A\) there is a universal \(A\)-module \(\Omega_{A/k}\) of “formal derivatives.” On a smooth curve \(X\), globalizing this construction yields a rank-1 locally free sheaf \(\omega_X = \Omega_{X/k}\) — the canonical bundle — whose degree is \(2g - 2\).

2. Riemann-Roch. For any divisor \(D\) on \(X\): \[\ell(D) - \ell(K_X - D) = \deg D + 1 - g.\] This single identity simultaneously encodes the structure of every linear system on \(X\) and fixes \(g\) as the fundamental numerical invariant. Using it: genus-0 curves are \(\mathbb{P}^1\), genus-1 curves admit Weierstrass equations, smooth plane curves of degree \(d\) have genus \(\binom{d-1}{2}\).

3. Hurwitz’s formula. For a finite separable morphism \(f: X \to Y\) of degree \(n\) with ramification divisor \(R\): \[2g(X) - 2 = n\bigl(2g(Y) - 2\bigr) + \deg R.\] This propagates the genus formula to coverings and pins down the branch-point count for hyperelliptic curves: exactly \(2g+2\) branch points.

2. Kähler Differentials 🔬

2.1 The Module of Kähler Differentials

Definition (Module of Kähler Differentials). Let \(A\) be a commutative \(k\)-algebra. The module of Kähler differentials \(\Omega_{A/k}\) is the \(A\)-module generated by formal symbols \(\{da : a \in A\}\) modulo the relations: \[d(a + b) = da + db, \qquad d(ab) = a\,db + b\,da, \qquad d\lambda = 0 \text{ for all } \lambda \in k.\] The map \(d: A \to \Omega_{A/k}\), \(a \mapsto da\), is the universal derivation.

The second relation is the Leibniz rule (product rule) from calculus, now imposed algebraically. The condition \(d\lambda = 0\) forces constants to be “flat.”

Proposition (Universal property). For any \(A\)-module \(M\) and any \(k\)-linear map \(D: A \to M\) satisfying \(D(ab) = aD(b) + bD(a)\) (a \(k\)-derivation \(A \to M\)), there is a unique \(A\)-linear map \(\phi: \Omega_{A/k} \to M\) such that \(D = \phi \circ d\).

In categorical language: \(\mathrm{Hom}_A(\Omega_{A/k}, M) \cong \mathrm{Der}_k(A, M)\) naturally in \(M\).

Proof. Define \(\phi(da) = D(a)\) and extend \(A\)-linearly. The Leibniz rule for \(D\) ensures this is consistent on all generators, and linearity forces uniqueness. \(\square\)

Explicit presentation. For \(A = k[x_1, \ldots, x_n]/(f_1, \ldots, f_m)\): \[\Omega_{A/k} = \frac{\bigoplus_{i=1}^n A\,dx_i}{\left\langle \displaystyle\sum_{j=1}^n \frac{\partial f_i}{\partial x_j}\,dx_j : i = 1, \ldots, m \right\rangle}.\]

Proof sketch: For the polynomial ring \(k[x_1,\ldots,x_n]\), the module \(\Omega\) is freely generated by \(dx_1,\ldots,dx_n\) (since \(d(x_i x_j) = x_i\,dx_j + x_j\,dx_i\) is already determined, and every polynomial derivative follows by linearity). Passing to the quotient by \((f_1,\ldots,f_m)\) forces \(df_i = 0\), which imposes the relation \(\sum_j \tfrac{\partial f_i}{\partial x_j}\,dx_j = 0\). \(\square\)

2.2 Kähler Differentials for Varieties

The construction globalizes: for a \(k\)-scheme \(X\), the sheaf of Kähler differentials \(\Omega_{X/k}\) is defined on affine opens \(U = \mathrm{Spec}\,A\) by \(\Omega_{X/k}(U) = \Omega_{A/k}\), and these patch to a quasi-coherent sheaf on \(X\).

Theorem. If \(X\) is a smooth curve over \(k\), then \(\Omega_{X/k}\) is locally free of rank 1 (a line bundle).

Proof sketch: Smoothness at a point \(P\) means \(\mathcal{O}_{X,P}\) is a DVR, hence a regular local ring of dimension 1. The maximal ideal \(\mathfrak{m}_P\) is principal, generated by a local parameter \(t\) (a uniformizer). By the universal property, \(\Omega_{\mathcal{O}_{X,P}/k}\) is generated by \(dt\), and one checks (using the fact that \(\mathcal{O}_{X,P}\) is a flat \(k\)-algebra of relative dimension 1) that \(dt\) is a free generator. Thus \(\Omega_{X/k}|_U \cong \mathcal{O}_U \cdot dt\) on any chart \(U\) with local parameter \(t\). \(\square\)

Definition (Canonical bundle). The canonical bundle of a smooth variety \(X\) of dimension \(n\) is \(\omega_X := \bigwedge^n \Omega_{X/k} = \Omega^n_{X/k}\), the top exterior power of the cotangent sheaf. It is always a line bundle. We write \(K_X\) for the divisor class \([\omega_X] \in \mathrm{Pic}(X)\), the canonical class. For curves (\(n = 1\)), \(\bigwedge^1 \Omega_{X/k} = \Omega_{X/k}\), so the distinction disappears.

Transition functions. On an overlap \(U_\alpha \cap U_\beta\) with local parameters \(t_\alpha, t_\beta\) respectively: the two trivializations \(\omega_X|_{U_\alpha} \cong \mathcal{O}_{U_\alpha} \cdot dt_\alpha\) and \(\omega_X|_{U_\beta} \cong \mathcal{O}_{U_\beta} \cdot dt_\beta\) are related by \[dt_\alpha = \frac{dt_\alpha}{dt_\beta} \cdot dt_\beta,\] where \(dt_\alpha/dt_\beta \in \mathcal{O}_{U_\alpha \cap U_\beta}^\times\) is the Jacobian. These form the transition cocycle of \(\omega_X\).

2.3 Examples

Example 1: Affine line. On \(\mathbb{A}^1 = \mathrm{Spec}\,k[x]\), the Kähler differentials are \(\Omega_{\mathbb{A}^1/k} = k[x]\,dx\), free of rank 1 with generator \(dx\). Every 1-form is \(f(x)\,dx\) for some polynomial \(f\).

Example 2: \(\mathbb{P}^1\). As computed by transition functions, \(\omega_{\mathbb{P}^1} \cong \mathcal{O}(-2)\), a line bundle of degree \(-2\). In particular, \(H^0(\mathbb{P}^1, \omega_{\mathbb{P}^1}) = 0\): there are no nonzero global regular 1-forms on \(\mathbb{P}^1\). This matches \(g(\mathbb{P}^1) = 0\), since \(\ell(K) = g = 0\).

Example 3: Smooth plane curve via adjunction. Let \(C = V(f) \subset \mathbb{P}^2\) be smooth of degree \(d\). The adjunction formula gives: \[\omega_C \cong (K_{\mathbb{P}^2} + C)|_C \cong \mathcal{O}(d-3)|_C.\] Since \(K_{\mathbb{P}^2} = \mathcal{O}(-3)\) (computed in §5.3 below) and \([C] = \mathcal{O}(d)\), the adjunction formula combines these to give the canonical bundle of \(C\) as a restriction. In particular \(\deg \omega_C = d(d-3)\), and since \(\deg K_C = 2g-2\), we get \(g = \binom{d-1}{2}\).

3. The Canonical Divisor 🎯

3.1 Divisor of a Differential Form

A rational 1-form on \(X\) is a nonzero element \(\omega \in K(X) \otimes_{\mathcal{O}_X} \Omega_{X/k}\), where \(K(X)\) is the function field. Near any point \(P \in X\) with local parameter \(t\), we can write \(\omega = f\,dt\) with \(f \in K(X)^\times\).

Definition (Divisor of a differential). The order of \(\omega\) at \(P\) is: \[v_P(\omega) := v_P(f),\] where \(v_P\) is the discrete valuation on \(K(X)\) at \(P\). The divisor of \(\omega\) is: \[\mathrm{div}(\omega) := \sum_{P \in X} v_P(\omega)\,[P].\]

This is well-defined (independent of the choice of local parameter \(t\)) because: if \(t'\) is another local parameter, \(dt = (dt/dt')\,dt'\) where \(dt/dt' \in \mathcal{O}_{X,P}^\times\) (a unit, since both are uniformizers), so \(v_P(f\,dt) = v_P(f) + v_P(dt/dt') = v_P(f)\).

Proposition. The sum \(\mathrm{div}(\omega) = \sum_P v_P(\omega)[P]\) is finite (only finitely many \(P\) have \(v_P(\omega) \neq 0\)), so \(\mathrm{div}(\omega)\) is indeed a divisor.

Proof sketch: On any affine chart \(U = \mathrm{Spec}\,A\) intersecting \(X\), write \(\omega = f\,dt\). The function \(f\) has finitely many zeros and poles on the projective curve \(X\) (since \(\mathrm{div}(f)\) is a finite sum). \(\square\)

Lemma (Linear equivalence of canonical divisors). If \(\omega, \omega'\) are two nonzero rational 1-forms, then \(\mathrm{div}(\omega') = \mathrm{div}(\omega) + \mathrm{div}(g)\) for \(g = \omega'/\omega \in K(X)^\times\). In particular, \(\mathrm{div}(\omega') \sim \mathrm{div}(\omega)\).

Proof: Write \(\omega' = g\omega\); then \(v_P(\omega') = v_P(g) + v_P(\omega)\) for all \(P\). \(\square\)

Definition (Canonical class). The canonical class is \(K_X := [\mathrm{div}(\omega)] \in \mathrm{Pic}(X)\), for any choice of nonzero rational 1-form \(\omega\). By the lemma, this is independent of the choice.

The canonical class is the divisor-theoretic incarnation of the canonical bundle: the sheaf \(\mathcal{O}(K_X)\) is isomorphic to \(\omega_X\) as line bundles on \(X\).

3.2 Degree of the Canonical Class

Theorem. \(\deg K_X = 2g - 2\).

We give a proof that uses Riemann-Roch (which we prove independently in §4), avoiding circularity by first establishing the formula for \(\mathbb{P}^1\) directly.

Proof for \(\mathbb{P}^1\): By Exercise 4, \(\deg K_{\mathbb{P}^1} = -2 = 2(0) - 2\). ✓

Proof for general \(g\) using Riemann-Roch: Apply Riemann-Roch to \(D = K_X\): \[\ell(K) - \ell(K - K) = \deg K + 1 - g \implies \ell(K) - \ell(0) = \deg K + 1 - g.\] We know \(\ell(0) = 1\) (only constant functions are global regular functions on a projective curve) and \(\ell(K) = g\) (by definition of geometric genus, §3.3). Therefore: \[g - 1 = \deg K + 1 - g \implies \deg K = 2g - 2. \quad \square\]

3.3 Geometric Genus

Definition (Geometric genus). The geometric genus of \(X\) is: \[g = g(X) := \ell(K_X) = \dim_k H^0(X, \omega_X).\] Equivalently, \(g\) counts the dimension of the space of global regular 1-forms on \(X\).

The number \(g\) is a birational invariant of \(X\): any two smooth projective curves birational to each other have the same genus. It is also a topological invariant over \(\mathbb{C}\): \(g\) equals the topological genus (number of handles) of the Riemann surface \(X(\mathbb{C})\).

Genus of smooth plane curves. For \(C = V(f) \subset \mathbb{P}^2\) smooth of degree \(d\): \[g(C) = \frac{(d-1)(d-2)}{2} = \binom{d-1}{2}.\]

Derivation (from adjunction): By §2.3, \(\omega_C \cong \mathcal{O}(d-3)|_C\), so \(\deg K_C = d(d-3)\). Setting \(\deg K_C = 2g-2\): \[2g - 2 = d(d-3) = d^2 - 3d \implies g = \frac{d^2 - 3d + 2}{2} = \frac{(d-1)(d-2)}{2}.\]

4. The Riemann-Roch Theorem 🔑

4.1 Serre Duality

Theorem (Serre duality, curve version). Let \(X\) be a smooth projective curve over \(k\) and \(D\) any divisor on \(X\). There is a natural isomorphism: \[H^1(X, \mathcal{O}(D)) \cong H^0(X, \mathcal{O}(K_X - D))^\vee.\] In particular, \(h^1(\mathcal{O}(D)) = \ell(K_X - D)\).

We state this without proof. The key ingredients are: (i) identifying \(\omega_X\) as the dualizing sheaf of \(X\) (the unique sheaf for which Serre duality holds), and (ii) constructing the perfect pairing \(H^0(\mathcal{O}(K-D)) \times H^1(\mathcal{O}(D)) \to H^1(\omega_X) \cong k\) via cup product and a trace map. Full details appear in Hartshorne III.7 or Serre’s Algebraic Groups and Class Fields.

4.2 Statement and Proof Sketch

Theorem (Riemann-Roch). For any divisor \(D\) on a smooth projective curve \(X\) of genus \(g\): \[\ell(D) - \ell(K_X - D) = \deg D + 1 - g.\]

Proof sketch.

Step 1: Reformulate via Euler characteristic. Define \(\chi(\mathcal{O}(D)) = h^0(\mathcal{O}(D)) - h^1(\mathcal{O}(D))\). By Serre duality, \(h^1(\mathcal{O}(D)) = \ell(K - D)\). So the theorem states: \[\chi(\mathcal{O}(D)) = \deg D + \chi(\mathcal{O}).\]

Step 2: Base case. For \(D = 0\): \(\chi(\mathcal{O}) = h^0(\mathcal{O}) - h^1(\mathcal{O}) = 1 - g\) (using \(h^0(\mathcal{O}) = 1\) and \(h^1(\mathcal{O}) = g\)). The formula gives \(\ell(0) - \ell(K) = 0 + 1 - g\), i.e., \(1 - g = 1 - g\). ✓

Step 3: Inductive step. For any closed point \(P \in X\), there is an exact sequence of sheaves on \(X\): \[0 \to \mathcal{O}(D - P) \to \mathcal{O}(D) \to k_P \to 0,\] where \(k_P\) is the skyscraper sheaf at \(P\) with stalk \(k\). Taking the long exact sequence in cohomology: \[0 \to H^0(\mathcal{O}(D-P)) \to H^0(\mathcal{O}(D)) \to k \to H^1(\mathcal{O}(D-P)) \to H^1(\mathcal{O}(D)) \to 0.\] The alternating sum gives \(\chi(\mathcal{O}(D)) = \chi(\mathcal{O}(D-P)) + 1 = \chi(\mathcal{O}(D-P)) + \deg(D) - \deg(D-P)\).

Step 4: Conclusion. By induction (going from \(D = 0\) to any \(D\) by adding or removing points): \[\chi(\mathcal{O}(D)) = \chi(\mathcal{O}) + \deg D = (1 - g) + \deg D. \quad \square\]

4.3 Key Corollaries

The Riemann-Roch theorem \(\ell(D) - \ell(K - D) = \deg D + 1 - g\) has several immediate corollaries that together give a nearly complete picture of linear systems on a curve.

Corollary 1 (Basic invariants). - \(\ell(0) = 1\): the only global regular functions on \(X\) are constants. - \(h^1(\mathcal{O}_X) = g\): the first cohomology of the structure sheaf has dimension \(g\). - \(\ell(K) = g\): there are \(g\) independent global 1-forms. - \(\deg K = 2g - 2\).

Corollary 2 (Non-special divisors). If \(\deg D > 2g - 2\), then \(\deg(K - D) < 0\), so \(\ell(K-D) = 0\) (a line bundle of negative degree has no nonzero global sections). Hence: \[\ell(D) = \deg D + 1 - g \quad \text{for } \deg D > 2g - 2.\] This is the regime where \(\ell(D)\) is determined entirely by \(\deg D\) and \(g\).

Corollary 3 (Negative degree). If \(\deg D < 0\), then \(\ell(D) = 0\).

Corollary 4 (Genus 0). \(g = 0 \implies \ell(D) = \max(0,\, \deg D + 1)\).

Corollary 5 (Riemann’s inequality). Since \(\ell(K-D) \geq 0\): \(\ell(D) \geq \deg D + 1 - g\).

Definition (Special divisor). A divisor \(D\) with \(\ell(K - D) > 0\) (equivalently, \(\deg D \leq 2g-2\) and \(\ell(D) > \deg D + 1 - g\)) is called special. The correction term \(\ell(K-D)\) is the Roch correction.

5. Applications of Riemann-Roch 🌟

5.1 Genus 0: Classification

Theorem. Every smooth projective curve \(X\) of genus \(g = 0\) over an algebraically closed field \(k\) is isomorphic to \(\mathbb{P}^1_k\).

Proof. Since \(g = 0\), we have \(\deg K = -2\). Pick any point \(P \in X\) and consider $D = -K = $ a degree-2 divisor (we use the fact that for \(g = 0\), \(-K\) is effective: \(\ell(-K) = \deg(-K) + 1 - 0 = 3\), so \(|-K|\) is non-empty). Actually, take \(D = P\): \[\ell(P) = \deg P + 1 - g + \ell(K - P) = 1 + 1 - 0 + \ell(K - P).\] Since \(\deg(K-P) = -2 - 1 = -3 < 0\), \(\ell(K-P) = 0\). So \(\ell(P) = 2\).

A 2-dimensional linear system \(|P|\) on \(X\) gives a non-constant map \(\phi_P: X \to \mathbb{P}^1\) (the map associated to the two-dimensional space \(H^0(\mathcal{O}(P))\)). Since \(\deg P = 1\), the map \(\phi_P\) has degree 1, hence is birational. Since both \(X\) and \(\mathbb{P}^1\) are smooth projective curves, a birational morphism is an isomorphism. \(\square\)

5.2 Genus 1: Weierstrass Form from Riemann-Roch

Let \(E\) be a smooth projective curve of genus \(g = 1\) (an elliptic curve) with a marked point \(O \in E\). We derive the Weierstrass equation from the dimensions \(\ell(nO)\).

Computing \(\ell(nO)\) for \(n = 0, 1, 2, 3, \ldots\):

Since \(g = 1\), Riemann-Roch gives \(\ell(D) - \ell(K - D) = \deg D + 1 - 1 = \deg D\). Also \(\deg K = 0\) and \(K \sim 0\) (since \(\omega_E \cong \mathcal{O}_E\)), so \(\ell(K) = 1\).

Note: For \(n \geq 1\): \(\deg(K - nO) = -n < 0\), so \(\ell(K - nO) = 0\), giving \(\ell(nO) = n\).

Constructing Weierstrass form:

• \(H^0(E, \mathcal{O}(2O))\) has dimension 2: basis \(\{1, x\}\) where \(x\) has a double pole at \(O\) and no other poles.
• \(H^0(E, \mathcal{O}(3O))\) has dimension 3: basis \(\{1, x, y\}\) where \(y\) has a triple pole at \(O\).
• Consider the 7 elements \(\{1, x, y, x^2, xy, y^2, x^3\} \subset H^0(E, \mathcal{O}(6O))\).
• \(\ell(6O) = 6\), so these 7 elements are linearly dependent.

The pole orders at \(O\) of these elements are: \(0, 2, 3, 4, 5, 6, 6\) respectively. Any nontrivial linear relation among them must involve both \(y^2\) and \(x^3\) (the two elements with pole order 6), since otherwise we would get a relation among functions with distinct pole orders, forcing all coefficients zero. Thus the relation takes the form:

\[y^2 + a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6, \quad a_i \in k. \tag{W}\]

This is the general Weierstrass form of an elliptic curve. Over \(\mathrm{char}(k) \neq 2, 3\): complete the square in \(y\) and cube in \(x\) to reduce to \(y^2 = x^3 + ax + b\).

5.3 Genus Formula for Smooth Plane Curves

Theorem. A smooth projective curve \(C = V(f) \subset \mathbb{P}^2\) of degree \(d\) has genus \(g = \binom{d-1}{2}\).

Proof via adjunction.

Step 1: Canonical class of \(\mathbb{P}^2\). We compute \(K_{\mathbb{P}^2}\). On the affine chart \(U_{[x:y:z]}\) with \(z \neq 0\): coordinates \((s, t) = (x/z, y/z)\), and the 2-form \(\omega = ds \wedge dt\) is a nonzero rational 2-form on \(\mathbb{P}^2\). In the chart \(U_z' = \{x \neq 0\}\) with \((u,v) = (y/x, z/x)\): \(s = 1/v\), \(t = u/v\), so \(ds = -v^{-2}\,dv\) and \(dt = (v\,du - u\,dv)/v^2\). Thus: \[ds \wedge dt = \frac{-1}{v^3}\,dv \wedge du = \frac{1}{v^3}\,du \wedge dv.\] At \(v = 0\) (the line \(\{z = 0\} = \mathbb{P}^1\) at infinity): \(\omega\) has a triple pole. So \(K_{\mathbb{P}^2} = -3[H]\) where \([H]\) is the class of a hyperplane. Equivalently, \(\omega_{\mathbb{P}^2} \cong \mathcal{O}(-3)\).

Step 2: Adjunction formula. For a smooth hypersurface \(C \subset \mathbb{P}^2\) of degree \(d\), the adjunction formula gives: \[\omega_C \cong (\omega_{\mathbb{P}^2} \otimes \mathcal{O}(C))|_C \cong (\mathcal{O}(-3) \otimes \mathcal{O}(d))|_C \cong \mathcal{O}(d-3)|_C.\] Thus \(\deg K_C = d(d-3)\) (since \(\mathcal{O}(d-3)|_C\) has degree \((d-3) \cdot d\) by Bézout: the restriction of a degree-\((d-3)\) class to a degree-\(d\) curve).

Step 3: Genus. From \(\deg K_C = 2g - 2\): \[2g - 2 = d(d-3) = d^2 - 3d \implies g = \frac{d^2 - 3d + 2}{2} = \frac{(d-1)(d-2)}{2} = \binom{d-1}{2}. \quad \square\]

6. Hurwitz’s Formula 🔀

6.1 Finite Morphisms and Ramification

Setup. Let \(f: X \to Y\) be a finite separable morphism of smooth projective curves over \(k\), with \(\deg f = n\). Separability means the induced extension of function fields \(k(X)/k(Y)\) is a finite separable field extension (equivalently, the derivative map \(df: f^*\Omega_{Y/k} \to \Omega_{X/k}\) is generically nonzero).

Definition (Ramification index). At each point \(P \in X\): choose a local parameter \(t_Q\) at \(Q = f(P) \in Y\). The ramification index of \(f\) at \(P\) is: \[e_P = v_P(f^* t_Q) = v_P(t_Q \circ f),\] where \(v_P\) is the discrete valuation at \(P\) on \(X\). Since \(f\) is finite, \(e_P \geq 1\) always. We say: - \(f\) is unramified at \(P\) if \(e_P = 1\). - \(f\) is ramified at \(P\) if \(e_P \geq 2\).

Degree formula. For any \(Q \in Y\): \[\sum_{P \in f^{-1}(Q)} e_P = n.\] Proof: This follows from the fact that \(\mathcal{O}_{Y,Q} \to \prod_{P | Q} \mathcal{O}_{X,P}\) has degree \(n\) counting multiplicities. \(\square\)

Definition (Branch point). \(Q \in Y\) is a branch point of \(f\) if \(|f^{-1}(Q)| < n\), equivalently if some \(e_P \geq 2\) for \(P \in f^{-1}(Q)\). There are finitely many branch points.

6.2 Ramification Divisor and Hurwitz

Definition (Ramification divisor). The ramification divisor of \(f: X \to Y\) is: \[R = \sum_{P \in X} (e_P - 1)\,[P] \in \mathrm{Div}(X).\] Note \(R \geq 0\) (effective) and \(\mathrm{supp}(R)\) is finite.

Remark on tame vs. wild ramification. In characteristic \(p > 0\): if \(p \mid e_P\), the ramification is wild and the contribution to \(R\) may be larger than \(e_P - 1\) (the correct weight is the different exponent \(d_P \geq e_P - 1\)). For the rest of this section, we assume tame ramification (\(\mathrm{char}(k) = 0\) or \(\mathrm{char}(k) = p\) with \(p \nmid e_P\) for all \(P\)).

Pullback of differentials. For a separable morphism \(f: X \to Y\): there is a canonical map \(f^*\Omega_{Y/k} \to \Omega_{X/k}\), given locally by \(f^*\omega = f^*(\alpha\,dt_Q) = (f^*\alpha)\,d(f^*t_Q)\). Near a point \(P\) with local parameter \(t_P\) and \(f^* t_Q = u \cdot t_P^{e_P}\) (for a unit \(u \in \mathcal{O}_{X,P}^\times\)): \[d(f^* t_Q) = d(u \cdot t_P^{e_P}) = e_P \cdot u \cdot t_P^{e_P - 1}\,dt_P + t_P^{e_P}\,du.\] In tame characteristic, \(e_P \neq 0\) in \(k\), so the leading term is \(e_P u t_P^{e_P-1} dt_P\), which vanishes to order \(e_P - 1\). Thus: \[v_P(f^*\omega) = e_P \cdot v_Q(\omega) + (e_P - 1).\]

Lemma. \(\mathrm{div}(f^*\omega) = f^*(\mathrm{div}(\omega)) + R\), where \(f^*(\mathrm{div}(\omega)) = \sum_{P} e_P \cdot v_{f(P)}(\omega)\,[P]\).

Proof: At each \(P\): \(v_P(f^*\omega) = e_P \cdot v_{f(P)}(\omega) + (e_P - 1)\). The term \(e_P \cdot v_{f(P)}(\omega)\) is \([f^*\mathrm{div}(\omega)]_P\) and \((e_P - 1)\) contributes \(R\). \(\square\)

Theorem (Hurwitz’s Formula). For \(f: X \to Y\) finite separable of degree \(n\) (tamely ramified): \[2g(X) - 2 = n\bigl(2g(Y) - 2\bigr) + \deg R.\]

Proof. Take any nonzero rational 1-form \(\omega\) on \(Y\). Then \(f^*\omega\) is a nonzero rational 1-form on \(X\). Taking degrees of the divisor identity \(\mathrm{div}(f^*\omega) = f^*\mathrm{div}(\omega) + R\): \[\deg K_X = \deg(f^*\mathrm{div}(\omega)) + \deg R.\] Now \(\deg(f^*\mathrm{div}(\omega)) = n \cdot \deg\mathrm{div}(\omega) = n \cdot \deg K_Y\) (the pullback of a divisor on \(Y\) to \(X\) multiplies degree by \(n = \deg f\)). Substituting \(\deg K_X = 2g(X) - 2\) and \(\deg K_Y = 2g(Y) - 2\): \[2g(X) - 2 = n(2g(Y) - 2) + \deg R. \quad \square\]

The formula as a relation on canonical classes. The lemma gives a linear equivalence: \[K_X \sim f^* K_Y + R.\] This is an equality in \(\mathrm{Pic}(X)\), not just a degree statement.

6.3 Applications

Hyperelliptic curves. A hyperelliptic curve is a smooth projective curve \(X\) admitting a degree-2 map \(f: X \to \mathbb{P}^1\). Taking \(n = 2\), \(g(Y) = 0\) in Hurwitz: \[2g(X) - 2 = 2(-2) + \deg R = -4 + \deg R.\] So \(\deg R = 2g(X) + 2\). Since \(R = \sum_P (e_P - 1)[P]\) and each \(e_P \in \{1, 2\}\): \(\deg R\) equals the number of ramification points, each contributing 1. Hence:

\(X\) has exactly \(2g(X) + 2\) branch points in \(\mathbb{P}^1\).

For \(y^2 = f(x)\) with \(f\) squarefree of degree \(2g+2\): the \(2g+2\) roots of \(f\) are the affine branch points, and (for even degree) \(\infty\) is unramified. For \(f\) of degree \(2g+1\): there are \(2g+1\) finite branch points plus \(\infty\) (with \(e_\infty = 2\)), again \(2g+2\) total.

Plane curves via projection. Let \(C \subset \mathbb{P}^2\) be smooth of degree \(d\). Project from a general point \(Q \notin C\) to give \(f: C \to \mathbb{P}^1\) of degree \(d\). The branch points are the values where the line through \(Q\) is tangent to \(C\). By the Plücker formula, the number of such tangent lines (counted with multiplicity) is \(d(d-1)\). Each tangent line gives one ramification point with \(e_P = 2\) (simple tangency), contributing 1 to \(\deg R\). So \(\deg R = d(d-1)\). Hurwitz gives: \[2g - 2 = d(-2) + d(d-1) = -2d + d^2 - d = d^2 - 3d,\] recovering \(g = \binom{d-1}{2}\) — consistent with the adjunction formula computation.

Characteristic \(p\): failure of naive Hurwitz. The Frobenius morphism \(\mathrm{Fr}: X \to X^{(p)}\) (in characteristic \(p > 0\)) is a purely inseparable map of degree \(p\). Since the field extension is purely inseparable, \(\Omega_{X/X^{(p)}} = 0\) — there is no ramification in the sense of differentials. Surprisingly, Hurwitz in its naive form gives \(2g(X) - 2 = p(2g(X^{(p)}) - 2) + 0\), which would force \(g(X) = g(X^{(p)})\) (true!) and \(p = 1\) unless \(g = 1\) — a contradiction for \(p > 1\) in general. The resolution: the Frobenius has \(g(X) = g(X^{(p)})\) (since \(X\) and \(X^{(p)}\) are abstractly isomorphic as varieties), and the formula holds vacuously since purely inseparable maps have \(\Omega_{X/X^{(p)}} = 0\) by definition, so \(\deg R = 0\) and \(g(X) = g(X^{(p)})\) consistently. The Hurwitz formula requires separability to be non-trivial; for purely inseparable maps, it carries no information.

References