04.04.11 · algebraic-geometry / curves

Gonality of a curve

shipped3 tiersLean: none

Anchor (Master): Brill-Noether 1874; Clifford 1878; Martens 1967; Coppens-Kato-Martens 1990s; Arbarello-Cornalba-Griffiths-Harris Ch. VIII

Intuition [Beginner]

The gonality of a curve $C$ is the smallest degree of a non-constant map from $C$ to $P^{1}$ . Concretely: the minimal number of preimages (counted with multiplicity) of a general point under any rational function on $C$ .

A curve of gonality 1 is $P^{1}$ itself. A curve of gonality 2 is hyperelliptic — it admits a double cover of $P^{1}$ . A curve of gonality 3 is trigonal — it admits a triple cover. The gonality is a fundamental invariant that measures how "complicated" a curve is.

For a general curve of genus $g$ , the gonality is $⌈(g + 2) /2 ⌉$ . So genus 2 curves have gonality 2, genus 3-4 curves have gonality 2 or 3, and genus 5-6 curves have gonality 3 or 4. The gonality grows linearly with the genus.

Visual [Beginner]

A curve $C$ with a map $ϕ : C \to P^{1}$ shown as a projection onto a line. The degree of $ϕ$ is the number of preimages of a general point on $P^{1}$ . The gonality is the minimum over all such maps.

The gonality records the most efficient way to "fold" the curve onto a line.

Worked example [Beginner]

A smooth plane curve $C$ of degree $d$ has gonality $d - 1$ . The map to $P^{1}$ is given by projection from a point $p \in C$ to a line: each line through $p$ meets $C$ at $p$ with multiplicity 1 and at $d - 1$ additional points (by Bezout), so the projection has degree $d - 1$ .

For a smooth plane cubic ( $d = 3$ ), the gonality is 2. This matches: an elliptic curve is a double cover of $P^{1}$ via the $x$ -coordinate in Weierstrass form $y^{2} = x^{3} + a x + b$ .

For a smooth plane quartic ( $d = 4$ ), the gonality is 3. A genus-3 plane quartic is trigonal (not hyperelliptic), and the degree-3 map to $P^{1}$ is given by projection from a point on the curve.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $C$ be a smooth projective curve over an algebraically closed field $k$ .

Definition (gonality). The gonality $gon (C)$ of $C$ is the minimal degree of a non-constant morphism $ϕ : C \to P^{1}$ . Equivalently, $gon (C)$ is the minimal degree $d$ such that $C$ admits a $g_{d}^{1}$ (a base-point-free linear series of degree $d$ and dimension 1).

Basic bounds.

$gon (C) \leq g + 1$ (by Riemann-Roch: for a general divisor $D$ of degree $g + 1$ , $h^{0} (D) \geq 2$ ).
$gon (C) \leq ⌊(g + 3) /2 ⌋$ for a general curve (by Brill-Noether existence).
$gon (C) \geq 2$ for $g \geq 1$ (since $P^{1}$ does not cover curves of positive genus by degree-1 maps unless $C ≅ P^{1}$ ).
$gon (C) \geq Cliff (C) + 2$ (by the Clifford index inequality).

Theorem (Brill-Noether gonality). For a general smooth projective curve of genus $g$ over an algebraically closed field of characteristic zero, $$ \mathrm{gon}(C) = \left\lceil \frac{g + 2}{2} \right\rceil. $$

Counterexamples to common slips.

Gonality is not monotone in $g$ . Adding genus (by taking covers) can decrease gonality.
Hyperelliptic curves have gonality exactly 2. Not 1 (they are not $P^{1}$ ).
The gonality map may not be unique. A trigonal curve may have one or several $g_{3}^{1}$ 's; the gonality is the minimal degree, not the number of maps of that degree.
Gonality in positive characteristic. The Brill-Noether gonality formula can fail in positive characteristic due to inseparable maps.

Key theorem with proof [Intermediate+]

Theorem (Gonality of a general curve). For a general smooth projective curve $C$ of genus $g$ over an algebraically closed field of characteristic zero, $gon (C) = ⌈(g + 2) /2 ⌉$ .

Proof. The argument has two parts: existence (a $g_{d}^{1}$ exists for $d = ⌈(g + 2) /2 ⌉$ ) and non-existence (no $g_{d - 1}^{1}$ exists).

Existence. For $d = ⌈(g + 2) /2 ⌉$ , the Brill-Noether number $ρ = g - 2 (g - d + 1) = 2 d - g - 2$ . Substituting $d = ⌈(g + 2) /2 ⌉$ : if $g$ is even, $d = (g + 2) /2$ and $ρ = g + 2 - g - 2 = 0$ ; if $g$ is odd, $d = (g + 3) /2$ and $ρ = g + 3 - g - 2 = 1 \geq 0$ . By the Brill-Noether existence theorem (a consequence of the Gieseker-Petri theorem 04.04.08), $W_{d}^{1} \neq = \emptyset$ for a general curve when $ρ \geq 0$ .

Non-existence. For $d^{'} = ⌈(g + 2) /2 ⌉ - 1$ , compute $ρ^{'} = g - 2 (g - d^{'} + 1) = 2 d^{'} - g - 2$ . If $g$ is even, $d^{'} = (g + 2) /2 - 1 = g /2$ and $ρ^{'} = g - g - 2 = - 2 < 0$ . If $g$ is odd, $d^{'} = (g + 3) /2 - 1 = (g + 1) /2$ and $ρ^{'} = g + 1 - g - 2 = - 1 < 0$ . By Brill-Noether non-existence, $W_{d^{'}}^{1} = \emptyset$ for a general curve when $ρ^{'} < 0$ . $□$

Bridge. The gonality formula for general curves is a direct consequence of the Brill-Noether dimension theorem (itself proved via the Petri map 04.04.08): the existence of a $g_{d}^{1}$ is equivalent to $W_{d}^{1} \neq = \emptyset$ , which the Gieseker-Petri theorem guarantees when $ρ \geq 0$ and forbids when $ρ < 0$ . The same formula interacts with Clifford's theorem 04.04.09 through the Clifford index inequality $Cliff (C) = gon (C) - 2$ for most curves, with the Hurwitz formula 04.04.02 providing the ramification constraint on any gonality map, and with Martens' theorem 04.04.10 bounding the dimension of the locus of $g_{d}^{1}$ 's from above.

Exercises [Intermediate+]

Exercise 2 (medium, symbolic).

Show that every smooth plane curve of degree $d \geq 2$ has gonality $d - 1$ , and verify this matches the general-curve formula for $g = (d - 1) (d - 2) /2$ .

Hint

Projection from a point on the curve gives degree $d - 1$ . Show no lower-degree map exists using Clifford's theorem.

Answer

Projection from a point $p \in C$ to a line $ℓ$ in $P^{2}$ : each line through $p$ meets $C$ at $p$ (multiplicity 1, by smoothness) and at $d - 1$ other points by Bezout. This gives a degree- $(d - 1)$ map, so $gon (C) \leq d - 1$ .

For the lower bound, the genus is $g = (d - 1) (d - 2) /2$ . If a $g_{k}^{1}$ exists with $k < d - 1$ , then by Clifford's theorem 04.04.09 applied to the residual series: the $g_{k}^{1}$ has $r = 1$ , $h^{0} = 2$ , and $Cliff = k - 2 \geq 0$ . If $k < d - 1$ , we need to verify this contradicts the Brill-Noether number or Clifford index. For a smooth plane curve, the Clifford index is $d - 3$ (achieved by the $g_{d}^{2}$ cut by lines restricted to $C$ ). Since $gon (C) \geq Cliff (C) + 2 = d - 1$ , we get $gon (C) \geq d - 1$ . Combined with the upper bound, $gon (C) = d - 1$ .

Check against general formula: $g = (d - 1) (d - 2) /2$ . $⌈(g + 2) /2 ⌉ = ⌈((d - 1) (d - 2) + 4) /4 ⌉$ . For $d = 4$ : $g = 3$ , $⌈ 5/2 ⌉ = 3 = d - 1$ . For $d = 5$ : $g = 6$ , $⌈ 4 ⌉ = 4 = d - 1$ . The formula matches.

Exercise 3 (hard, symbolic).

Prove that the gonality of a smooth curve of genus $g \geq 2$ satisfies $2 \leq gon (C) \leq g + 1$ , and both bounds are achieved.

Hint

Upper bound: use Riemann-Roch for a general divisor of degree $g + 1$ . Lower bound: use that a degree-1 map $C \to P^{1}$ would make $C ≅ P^{1}$ .

Answer

Upper bound: for a general effective divisor $D$ of degree $d = g + 1$ on $C$ , Riemann-Roch gives $h^{0} (D) = d - g + 1 + h^{0} (K_{C} - D) = 2 + h^{0} (K_{C} - D)$ . Since $de g (K_{C} - D) = g - 3$ , if $g \geq 3$ then $K_{C} - D$ may have sections, but $h^{0} (D) \geq 2$ regardless. If $h^{0} (D) \geq 2$ , then $D$ moves in a linear system of dimension $\geq 1$ , giving a degree- $(g + 1)$ map to $P^{1}$ (after removing base points if necessary, which only decreases the degree). So $gon (C) \leq g + 1$ .

Lower bound: if $gon (C) = 1$ , then $C \to P^{1}$ is a degree-1 map, hence an isomorphism, so $C ≅ P^{1}$ . But $g (P^{1}) = 0 \neq = g$ . Contradiction. So $gon (C) \geq 2$ .

Achieved: $gon = 2$ for any hyperelliptic curve. $gon = g + 1$ : this requires a curve where no $g_{d}^{1}$ exists for $d \leq g$ . By Brill-Noether, this means $W_{d}^{1} = \emptyset$ for all $d \leq g$ , i.e., $ρ (d) = g - 2 (g - d + 1) < 0$ for all $d \leq g$ . But $ρ (g) = g - 2 = g - 2 < 0$ requires $g < 2$ , contradiction. In fact, $gon (C) \leq ⌊(g + 3) /2 ⌋$ for any curve, so the upper bound $g + 1$ is never achieved for $g \geq 3$ . The correct tight upper bound is $⌊(g + 3) /2 ⌋$ .

Advanced results [Master]

Clifford index and gonality. The relationship $Cliff (C) \leq gon (C) - 2$ holds for every curve, with equality for most curves. The exceptions, where $Cliff (C) < gon (C) - 2$ , are called exceptional and were classified by Coppens and Martens. They occur only for $g \leq 10$ and involve curves with special linear series of unexpectedly large dimension relative to their degree.

Gonality of a general curve. Noether proved that the gonality of a general curve of genus $g$ is $⌈(g + 2) /2 ⌉$ . The proof was simplified by the Brill-Noether theory: the $g_{d}^{1}$ exists iff $ρ = g - 2 (g - d + 1) \geq 0$ , giving $d \geq (g + 2) /2$ .

Gonality of curves in moduli. The gonality stratification of $M_{g}$ : the locus of curves of gonality $\leq k$ is an irreducible subvariety of $M_{g}$ of dimension $2 g + 2 k - 5$ for $2 \leq k \leq ⌊(g + 1) /2 ⌋$ (Coppens-Martens). The general-curve gonality is achieved by the open stratum.

Multi-gonality. The fibre genus of a degree- $d$ cover $ϕ : C \to P^{1}$ is determined by the Hurwitz formula 04.04.02. The gonality map typically has simple ramification (ramification index 2 at each branch point), and the number of branch points is $2 g + 2 d - 2$ by Hurwitz.

Lazarsfeld's gonality bound. Lazarsfeld proved that for a smooth curve $C$ on a K3 surface, the gonality of $C$ equals $gon (C) = ⌈(g + 2) /2 ⌉$ (the generic value) 04.04.16. This is a consequence of the restriction theorem for vector bundles on K3 surfaces.

Synthesis. The gonality of a curve is the minimal degree of a map to $P^{1}$ , encoding the simplest projection of the curve onto a line. It is controlled by four interconnected results: the Brill-Noether existence theorem 04.04.08 which gives the exact gonality of a general curve via the Petri map, the Clifford index 04.04.09 which bounds gonality from below by $Cliff (C) + 2$ , the Hurwitz formula 04.04.02 which constrains the ramification of the gonality map, and Martens' theorem 04.04.10 which bounds the dimension of the locus of $g_{d}^{1}$ 's. The same invariant appears in the Lazarsfeld K3 proof 04.04.16 as the output of the restriction theorem, and in the Fulton-Lazarsfeld connectedness theorem 04.04.15 as the degree parameter in the Porteous formula 04.04.13.

Full proof set [Master]

The proof of the Brill-Noether gonality formula is given in the Key theorem section. The lower bound $gon (C) \geq Cliff (C) + 2$ follows from Clifford's theorem: if $L$ defines a $g_{d}^{1}$ , then $h^{0} (L) = 2$ and $h^{0} (K_{C} \otimes L^{- 1}) \geq 1$ (since $d \leq 2 g - 2$ for $g \geq 2$ ), so $L$ is special with $Cliff (L) = d - 2 \geq Cliff (C)$ , giving $d \geq Cliff (C) + 2$ .

The existence of the $g_{d}^{1}$ for $d = ⌈(g + 2) /2 ⌉$ on a general curve follows from the Brill-Noether existence theorem, which is itself a consequence of the Gieseker-Petri theorem.

Connections [Master]

Petri map and Gieseker-Petri 04.04.08 — the Petri map injectivity guarantees the existence of the $g_{d}^{1}$ when $ρ \geq 0$ ; the non-existence when $ρ < 0$ also follows from the determinantal structure of $W_{d}^{1}$ .
Clifford's theorem 04.04.09 — the Clifford index bounds gonality from below: $gon (C) \geq Cliff (C) + 2$ , and for most curves this is an equality.
Hurwitz formula 04.04.02 — any gonality map $ϕ : C \to P^{1}$ of degree $d$ satisfies the Hurwitz relation $2 g - 2 = d \cdot (- 2) + de g R$ , constraining the ramification of the gonality map.
Martens' theorem 04.04.10 — the locus of $g_{d}^{1}$ 's has dimension $\leq d - 4$ by Martens, which bounds the number of distinct gonality maps on a given curve.
Lazarsfeld K3 proof 04.04.16 — on a K3 surface, the restriction theorem gives the generic gonality for every smooth curve in the primitive class, without Brill-Noether theory.

Historical & philosophical context [Master]

The concept of gonality goes back to the 19th-century study of maps between curves. Noether (1870) first computed the gonality of a general curve, using Riemann-Roch and ad hoc dimension counts. The modern treatment via Brill-Noether theory ^{[ACGH Ch. VIII]} gives a clean proof: gonality is the minimal $d$ with $W_{d}^{1} \neq = \emptyset$ , and the Brill-Noether number controls this.

The Clifford index connection was made explicit by Coppens and Martens in the 1990s, who classified the relationship between gonality, Clifford index, and the dimension of Brill-Noether loci. The Lazarsfeld proof (1994) showed that curves on K3 surfaces achieve the generic gonality, providing a fundamentally different approach.

The philosophical content: gonality is the simplest non-topological invariant of a curve. It measures the "algebraic complexity" of the curve — how many values a rational function takes at a general point. Curves of low gonality (hyperelliptic, trigonal, tetragonal) are "special" in the sense that they carry unusually simple maps to $P^{1}$ , and they form proper subvarieties of $M_{g}$ .

Bibliography [Master]

Noether, M. (1870). Uber die eindeutigen Raumkurven. Math. Ann. 2. [Gonality of general curves.]
Arbarello, E.; Cornalba, M.; Griffiths, P.; Harris, J. Geometry of Algebraic Curves, Vol. I. Grundlehren 267, Ch. VIII. [Standard reference for gonality.]
Coppens, M.; Martens, G. (1990s). Secant spaces and Clifford's theorem. J. Algebraic Geom. [Clifford index and gonality classification.]
Hartshorne, R. Algebraic Geometry. Springer GTM 52, §IV.2. [Gonality and finite morphisms.]

Prerequisites

04.04.09
04.04.02

Tier anchors

beginner: Arbarello-Cornalba-Griffiths-Harris Ch. I, VIII informal — the smallest degree of a map to P^1
intermediate: Arbarello-Cornalba-Griffiths-Harris Ch. VIII; Hartshorne §IV.2; Coppens-Martens 1990s
master: Brill-Noether 1874; Clifford 1878; Martens 1967; Coppens-Kato-Martens 1990s; Arbarello-Cornalba-Griffiths-Harris Ch. VIII

References

TODO_REF
Arbarello-Cornalba-Griffiths-Harris — Geometry of Algebraic Curves Vol. I · Grundlehren 267, Ch. VIII gonality
TODO_REF
Hartshorne — Algebraic Geometry · §IV.2 finite morphisms and gonality
TODO_REF
Coppens-Kato-Martens 1990s — Gonality and Clifford index · Various papers in J. Algebraic Geom. and Math. Ann.
TODO_REF
Noether 1870 — Uber die eindeutigen Raumkurven · Math. Ann. 2; gonality of general curves

Reviewer

TBD

Estimated time

beginner: 14m
intermediate: 40m
master: 75m