Over a field of characteristic $p$, the Artin--Schreier cover $y^p - y = f(x)$ defines a degree-$p$ cover $C \to \mathbb{P}^1$.

Consider $y^p - y = x^m$ with $\gcd(m, p) = 1$ over $\mathbb{F}_p$. The map $C \to \mathbb{P}^1$ has degree $p$, and:

• Over $x = \infty$, there is a single point with $e = p$ and wild ramification. The different exponent is $e - 1 + \delta = (p-1) + (m+1)(p-1) - (p-1) = (m+1)(p-1)$ when computed carefully.
• All other fibers are unramified (for generic $m$).

By the Hurwitz formula: $2g_C - 2 = p(2 \cdot 0 - 2) + (m+1)(p-1)$, giving $2g_C - 2 = -2p + (m+1)(p-1)$, so $g_C = \frac{(p-1)(m-1)}{2}$ when $m$ is chosen appropriately.

Compare this with the tame case: for $y^n = x^m$ in characteristic $0$ with $\gcd(m,n) = 1$, the genus formula yields a different answer because there is no wild contribution.

A hyperelliptic curve of genus $g$ is a smooth projective curve $C$ admitting a degree-$2$ map $f: C \to \mathbb{P}^1$. Applying Hurwitz with $n = 2$, $g_D = 0$:

$2g_C - 2 = 2(2 \cdot 0 - 2) + \deg R = -4 + \deg R.$

Each ramification point has $e_P = 2$ (since the degree is $2$, and any ramification index must divide the degree), so $e_P - 1 = 1$ at each ramification point. If there are $r$ ramification points:

$2g - 2 = -4 + r, \quad \text{hence } r = 2g + 2.$

Conclusion: A hyperelliptic curve of genus $g$ (in characteristic $\neq 2$) has exactly $2g + 2$ branch points in $\mathbb{P}^1$.

For example: genus $1$ (elliptic) has $4$ branch points, genus $2$ has $6$, genus $3$ has $8$.

Consider $C: y^2 = f(x)$ where $f(x) \in k[x]$ is a squarefree polynomial of degree $d$, with $\operatorname{char}(k) \neq 2$. The projection $(x,y) \mapsto x$ gives a degree-$2$ map $C \to \mathbb{P}^1$.

Ramification occurs at the roots of $f(x)$ (where $y = 0$, so there is a single point above each root) and possibly at $\infty$:

• If $d$ is odd: there is one point above $\infty$ with $e = 2$, so $r = d + 1$ ramification points.
• If $d$ is even: there are two points above $\infty$ with $e = 1$, so $r = d$ ramification points.

Hurwitz gives $2g - 2 = -4 + r$, so:

• $d$ odd: $g = (d - 1)/2$,
• $d$ even: $g = (d - 2)/2$.

In both cases, $g = \lfloor (d-1)/2 \rfloor$. For instance: $y^2 = x^5 + 1$ has genus $2$; $y^2 = x^6 + 1$ also has genus $2$.

Let $C: y^n = f(x)$ where $f(x) = \prod_{i=1}^{d}(x - a_i)$ is squarefree, $\operatorname{char}(k) \nmid n$, and $\gcd(n, d)$ divides $n$. The projection to $x$ gives a degree-$n$ map $f: C \to \mathbb{P}^1$.

At each root $a_i$, there is a unique point above with $e = n$ (when $\gcd(n, d) = 1$ or handled carefully). The total ramification over a finite branch point $a_i$ contributes $n/e_j \cdot (e_j - 1)$ for points above it, where we need to track the local structure.

In the simplest case where $n \mid d$ and all ramification is total (single point above each $a_i$ with $e = n$): each branch point contributes $n - 1$ to $\deg R$, and over $\infty$ the fiber is unramified. Then:

$2g_C - 2 = n(-2) + d(n - 1),$

giving $g_C = \frac{(n-1)(d-2)}{2} + 1 - \frac{d(n-1)}{2} + \frac{d(n-1)}{2}$. More carefully:

$g_C = \frac{(n-1)(d-2)}{2}$

when $n \mid d$ and $d \geq 2$. For $n = 2$, this recovers $g = (d-2)/2$ for even $d$.

The Fermat curve $F_n: X^n + Y^n = Z^n$ in $\mathbb{P}^2$ is a smooth curve of degree $n$ (when $\operatorname{char}(k) \nmid n$).

Method 1 (degree-genus formula): For a smooth plane curve of degree $n$:

$g = \frac{(n-1)(n-2)}{2}.$

Method 2 (Hurwitz formula): Consider the projection $[X:Y:Z] \mapsto [X:Z]$ from $F_n$ to $\mathbb{P}^1$. For each value of $X/Z = a$, we solve $Y^n = Z^n - X^n = Z^n(1 - a^n)$.

• If $a^n \neq 1$: there are $n$ distinct solutions, so the fiber is unramified.
• If $a^n = 1$ (the $n$-th roots of unity): then $Y^n = 0$, so there is a single point with $e = n$.
• At $a = \infty$ (i.e., $Z = 0$): the equation becomes $X^n + Y^n = 0$, giving $n$ distinct points (since $\operatorname{char}(k) \nmid n$).

There are $n$ branch points (the $n$-th roots of unity), each with a single point above having $e = n$, contributing $n - 1$ each. Hurwitz gives:

$2g - 2 = n(-2) + n(n-1) = n(n - 3),$

so $g = \frac{n^2 - 3n + 2}{2} = \frac{(n-1)(n-2)}{2}$, confirming the degree-genus formula.

Let $f: C \to \mathbb{P}^1$ be a degree-$3$ cover (in characteristic $\neq 2, 3$). Then:

$2g_C - 2 = 3(-2) + \deg R = -6 + \deg R.$

Possible ramification types over a branch point $Q$:

• (3): One point above $Q$ with $e = 3$, contributing $3 - 1 = 2$ to $\deg R$.
• (2, 1): One point with $e = 2$ and one with $e = 1$, contributing $2 - 1 = 1$ to $\deg R$.

If we have $a$ branch points of type $(3)$ and $b$ of type $(2,1)$, then $\deg R = 2a + b$ and:

$g_C = \frac{-6 + 2a + b + 2}{2} = \frac{2a + b - 4}{2} = a + \frac{b}{2} - 2.$

Since $g_C \geq 0$ is an integer, $b$ must be even. For genus $0$: need $2a + b = 4$, giving $(a, b) \in \{(0, 4), (1, 2), (2, 0)\}$. For genus $1$: need $2a + b = 6$.

Concrete example: $y^3 = x(x-1)$. This is a degree-$3$ cover of $\mathbb{P}^1$ totally ramified over $0$, $1$, and $\infty$, so $a = 3$, $b = 0$, giving $g = 3 + 0 - 2 = 1$. It is an elliptic curve.

An elliptic curve over $k$ (with $\operatorname{char}(k) \neq 2$) is always a double cover of $\mathbb{P}^1$ branched at $4$ points. In Weierstrass form $y^2 = (x - e_1)(x - e_2)(x - e_3)$:

• The map $(x, y) \mapsto x$ is a degree-$2$ map $E \to \mathbb{P}^1$.
• Ramification at $x = e_1, e_2, e_3$ (where $y = 0$) and at $x = \infty$ (after compactification).
• Hurwitz: $2(1) - 2 = 2(0 - 2) + 4 \cdot 1$, i.e., $0 = -4 + 4$ ✓.

The $4$ branch points $\{e_1, e_2, e_3, \infty\}$ determine $E$ up to isomorphism (modulo automorphisms of $\mathbb{P}^1$). Since $\operatorname{Aut}(\mathbb{P}^1)$ is $3$-dimensional and we have $4$ points, the moduli space of elliptic curves is $1$-dimensional, parametrized by the $j$-invariant.

Hurwitz gives strong non-existence results:

A genus-$2$ curve cannot cover a genus-$2$ curve non-trivially. If $f: C \to D$ with $g_C = g_D = 2$ and $\deg f = n$, Hurwitz gives $2 = n \cdot 2 + \deg R$. Since $\deg R \geq 0$, we need $n \leq 1$. So $n = 1$, i.e., $f$ is an isomorphism.

A genus-$3$ curve can cover a genus-$2$ curve only by a degree-$1$ map (isomorphism is impossible since genera differ) or... Hurwitz: $4 = 2n + \deg R$, so $\deg R = 4 - 2n$. For $n = 2$: $\deg R = 0$, so $f$ must be unramified of degree $2$. Such maps exist (unramified double covers correspond to $2$-torsion in $\operatorname{Pic}^0(D) \cong \operatorname{Jac}(D)[2]$, which is $(\mathbb{Z}/2)^4$).

No map from $\mathbb{P}^1$ to a curve of genus $\geq 1$. If $g_C = 0$, then $-2 = n(2g_D - 2) + \deg R$. For $g_D \geq 1$: the right side is $\geq n \cdot 0 + 0 = 0 > -2$, contradiction.

If $g_C = g_D = 2$, then Hurwitz gives $n \leq \frac{2}{2} = 1$ for an unramified map, and for ramified maps $\deg R = 2 - 2n$, which forces $n = 1$. So the only maps from a genus-$2$ curve to another genus-$2$ curve are isomorphisms, and there are finitely many (in fact, the automorphism group of a genus-$2$ curve is finite, of order at most $48$ in characteristic $0$).

An unramified double cover $f: C \to D$ with $\deg R = 0$ gives $2g_C - 2 = 2(2g_D - 2)$, so $g_C = 2g_D - 1$.

• Double cover of genus $1$: $g_C = 1$ (an isogeny between elliptic curves).
• Double cover of genus $2$: $g_C = 3$.
• Double cover of genus $3$: $g_C = 5$.

Unramified double covers of $D$ are classified by $H^1(D, \mathbb{Z}/2) \cong (\mathbb{Z}/2)^{2g_D}$, giving $2^{2g_D} - 1$ non-trivial covers (up to isomorphism as covers). For $g_D = 2$: there are $15$ unramified double covers, each a genus-$3$ curve.

Let $C$ be a curve of genus $g$ admitting an involution $\sigma: C \to C$ (an automorphism of order $2$, with $\operatorname{char}(k) \neq 2$). The quotient $D = C/\langle \sigma \rangle$ is a smooth curve, and $f: C \to D$ is a degree-$2$ map. Let $g_D$ be the genus of $D$ and let $r$ be the number of fixed points of $\sigma$ (which are exactly the ramification points). By Hurwitz:

$2g - 2 = 2(2g_D - 2) + r, \quad \text{hence } r = 2g - 4g_D + 2.$

Since $r \geq 0$: $g_D \leq (g + 1)/2$, i.e., $g_D \leq \lfloor (g+1)/2 \rfloor$.

Special cases:

• $g_D = 0$ (hyperelliptic involution): $r = 2g + 2$.
• $g_D = 1$: $r = 2g - 2$, so $g \geq 1$. For $g = 1$, $r = 0$ (isogeny).
• $g_D = g/2$: $r = 2$, a "bi-elliptic" situation when $g_D = 1$ and $g = 2$.

Consider the tower $C \xrightarrow{f} D \xrightarrow{g} E$ with $\deg f = m$, $\deg g = n$. Applying Hurwitz twice:

$2g_C - 2 = m(2g_D - 2) + \deg R_f,$ $2g_D - 2 = n(2g_E - 2) + \deg R_g.$

Substituting: $2g_C - 2 = m \cdot n(2g_E - 2) + m \cdot \deg R_g + \deg R_f$.

Alternatively, applying Hurwitz to $g \circ f: C \to E$ of degree $mn$:

$2g_C - 2 = mn(2g_E - 2) + \deg R_{g \circ f}.$

Comparing: $\deg R_{g \circ f} = m \cdot \deg R_g + \deg R_f$. This reflects the multiplicativity of ramification: $e_P(g \circ f) = e_P(f) \cdot e_{f(P)}(g)$.

Belyi's theorem states that a smooth projective curve $C$ over $\overline{\mathbb{Q}}$ admits a morphism $f: C \to \mathbb{P}^1$ branched over at most $3$ points (which may be taken to be $\{0, 1, \infty\}$).

For such a Belyi map $f: C \to \mathbb{P}^1$ of degree $n$ branched over $\{0, 1, \infty\}$, the Hurwitz formula gives:

$2g - 2 = -2n + \deg R.$

The ramification data over each branch point defines a partition of $n$: if the fiber over $0$ has ramification indices $(e_1, \ldots, e_r)$, this gives a partition $e_1 + \cdots + e_r = n$ with contribution $\sum(e_i - 1) = n - r$ to $\deg R$.

If the three partitions have $r_0$, $r_1$, $r_\infty$ parts respectively, then $\deg R = (n - r_0) + (n - r_1) + (n - r_\infty) = 3n - r_0 - r_1 - r_\infty$, giving:

$g = \frac{n + 2 - r_0 - r_1 - r_\infty}{2}.$

For genus $0$ Belyi maps: $r_0 + r_1 + r_\infty = n + 2$. For genus $1$: $r_0 + r_1 + r_\infty = n$.

Let $G$ be a finite group acting on a smooth projective curve $C$ over $k$ (with $\operatorname{char}(k) \nmid |G|$). The quotient $D = C/G$ is a smooth curve, and $f: C \to D$ is a Galois cover of degree $|G|$.

For a point $Q \in D$, the stabilizer $G_P$ at any $P \in f^{-1}(Q)$ has order $e_P$, and $|f^{-1}(Q)| = |G|/e_P$. The Hurwitz formula becomes:

$2g_C - 2 = |G|(2g_D - 2) + |G| \sum_{Q \in D} \left(1 - \frac{1}{e_Q}\right),$

where $e_Q = e_P$ for any $P$ above $Q$ (well-defined since the cover is Galois).

This form is especially useful: the sum $\sum_Q (1 - 1/e_Q)$ is taken over branch points of $D$, and the contribution of each branch point depends only on the ramification index.

The Klein quartic is the curve $C: X^3 Y + Y^3 Z + Z^3 X = 0$ in $\mathbb{P}^2$, which is smooth of degree $4$ and genus $g = 3$.

It has automorphism group $G = \operatorname{PSL}(2, \mathbb{F}_7)$ of order $168$, the maximum for genus $3$ (by the Hurwitz bound $|\operatorname{Aut}(C)| \leq 84(g - 1) = 168$).

The quotient $D = C/G$ has genus $g_D$, and the Hurwitz formula gives:

$2(3) - 2 = 168(2g_D - 2) + 168 \sum_Q \left(1 - \frac{1}{e_Q}\right).$

So $4 = 168(2g_D - 2) + 168 \cdot S$ where $S = \sum_Q (1 - 1/e_Q)$. For $g_D = 0$: $4 = -336 + 168S$, giving $S = 340/168 = 85/42$. One checks that $S = (1 - 1/2) + (1 - 1/3) + (1 - 1/7) = 1/2 + 2/3 + 6/7 = 85/42$ ✓.

So $C \to \mathbb{P}^1$ is branched at $3$ points with ramification indices $2, 3, 7$. This is the origin of the famous $(2, 3, 7)$ triangle group.

Let $C \subset \mathbb{P}^2$ be a smooth curve of degree $d$. Projecting from a point $P \notin C$ gives $f: C \to \mathbb{P}^1$ of degree $d$ (a generic line through $P$ meets $C$ in $d$ points).

For a general projection, the ramification occurs at the points where the tangent line to $C$ passes through $P$. By a class computation (or Plucker formulas), the number of tangent lines from a general point to a smooth degree-$d$ curve is $d(d-1)$, and each tangent point has $e = 2$. So $\deg R = d(d-1) \cdot 1 = d(d-1)$, and:

$2g - 2 = d(-2) + d(d - 1) = d^2 - 3d,$

giving $g = \frac{(d-1)(d-2)}{2}$, the degree-genus formula.

A smooth complete intersection $C = V(F_1, \ldots, F_{n-1}) \subset \mathbb{P}^n$ of hypersurfaces of degrees $d_1, \ldots, d_{n-1}$ is a curve of degree $d = d_1 \cdots d_{n-1}$ and genus:

$g = 1 + \frac{d}{2}\left(\sum_{i=1}^{n-1} d_i - n - 1\right).$

This follows from the adjunction formula rather than Hurwitz directly, but one can verify it via Hurwitz by projecting to $\mathbb{P}^1$ and computing ramification.

Examples:

• Curve of type $(2, 2)$ in $\mathbb{P}^3$: $d = 4$, $g = 1 + 2(2 + 2 - 4) = 1$. An elliptic curve!
• Curve of type $(2, 3)$ in $\mathbb{P}^3$: $d = 6$, $g = 1 + 3(2 + 3 - 4) = 4$.
• Curve of type $(2, 2, 2)$ in $\mathbb{P}^4$: $d = 8$, $g = 1 + 4(2 + 2 + 2 - 5) = 5$.