04.10.33 · algebraic-geometry / moduli

Explicit low-genus moduli: Igusa-Clebsch invariants for M_2 and the plane-quartic model of M_3

shipped3 tiersLean: none

Anchor (Master): Igusa 1960 *Ann. of Math.* 72, 612-649; Clebsch 1872 *Theorie der binären algebraischen Formen*; Dixmier 1987 *On the projective invariants of quartic plane curves* (Adv. Math. 64); Ohno (unpublished, see Elsenhans 2015 and Lercier-Ritzenthaler 2012 *J. Algebra* for the Dixmier-Ohno reconstruction); Bolza 1887 *Amer. J. Math.* 10 (the genus-2 automorphism stratification); Mestre 1991 (reconstruction of a genus-2 curve from its Igusa invariants)

Intuition Beginner

A smooth curve of genus $2$ always arises in the same way: take the line, choose six points on it, and build a double cover branched over exactly those six points. The resulting surface is a genus- $2$ curve, and every genus- $2$ curve appears this way. So a genus- $2$ curve is the same data as six points on a line, up to the freedom of redrawing the line by a fractional-linear change of coordinates.

This gives a beautiful picture of the moduli space. The space of all genus- $2$ curves is the space of six unordered points on a line, divided by the coordinate changes of the line. The hard part is turning "divided by coordinate changes" into honest numbers. The answer is to write down combinations of the six points that do not change when you redraw the line. Such combinations are called invariants. For genus $2$ the right list is four numbers, of weights $2, 4, 6, 10$ , named after Igusa and Clebsch. Knowing these four numbers up to a common rescaling tells you the curve.

Genus $3$ has two flavours. A generic genus- $3$ curve sits in the plane as a curve cut out by one degree-four equation, a plane quartic. The invariants are now built from the coefficients of that quartic equation, and they are unchanged under linear coordinate changes of the plane. The special genus- $3$ curves are the hyperelliptic ones, double covers of the line branched over eight points, and they form a thin boundary inside the genus- $3$ moduli space. The point of this unit is that for these lowest genera the abstract moduli space becomes a concrete list of numbers you can compute.

Visual Beginner

A two-panel diagram. The left panel shows a line with six marked branch points and, drawn above it, the genus- $2$ curve as a two-sheeted cover that joins the sheets at those six points; an arrow labelled "invariants" points to a box listing four weighted coordinates of weights $2, 4, 6, 10$ . The right panel shows the projective plane with a smooth quartic curve drawn inside it and twenty-eight short line segments each touching the quartic at two points (the bitangents); an arrow labelled "invariants" points to a box of plane-quartic invariants, and a thin shaded strip at the edge is labelled "hyperelliptic locus: eight points on a line".

The picture records the central contrast of the unit. Both moduli spaces were already described in the corpus by a quotient slogan: six or eight points on a line modulo coordinate change. Here that slogan is replaced by explicit coordinate rings of invariants, which is what makes these spaces computable in practice.

Worked example Beginner

Build the weighted coordinates for a genus- $2$ curve from its six branch points, in the simplest stages.

Step 1. Start with a genus- $2$ curve written as $y^{2} = f (x)$ , where $f$ is a degree-six polynomial with six distinct roots. Those six roots are the six branch points on the line. Two such curves are the same curve exactly when one set of six points is carried to the other by a coordinate change of the line.

Step 2. To compare them, form numbers out of the six roots that do not move under coordinate changes. The simplest such number measures whether the six points are all distinct: it is the discriminant, which is built from the products of differences of the roots and vanishes exactly when two branch points collide. A collision means the curve degenerates, so the discriminant staying away from zero is the condition for a smooth genus- $2$ curve.

Step 3. The full list of independent numbers has four entries, traditionally of weights $2, 4, 6, 10$ . The weight records how each number rescales when you scale the polynomial $f$ by a constant. The weight- $10$ entry is the discriminant. The other three are lower-weight averages of the root configuration.

Step 4. Because rescaling $f$ rescales each weight- $w$ entry by the scale to the power $w$ , only the ratios that cancel the scaling carry geometric meaning. So the curve is named not by the four numbers themselves but by the point they define in a weighted projective space, where a coordinate of weight $w$ may be rescaled by the $w$ -th power of a common factor.

What this shows: the genus- $2$ moduli space is a weighted projective space of these four coordinates, with the locus where the weight- $10$ discriminant vanishes removed. Computing the four numbers from a given equation is a finite, mechanical task.

Check your understanding Beginner

Formal definition Intermediate+

Work over an algebraically closed field $k$ of characteristic zero (or of characteristic away from the relevant small primes; Igusa's 1960 treatment handles all characteristics). Let $V_{d} = Sym^{d} (k^{2})^{\lor}$ be the space of binary forms of degree $d$ in variables $x, z$ , a representation of $SL_{2} = SL (k^{2})$ .

Definition (binary sextic and its invariant ring). A binary sextic is a nonzero element $$ f(x, z) = \sum_{i = 0}^{6} \binom{6}{i} a_i, x^{6 - i} z^{i} \in V_6. $$ The group $SL_{2}$ acts on $V_{6}$ by substitution. An invariant of degree $n$ is a polynomial $Φ$ in the coefficients $a_{0}, \dots, a_{6}$ with $Φ (g \cdot f) = Φ (f)$ for all $g \in SL_{2}$ . The graded ring of all invariants is denoted $R = k [V_{6}]^{SL_{2}}$ . It is finitely generated (Gordan 1868 for binary forms; Hilbert's basis theorem in general), and a homogeneous invariant of degree $n$ in the coefficients has weight $w = 3 n$ under the scaling $f \mapsto t \cdot f$ in the sense that it scales by $t^{n}$ while a coordinate change of the line of determinant $λ$ multiplies it by $λ^{w}$ with $w = 3 n$ .

Definition (Igusa-Clebsch invariants). The Clebsch invariants $A, B, C, D$ of degrees $2, 4, 6, 10$ in the coefficients generate the even part of $R$ . Igusa's normalised generators, the Igusa-Clebsch invariants, are $$ I_2, \quad I_4, \quad I_6, \quad I_{10}, $$ of degrees $2, 4, 6, 10$ in the $a_{i}$ (equivalently weights $6, 12, 18, 30$ under the determinant scaling, but they are uniformly written with subscript equal to the degree in the coefficients). Igusa also introduces $J_{2}, J_{4}, J_{6}, J_{8}, J_{10}$ , polynomial reexpressions adapted to all characteristics, with $J_{10}$ the discriminant $$ J_{10} = \mathrm{disc}(f) = a_0^{10} \prod_{i < j} (\alpha_i - \alpha_j)^2 \quad (\text{up to a constant}), $$ where $α_{1}, \dots, α_{6}$ are the roots of $f$ . The invariant $J_{10}$ vanishes exactly when $f$ has a repeated root.

Definition (moduli space of genus-2 curves). Every smooth projective curve $C$ of genus $2$ is hyperelliptic (Clifford's theorem and the canonical map force the degree- $2$ map to $P^{1}$ ; see 04.04.09), so $C : y^{2} = f (x)$ for a binary sextic $f$ with distinct roots, well defined up to the $PGL_{2}$ action on $f$ and the scaling that absorbs $y$ . The coarse moduli space is the quotient $$ \mathcal{M}2 ;=; \big{, f \in V_6 : J{10}(f) \neq 0 ,\big} \big/ \mathrm{GL}2, $$ and Igusa's theorem realises it as the open subscheme ${J{10} \neq 0} $o f t h e w e i g h t e d p r o j ec t i v e$ \mathrm{Proj}$ of the invariant ring, $$ \overline{\mathcal{M}}2^{\mathrm{Ig}} = \mathrm{Proj}, R ;\cong; \mathbb{P}(2, 4, 6, 10) \big/ (\text{one relation}), $$ with weighted coordinates $[I_2 : I_4 : I_6 : I{10}] $s u bj ec tt o t h e w e i g h t s$ 2, 4, 6, 10 $. T h e p o in t o f$ \mathcal{M}_2 $a tt a c h e d t o$ C$ is the weighted tuple of its Igusa-Clebsch invariants.

Definition (genus-3 plane-quartic model). Let $W = Sym^{4} (k^{3})^{\lor}$ be the space of ternary quartics, a representation of $SL_{3}$ . A non-hyperelliptic genus- $3$ curve embeds by its canonical map 04.08.02 as a smooth quartic ${F = 0} \subset P^{2}$ with $F \in W$ , and two such quartics give isomorphic curves iff they differ by $PGL_{3}$ . The Dixmier-Ohno invariants are an explicit generating set of $k [W]^{SL_{3}}$ : Dixmier's seven invariants $I_{3}, I_{6}, I_{9}, I_{12}, I_{15}, I_{18}, I_{27}$ form a homogeneous system of parameters, and Ohno completed the list to thirteen generators $J_{9}, J_{12}, J_{15}, J_{18}, I_{21}, J_{21}$ (and $I_{27}$ the discriminant). The non-hyperelliptic part of $M_{3}$ is the GIT quotient $$ \mathcal{M}_3^{\mathrm{nh}} ;=; {\text{smooth quartics}} \big/ \mathrm{PGL}_3, $$ and the hyperelliptic locus is the codimension-one stratum $$ \mathcal{H}_3 ;=; \mathrm{Sym}^8 \mathbb{P}^1 \big/ \mathrm{PGL}_2 ;\subset; \mathcal{M}_3, $$ the closure of the non-hyperelliptic part meeting $H_{3}$ along the locus where the canonical model degenerates from a quartic to a conic-double-cover.

Counterexamples to common slips

The Igusa-Clebsch invariants $(I_{2}, I_{4}, I_{6}, I_{10})$ are not coordinates on an affine space; they live in a weighted projective space, and only weighted ratios such as $I_{2}^{5} / I_{10}$ or $I_{4} I_{6} / I_{10}$ are honest functions on $M_{2}$ . Treating the four numbers as an ordinary point of $A^{4}$ mis-identifies isomorphic curves.
The map "curve to invariant tuple" is not injective on the four raw numbers but is injective on the weighted-projective point, away from the extra-automorphism strata where a separate Bolza analysis is needed.
The hyperelliptic locus in $M_{3}$ is not defined by the vanishing of the discriminant of a plane quartic; it is the locus where the canonical model is not a plane quartic at all. The discriminant invariant $I_{27}$ vanishes on singular quartics, a different (boundary) phenomenon.
The slogan $M_{2} ≅ Sym^{6} P^{1} / PGL_{2}$ is correct as a set-level quotient but says nothing about coordinates; the content of Igusa's theorem is the ring $R$ that turns the quotient into a weighted projective variety with computable equations.

Key theorem with proof Intermediate+

Theorem (Igusa, 1960). Over a field of characteristic zero, the ring of invariants $R = k [V_{6}]^{SL_{2}}$ of binary sextics is generated by $J_{2}, J_{4}, J_{6}, J_{8}, J_{10}$ with a single relation expressing $J_{8}$ as a polynomial in $J_{2}, J_{4}, J_{6}, J_{10}$ , so that $Proj R ≅ P (2, 4, 6, 10)$ . The coarse moduli space of smooth genus-2 curves is the open subscheme $$ \mathcal{M}2 ;=; {, [I_2 : I_4 : I_6 : I{10}] \in \mathbb{P}(2, 4, 6, 10) : I_{10} \neq 0 ,}, $$ and two smooth genus-2 curves are isomorphic iff their weighted Igusa-Clebsch tuples coincide.

Proof. The argument has four steps: reduce isomorphism of curves to $GL_{2}$ -equivalence of sextics, identify functions on the quotient with invariants, exhibit the generators, and cut out the smooth locus by the discriminant.

Step 1: curves to sextics. A smooth genus- $2$ curve $C$ has canonical bundle of degree $2$ with $h^{0} (ω_{C}) = 2$ , and the canonical map $C \to P^{1}$ is the degree- $2$ hyperelliptic map (Clifford's theorem rules out a degree- $1$ canonical map, see 04.04.09). Its branch divisor is $2 g + 2 = 6$ distinct points of $P^{1}$ , recorded by a binary sextic $f$ with distinct roots, and $C : y^{2} = f$ . Two curves $C, C^{'}$ are isomorphic iff their branch divisors differ by an automorphism of $P^{1}$ , that is iff $f^{'} = λ \cdot (g \cdot f)$ for $g \in PGL_{2}$ and a scalar $λ$ ; absorbing the scalar, isomorphism classes of smooth genus- $2$ curves are the $GL_{2}$ -orbits of sextics with distinct roots.

Step 2: quotient functions are invariants. Regular functions on the affine GIT quotient ${J_{10} \neq = 0} / GL_{2}$ are exactly the $GL_{2}$ -invariant functions, that is the degree-zero weighted ratios of elements of $R = k [V_{6}]^{SL_{2}}$ after accounting for the central scaling. The numerical criterion 04.10.03 identifies the semistable locus: a binary sextic is semistable for the $SL_{2}$ -action iff no root has multiplicity exceeding $3$ , and stable iff no root has multiplicity $\geq 3$ ; the locus of distinct roots is contained in the stable locus, so the quotient of ${J_{10} \neq = 0}$ is a geometric quotient with coordinate ring the degree-zero part of the localisation $R [J_{10}^{- 1}]$ .

Step 3: the generators and the relation. Clebsch (1872) constructed invariants $A, B, C, D$ of degrees $2, 4, 6, 10$ by repeated transvection of $f$ with itself, and these generate the ring of invariants of a binary sextic over $C$ . Igusa normalised them to $J_{2}, J_{4}, J_{6}, J_{8}, J_{10}$ valid in all characteristics and proved by an explicit elimination that $R$ is generated by these five with one relation, the expression of $J_{8}$ as a polynomial in the others; geometrically $Proj R$ is the weighted projective space $P (2, 4, 6, 10)$ on $[I_{2} : I_{4} : I_{6} : I_{10}]$ .

Step 4: the smooth locus. The discriminant $J_{10}$ vanishes exactly on sextics with a repeated root, that is on the degenerate (non-smooth or non-genus- $2$ ) fibres. Removing ${J_{10} = 0}$ leaves the locus of smooth genus- $2$ curves, and Step 1 makes the assignment $C \mapsto [I_{2} : I_{4} : I_{6} : I_{10}]$ a bijection from isomorphism classes of smooth genus- $2$ curves to the open subscheme ${I_{10} \neq = 0}$ . The weighted-projective point is a complete isomorphism invariant. $□$

Bridge. This explicit invariant ring builds toward the general GIT construction of $\overline{M}_{g}$ and appears again in 04.10.01, where the same curve $M_{2}$ is produced abstractly as a GIT quotient of a Hilbert scheme; the foundational reason the two constructions agree is that both compute the ring of $SL_{2}$ -invariant functions on the parameter space of sextics, and this is exactly the slogan $M_{2} ≅ Sym^{6} P^{1} / PGL_{2}$ made into a coordinate ring. The bridge is that the abstract quotient and the concrete weighted projective variety are the same scheme, computed two ways: the numerical criterion of 04.10.03 selects the stable sextics, and the invariant ring of 04.10.02 supplies the coordinates. Putting these together, Igusa's weighted projective description generalises to the Dixmier-Ohno description of the genus- $3$ plane-quartic moduli, where ternary quartics replace binary sextics and $PGL_{3}$ replaces $PGL_{2}$ , and the central insight that the explicit invariants compute the GIT quotient is dual to the abstract Hilbert-scheme construction recorded in 04.10.01.

Exercises Intermediate+

Exercise 3 (medium, symbolic).

Compute the discriminant $disc (f)$ of the binary sextic $f (x) = \prod_{i = 1}^{6} (x - α_{i})$ up to a constant, and explain why it is the top Igusa invariant $J_{10}$ .

Hint

The discriminant of a degree- $d$ polynomial is the resultant of $f$ and its derivative, equal up to a constant to $\prod_{i < j} (α_{i} - α_{j})^{2}$ .

Answer

$disc (f) = \prod_{i < j} (α_{i} - α_{j})^{2}$ up to the leading-coefficient power $a_{0}^{2 d - 2} = a_{0}^{10}$ for $d = 6$ . There are $(2 6) = 15$ pairs, each contributing a squared difference, so the product has degree $2 \cdot 15 = 30$ in the roots, equivalently degree $10$ in the coefficients $a_{0}, \dots, a_{6}$ (since each $a_{i}$ is an elementary symmetric polynomial of degree $1$ in scaled roots, and the homogeneity in coefficients is $2 d - 2 = 10$ ). This degree- $10$ invariant is $J_{10}$ . It is $SL_{2}$ -invariant because the resultant of $f$ and $f^{'}$ transforms by a power of the determinant, equal to $1$ on $SL_{2}$ , and it vanishes exactly when two roots coincide, that is when the genus- $2$ curve $y^{2} = f$ degenerates. Rubric: full credit for the squared-difference product, the degree- $10$ count, and the $SL_{2}$ -invariance via the resultant.

Exercise 4 (medium, short-answer).

Why is every smooth genus- $2$ curve hyperelliptic, while a generic genus- $3$ curve is not? Use the canonical map.

Hint

The canonical map of a genus- $g$ curve sends $C$ to $P^{g - 1}$ . Compare $g = 2$ (target $P^{1}$ ) with $g = 3$ (target $P^{2}$ ).

Answer

For $g = 2$ the canonical map goes to $P^{g - 1} = P^{1}$ , and a non-constant map of a genus- $2$ curve to $P^{1}$ of degree $de g ω_{C} = 2 g - 2 = 2$ is exactly a hyperelliptic structure; so every genus- $2$ curve is hyperelliptic. For $g = 3$ the canonical map goes to $P^{g - 1} = P^{2}$ and has degree $2 g - 2 = 4$ . For a non-hyperelliptic curve it is an embedding onto a smooth plane quartic 04.08.02; for a hyperelliptic curve it factors through the degree- $2$ hyperelliptic map to a rational normal curve (here a conic), so the canonical image is a conic covered two-to-one rather than a quartic. The generic genus- $3$ curve has an embedding canonical map and is therefore a plane quartic, while the hyperelliptic ones form the codimension-one locus $Sym^{8} P^{1} / PGL_{2}$ . Rubric: full credit for the canonical-map degree argument at both genera and the embedding-versus-double-cover dichotomy.

Exercise 5 (medium, short-answer).

State the Hilbert-Mumford stability condition for a binary sextic under $SL_{2}$ , and explain why smooth genus- $2$ curves land in the stable locus.

Hint

A binary form is unstable iff some root has multiplicity strictly larger than half the degree; the numerical criterion of 04.10.03 measures this by one-parameter subgroups concentrating mass at a single root.

Answer

For the $SL_{2}$ -action on degree- $d$ binary forms, the Hilbert-Mumford numerical criterion 04.10.03 gives: a form is semistable iff no root has multiplicity $> d /2$ , and stable iff no root has multiplicity $\geq d /2$ . For $d = 6$ this reads: stable iff every root has multiplicity $\leq 2$ , semistable iff every root has multiplicity $\leq 3$ . A smooth genus- $2$ curve $y^{2} = f$ requires $f$ to have six distinct roots, all of multiplicity $1 < 3$ , so $f$ is stable; in fact the whole locus ${J_{10} \neq = 0}$ of distinct-root sextics is stable, which is why the GIT quotient there is a geometric quotient and a fine description of $M_{2}$ as a variety. The destabilising one-parameter subgroup for a multiplicity- $\geq 4$ root is the diagonal torus fixing that root. Rubric: full credit for the multiplicity- $\leq 2$ stable / $\leq 3$ semistable thresholds and the distinct-roots consequence.

Exercise 6 (hard, short-answer).

The smooth plane quartic carries $28$ bitangent lines. Derive the count $28$ from the theory of theta-characteristics, and relate it to the genus- $3$ invariant theory.

Hint

Bitangents of a canonical genus- $3$ quartic correspond to the odd theta-characteristics, of which there are $2^{g - 1} (2^{g} - 1)$ .

Answer

A bitangent line of the canonical quartic $C \subset P^{2}$ meets $C$ in a divisor $2 (p + q)$ , so $p + q$ is an effective theta-characteristic: a line bundle $L$ with $L^{\otimes 2} ≅ ω_{C}$ and $h^{0} (L) > 0$ , that is an odd theta-characteristic. The number of odd theta-characteristics on a genus- $g$ curve is $2^{g - 1} (2^{g} - 1)$ , which at $g = 3$ equals $4 \cdot 7 = 28$ . Each odd theta-characteristic of a non-hyperelliptic genus- $3$ curve has a unique section vanishing on a bitangent contact divisor, giving the classical $28$ bitangents. On the invariant side, the configuration of $28$ bitangents is acted on by the symplectic group $Sp (6, F_{2})$ , and the Dixmier-Ohno invariants of the ternary quartic are $PGL_{3}$ -invariant functions on the same space of quartics whose vanishing loci detect special bitangent configurations (the discriminant $I_{27}$ vanishes when two contact points collide and the quartic acquires a node). Rubric: full credit for the odd-theta-characteristic count $2^{g - 1} (2^{g} - 1) = 28$ and the link to the quartic invariants.

Exercise 7 (hard, short-answer).

Explain why $M_{2}$ has the structure of a weighted projective variety rather than an ordinary projective variety, and what goes wrong if one forgets the weights.

Hint

The generators of the invariant ring have unequal degrees $2, 4, 6, 10$ , so the $Proj$ is taken with respect to a non-standard grading.

Answer

The invariant ring $R = k [I_{2}, I_{4}, I_{6}, I_{10}]$ is graded with generators of degrees $2, 4, 6, 10$ , not all degree $1$ . The scaling $f \mapsto t \cdot f$ sends $I_{w} \mapsto t^{w} I_{w}$ , so the natural identification is of curves with points of the weighted projective space $P (2, 4, 6, 10) = Proj R$ , where $[I_{2} : I_{4} : I_{6} : I_{10}] = [λ^{2} I_{2} : λ^{4} I_{4} : λ^{6} I_{6} : λ^{10} I_{10}]$ for all $λ \neq = 0$ . If one forgets the weights and treats $(I_{2}, I_{4}, I_{6}, I_{10})$ as a point of ordinary $P^{3}$ with the uniform scaling $λ \cdot (I_{2}, I_{4}, I_{6}, I_{10})$ , then genuinely isomorphic curves (related by $f \mapsto t f$ ) map to different points whenever $t$ is not a tenth root of unity, so the ordinary-projective identification fails to be a moduli invariant. The weighted grading is exactly what records the way each invariant rescales, and $P (2, 4, 6, 10)$ is the correct ambient space. The absolute invariants such as $I_{2}^{5} / I_{10}$ are the degree-zero functions that descend to honest coordinates on the open chart. Rubric: full credit for the unequal-degree grading, the weighted rescaling law, and the failure of the uniform-scaling identification.

Advanced results Master

Theorem (Igusa, normalised invariants in all characteristics; Igusa 1960). There exist invariants $J_{2}, J_{4}, J_{6}, J_{8}, J_{10}$ of the binary sextic, polynomial in the coefficients with integer coefficients, such that over every field the ring of invariants is generated by them, $J_{10}$ is the discriminant, and the arithmetic variety of moduli of genus-2 curves is $Spec$ of the degree-zero subring of $Z [J_{2}, \dots, J_{10}] [J_{10}^{- 1}]$ . ^{[Igusa 1960]}

Igusa's achievement was to make the genus- $2$ moduli space an arithmetic object, a scheme over $Z$ , by replacing the Clebsch invariants $A, B, C, D$ (which have denominators in small primes) with integral generators. The absolute invariants $j_{1} = J_{2}^{5} / J_{10}$ , $j_{2} = J_{2}^{3} J_{4} / J_{10}$ , $j_{3} = J_{2}^{2} J_{6} / J_{10}$ coordinatise the moduli space over $Z [1/2]$ and are the genus- $2$ analogue of the $j$ -invariant of an elliptic curve, used in modern point-counting and complex-multiplication constructions of genus- $2$ Jacobians.

Theorem (Bolza stratification; Bolza 1887). The smooth genus-2 curves with extra automorphisms form a finite list of strata inside $M_{2}$ , indexed by the possible automorphism groups beyond the generic $Z /2$ : the strata with $Z /2 \times Z /2$ , $D_{4}$ , $D_{6}$ , $Z /2 \times S_{4}$ (the $y^{2} = x^{5} - x$ curve), and the $Z /10$ stratum of $y^{2} = x^{6} - 1$ . ^{[Bolza 1887]}

The automorphism strata are cut out inside $P (2, 4, 6, 10)$ by explicit vanishing conditions on the Igusa invariants. They are the genus- $2$ counterpart of the $j = 0$ and $j = 1728$ special points of the elliptic modular line, and the stack $M_{2}$ carries non-reduced automorphism gerbes along them. The reconstruction problem — recover a curve from its Igusa tuple — is solved generically by Mestre's algorithm ^{[Mestre 1991]} and requires case analysis precisely along the Bolza strata.

Theorem (Dixmier-Ohno generation; Dixmier 1987, Ohno). The invariant ring $k [Sym^{4} (k^{3})^{\lor}]^{SL_{3}}$ of ternary quartics is finitely generated; Dixmier's seven invariants $I_{3}, I_{6}, I_{9}, I_{12}, I_{15}, I_{18}, I_{27}$ form a homogeneous system of parameters, and Ohno's six further invariants complete a generating set of thirteen, with $I_{27}$ the discriminant cutting out the singular quartics. ^{[Dixmier 1987]}

The non-hyperelliptic locus $M_{3}^{nh}$ is the corresponding GIT quotient ${smooth quartics} / PGL_{3}$ , of dimension $14 - 8 = 6$ matching $dim M_{3} = 3 g - 3 = 6$ . The thirteen Dixmier-Ohno invariants give a closed embedding of this quotient into a weighted projective space, and the Lercier-Ritzenthaler reconstruction ^{[Lercier-Ritzenthaler 2012]} inverts the invariant map generically, the genus- $3$ analogue of Mestre's genus- $2$ algorithm. The hyperelliptic stratum $Sym^{8} P^{1} / PGL_{2}$ is governed instead by the invariant theory of binary octics, a separate and larger invariant ring.

Theorem (twenty-eight bitangents and the Aronhold sets). A smooth plane quartic has exactly $28$ bitangent lines, in bijection with the odd theta-characteristics; they carry an action of $Sp (6, F_{2})$ , and an Aronhold set of $7$ of them determines the remaining $21$ and reconstructs the quartic.

The classical Aronhold invariant and the configuration of $28$ bitangents are the projective-geometry incarnation of the genus- $3$ level- $2$ structure. The $36$ even theta-characteristics give the symmetric determinantal representations of the quartic, and the $28$ odd ones give the bitangents; the symmetry group $Sp (6, F_{2})$ of order $1451520$ permutes them. An Aronhold set is a maximal set of $7$ bitangents in general position whose contact data reconstructs the quartic, the genus- $3$ analogue of the six branch points naming a genus- $2$ curve, and the bridge from the synthetic plane-curve picture to the $PGL_{3}$ -invariant ring.

Synthesis. The central insight of explicit low-genus moduli is that the abstract GIT-quotient slogan becomes a computable coordinate ring exactly when the relevant invariant ring is written down, and this is the foundational reason the genus- $2$ and genus- $3$ moduli spaces are the only ones for which a complete, mechanical isomorphism test exists. Putting these together, three apparently distinct descriptions of $M_{2}$ — the set-level quotient $Sym^{6} P^{1} / PGL_{2}$ , the abstract GIT quotient of a Hilbert scheme, and Igusa's weighted projective variety $P (2, 4, 6, 10) ∖ {I_{10} = 0}$ — are one scheme computed three ways, and the bridge is the invariant ring $R$ that turns the quotient into coordinates. This is exactly the pattern that recurs at genus $3$ : the binary sextic is replaced by the ternary quartic, $PGL_{2}$ by $PGL_{3}$ , the four Igusa invariants by the thirteen Dixmier-Ohno invariants, and the six branch points by the seven-element Aronhold set of bitangents. The genus- $2$ construction is dual to the genus- $3$ construction: one realises a hyperelliptic moduli space through branch-point invariants and the other a canonical-embedding moduli space through plane-curve invariants, meeting along the hyperelliptic locus of $M_{3}$ where the genus- $3$ canonical model degenerates and binary-octic invariant theory takes over. This identification of explicit invariants with the abstract GIT quotient generalises to the modern arithmetic of low-genus Jacobians, where the Igusa and Dixmier-Ohno invariants are the working coordinates for complex multiplication, point-counting, and curves with prescribed reduction.

Full proof set Master

Proposition (weighted-projective structure of $M_{2}$ ). The coarse moduli space $M_{2}$ is isomorphic to the open subscheme ${I_{10} \neq = 0}$ of the weighted projective space $P (2, 4, 6, 10)$ , and is a rational affine variety of dimension $3$ .

Proof. By the Key Theorem, $M_{2} = Proj R \cap {I_{10} \neq = 0}$ with $R = k [I_{2}, I_{4}, I_{6}, I_{10}]$ free on generators of degrees $2, 4, 6, 10$ (the single Igusa relation expresses the redundant $J_{8}$ , so the four $I$ 's are algebraically independent). Inverting $I_{10}$ and passing to degree-zero elements gives the affine coordinate ring generated by $u = I_{2}^{5} / I_{10}$ , $v = I_{2}^{3} I_{4} / I_{10}$ , $w = I_{2}^{2} I_{6} / I_{10}$ , which are three algebraically independent absolute invariants; hence $M_{2}$ is affine of dimension $3$ , matching $dim M_{2} = 3 g - 3 = 3$ . Rationality is immediate since the absolute invariants give a birational map to $A^{3}$ . The weighted-projective ambient space records the rescaling $I_{w} \mapsto t^{w} I_{w}$ under $f \mapsto t f$ , and the open condition $I_{10} \neq = 0$ is the smoothness (distinct-roots) condition of the sextic. $□$

Proposition (dimension of the genus-3 plane-quartic moduli). The non-hyperelliptic locus $M_{3}^{nh} = {smooth quartics} / PGL_{3}$ has dimension $6$ , equal to $dim M_{3}$ , and the hyperelliptic locus has dimension $5$ .

Proof. The space of ternary quartics $Sym^{4} (k^{3})^{\lor}$ has dimension $(2 4 + 2) = 15$ , so $P (Sym^{4})$ has dimension $14$ . The group $PGL_{3}$ has dimension $8$ and acts with finite generic stabiliser on smooth quartics (a generic smooth plane quartic has only the identity automorphism), so the quotient has dimension $14 - 8 = 6 = 3 g - 3$ at $g = 3$ . The hyperelliptic locus is $Sym^{8} P^{1} / PGL_{2}$ of dimension $8 - 3 = 5$ , one less, so it is a codimension-one stratum of $M_{3}$ . The two strata together fill $M_{3}$ , with the non-hyperelliptic part open and dense. $□$

Proposition (bitangent count). A smooth plane quartic over an algebraically closed field of characteristic zero has exactly $28$ bitangents.

Proof. Bitangents of the canonical quartic $C$ correspond to effective odd theta-characteristics: a bitangent line $ℓ$ meets $C$ in $ℓ \cdot C = 2 (p + q)$ with $O_{C} (p + q)$ a square root of $ω_{C} = O_{C} (1)$ admitting a section, that is an odd theta-characteristic $L$ with $L^{\otimes 2} ≅ ω_{C}$ and $h^{0} (L)$ odd and positive. The set of theta-characteristics is a torsor under the $2$ -torsion $J (C) [2] ≅ (Z /2)^{2 g}$ , of order $2^{2 g} = 64$ at $g = 3$ , split by the Arf invariant of the Weil pairing into $2^{g - 1} (2^{g} + 1) = 36$ even and $2^{g - 1} (2^{g} - 1) = 28$ odd ones. Each odd theta-characteristic on a non-hyperelliptic genus- $3$ curve has $h^{0} (L) = 1$ with the unique section vanishing on a bitangent contact divisor, giving a bijection with the $28$ bitangents. $□$

Proposition (stability threshold for binary forms). Under the $SL_{2}$ -action on $V_{d}$ , a binary form is semistable iff every root has multiplicity $\leq d /2$ and stable iff every root has multiplicity $< d /2$ ; in particular distinct-root sextics are stable.

Proof. Apply the Hilbert-Mumford numerical criterion 04.10.03. A one-parameter subgroup $λ (t) = diag (t, t^{- 1})$ acts on the monomial $x^{d - i} z^{i}$ with weight $d - 2 i$ . The minimal weight on the support of $f$ measures instability: the criterion gives $μ (f, λ) = max_{i} {2 i - d : a_{i} \neq = 0}$ minimised over the $SL_{2}$ -orbit, which is negative (stable) iff no coordinate of $P^{1}$ can be moved to a root of multiplicity $\geq d /2$ . Concretely, $f$ is unstable iff some root has multiplicity $> d /2$ , strictly semistable iff the maximal root multiplicity equals $d /2$ , and stable iff all multiplicities are $< d /2$ . For $d = 6$ , distinct roots have multiplicity $1 < 3 = d /2$ , so distinct-root sextics are stable and the GIT quotient of ${J_{10} \neq = 0}$ is geometric. $□$

Connections Master

Moduli of curves 04.10.01. This unit supplies the explicit coordinate rings for the two lowest-genus cases of the general moduli space $M_{g}$ constructed abstractly in 04.10.01. Where 04.10.01 builds $M_{g}$ as a GIT quotient of a Hilbert scheme of tri-canonically embedded curves and computes $dim M_{2} = 3$ and $dim M_{3} = 6$ by deformation theory, the present unit realises those same spaces as $P (2, 4, 6, 10) ∖ {I_{10} = 0}$ and as ${quartics} / PGL_{3}$ with concrete invariants. The abstract slogan $M_{2} ≅ Sym^{6} P^{1} / PGL_{2}$ stated in 04.10.01 becomes a computable weighted-projective variety here.
Geometric invariant theory 04.10.02. Igusa's and Dixmier-Ohno's constructions are textbook instances of the GIT quotient of 04.10.02: the affine quotient $Spec (k [V]^{G})$ and projective quotient $Proj (⨁ k [V]_{n}^{G})$ for $G = SL_{2}$ acting on binary sextics and $G = SL_{3}$ acting on ternary quartics. The finite generation of the invariant ring (Hilbert, Gordan) that makes GIT quotients projective is exactly the finite generation of $R$ that makes $M_{2}$ a weighted projective variety. This unit is the explicit-coordinate face of the abstract GIT machine.
Hilbert-Mumford numerical criterion 04.10.03. The stability of binary forms — semistable iff no root has multiplicity $> d /2$ — is a direct computation with the numerical criterion of 04.10.03 via the diagonal one-parameter subgroup. The selection of the stable locus ${J_{10} \neq = 0}$ of distinct-root sextics, where the GIT quotient is geometric and equals the fine moduli interpretation, rests entirely on this criterion. The genus- $3$ case applies the same criterion to ternary quartics, where smooth quartics are stable for $PGL_{3}$ .
Clifford theorem / hyperelliptic curves 04.04.09. That every genus- $2$ curve is hyperelliptic, and that the genus- $3$ moduli space splits into a non-hyperelliptic plane-quartic part and a hyperelliptic stratum, both rest on the Clifford bound and the structure of the canonical map analysed in 04.04.09. The hyperelliptic involution and its $2 g + 2$ branch points are what make the binary-sextic and binary-octic invariant theories the right tool for the hyperelliptic loci.
Canonical sheaf 04.08.02. The plane-quartic model of a non-hyperelliptic genus- $3$ curve is its canonical embedding $ϕ_{K} : C ↪ P^{2} = P (H^{0} (ω_{C})^{\lor})$ studied in 04.08.02, whose image is the degree- $2 g - 2 = 4$ plane quartic. The dichotomy embedding-versus-double-cover of the canonical map is exactly what separates the plane-quartic locus from the hyperelliptic locus in $M_{3}$ , and the bitangents are the contact lines of this canonical model.

Historical & philosophical context Master

The invariant theory of binary forms was the central preoccupation of nineteenth-century algebra, and the genus- $2$ moduli problem was one of its first geometric payoffs. Alfred Clebsch in Theorie der binären algebraischen Formen (Teubner, 1872) ^{[Clebsch 1872]} constructed the invariants $A, B, C, D$ of a binary sextic by the symbolic method of transvectants, and the realisation that these invariants coordinatise the space of genus- $2$ curves was understood by Clebsch, Bolza, and the classical Italian and German schools. Oskar Bolza in On binary sextics with linear transformations into themselves (American Journal of Mathematics 10, 1887, 47-70) ^{[Bolza 1887]} classified the genus- $2$ curves with extra automorphisms, giving the stratification of the moduli space by automorphism group that remains in use; his special curves $y^{2} = x^{5} - x$ and $y^{2} = x^{6} - 1$ are the genus- $2$ analogues of the harmonic and equianharmonic elliptic curves.

The decisive modern step was Jun-ichi Igusa's Arithmetic variety of moduli for genus two (Annals of Mathematics 72, 1960, 612-649) ^{[Igusa 1960]}, which recast the classical invariant theory over $Z$ . Igusa replaced the Clebsch invariants, which have inconvenient denominators, with integral generators $J_{2}, \dots, J_{10}$ valid in all characteristics, proved that $M_{2}$ is the weighted projective $Proj$ of their ring with the discriminant $J_{10}$ removed, and thereby made the moduli of genus- $2$ curves an arithmetic scheme. This is the genus- $2$ counterpart of the classical fact that the moduli of elliptic curves is the affine $j$ -line, and Igusa's absolute invariants $j_{1}, j_{2}, j_{3}$ are the direct generalisation of the $j$ -invariant. Igusa's paper is a landmark in the transition from the synthetic invariant theory of the nineteenth century to the scheme-theoretic moduli theory of the twentieth, and it predates Mumford's general GIT construction of $M_{g}$ while computing the genus- $2$ case in closed form.

The genus- $3$ story splits the classical heritage in two. The non-hyperelliptic plane-quartic invariant theory descends from the work of Aronhold and Clebsch on ternary quartics and their $28$ bitangents, a configuration with the rich $Sp (6, F_{2})$ symmetry that fascinated nineteenth-century geometers and connects to the $E_{7}$ root system. The modern invariant-theoretic completion is due to Jacques Dixmier in On the projective invariants of quartic plane curves (Advances in Mathematics 64, 1987, 279-304) ^{[Dixmier 1987]}, who exhibited seven fundamental invariants forming a homogeneous system of parameters, and to Toshiaki Ohno, who completed the generating set to thirteen; the explicit reconstruction of a quartic from its Dixmier-Ohno invariants was developed by Reynald Lercier and Christophe Ritzenthaler in Hyperelliptic curves and their invariants (Journal of Algebra 372, 2012, 595-636) ^{[Lercier-Ritzenthaler 2012]} and collaborators. The philosophical point that recurs from genus $2$ to genus $3$ is that a moduli space described by an abstract quotient becomes an effective, arithmetic object only once its invariant ring is computed, and the computation is feasible precisely in these lowest genera; the contrast with the abstract GIT quotient, which exists for all $g$ but is explicit for almost none, is the organising lesson of Harris and Morrison's worked low-genus examples ^{[Harris-Morrison 1998]}.

Bibliography Master

@article{Igusa1960,
  author  = {Igusa, Jun-ichi},
  title   = {Arithmetic variety of moduli for genus two},
  journal = {Annals of Mathematics},
  volume  = {72},
  year    = {1960},
  pages   = {612--649}
}

@book{Clebsch1872,
  author    = {Clebsch, Alfred},
  title     = {Theorie der bin{\"a}ren algebraischen Formen},
  publisher = {B. G. Teubner},
  address   = {Leipzig},
  year      = {1872}
}

@article{Dixmier1987,
  author  = {Dixmier, Jacques},
  title   = {On the projective invariants of quartic plane curves},
  journal = {Advances in Mathematics},
  volume  = {64},
  year    = {1987},
  pages   = {279--304}
}

@article{Bolza1887,
  author  = {Bolza, Oskar},
  title   = {On binary sextics with linear transformations into themselves},
  journal = {American Journal of Mathematics},
  volume  = {10},
  year    = {1887},
  pages   = {47--70}
}

@incollection{Mestre1991,
  author    = {Mestre, Jean-Fran{\c{c}}ois},
  title     = {Construction de courbes de genre 2 {\`a} partir de leurs modules},
  booktitle = {Effective Methods in Algebraic Geometry (Castiglioncello 1990)},
  series    = {Progress in Mathematics},
  volume    = {94},
  publisher = {Birkh{\"a}user},
  year      = {1991},
  pages     = {313--334}
}

@article{LercierRitzenthaler2012,
  author  = {Lercier, Reynald and Ritzenthaler, Christophe},
  title   = {Hyperelliptic curves and their invariants: geometric, arithmetic and algorithmic aspects},
  journal = {Journal of Algebra},
  volume  = {372},
  year    = {2012},
  pages   = {595--636}
}

@book{HarrisMorrison1998,
  author    = {Harris, Joe and Morrison, Ian},
  title     = {Moduli of Curves},
  publisher = {Springer-Verlag},
  series    = {Graduate Texts in Mathematics},
  volume    = {187},
  year      = {1998}
}

@book{MukaiInvariants,
  author    = {Mukai, Shigeru},
  title     = {An Introduction to Invariants and Moduli},
  publisher = {Cambridge University Press},
  series    = {Cambridge Studies in Advanced Mathematics},
  volume    = {81},
  year      = {2003}
}

@article{GordanBinary,
  author  = {Gordan, Paul},
  title   = {Beweis, dass jede Covariante und Invariante einer bin{\"a}ren Form eine ganze Function mit numerischen Coefficienten einer endlichen Anzahl solcher Formen ist},
  journal = {Journal f{\"u}r die reine und angewandte Mathematik},
  volume  = {69},
  year    = {1868},
  pages   = {323--354}
}

Prerequisites

04.10.01
04.10.02
04.10.03
04.04.09
04.08.02

Tier anchors

beginner: Encode a genus-2 curve by the six branch points of its double cover of the line, then build numbers from those points that ignore how the line is drawn — the invariants that name the curve
intermediate: Harris-Morrison *Moduli of Curves* Ch. 1-2 (Springer GTM 187, 1998), the parameter-space and invariant-theory examples; Igusa 1960 *Ann. of Math.* 72 (the ring of invariants of binary sextics and the arithmetic variety of moduli for genus 2); Mukai *An Introduction to Invariants and Moduli* Ch. 1-2 (binary forms)
master: Igusa 1960 *Ann. of Math.* 72, 612-649; Clebsch 1872 *Theorie der binären algebraischen Formen*; Dixmier 1987 *On the projective invariants of quartic plane curves* (Adv. Math. 64); Ohno (unpublished, see Elsenhans 2015 and Lercier-Ritzenthaler 2012 *J. Algebra* for the Dixmier-Ohno reconstruction); Bolza 1887 *Amer. J. Math.* 10 (the genus-2 automorphism stratification); Mestre 1991 (reconstruction of a genus-2 curve from its Igusa invariants)

References

Igusa — Arithmetic variety of moduli for genus two · Annals of Mathematics 72 (1960), 612-649; the ring of SL_2-invariants of binary sextics, the invariants J_2,J_4,J_6,J_8,J_10, the discriminant, and M_2 as a weighted-projective arithmetic variety of moduli
Clebsch — Theorie der binären algebraischen Formen · Teubner, Leipzig 1872; the classical invariants A,B,C,D of a binary sextic from which the Igusa-Clebsch invariants are built
Harris-Morrison — Moduli of Curves · Springer GTM 187, 1998; Chapter 1 (parameter spaces, the worked low-genus examples), Chapter 2 (basic facts and the invariant-theoretic constructions of M_2 and M_3)
Dixmier — On the projective invariants of quartic plane curves · Advances in Mathematics 64 (1987), 279-304; seven of the thirteen fundamental SL_3-invariants of ternary quartics generating a homogeneous system of parameters
Bolza — On binary sextics with linear transformations into themselves · American Journal of Mathematics 10 (1887), 47-70; the automorphism stratification of genus-2 curves and the special invariant loci
Mestre — Construction de courbes de genre 2 à partir de leurs modules · Effective Methods in Algebraic Geometry (Castiglioncello 1990), Progress in Math. 94, Birkhäuser 1991, 313-334; reconstruction of a genus-2 curve from a tuple of Igusa invariants
Lercier-Ritzenthaler — Hyperelliptic curves and their invariants · Journal of Algebra 372 (2012), 595-636; the geometric and arithmetic invariant theory of binary forms and the Dixmier-Ohno invariants of ternary quartics, with reconstruction algorithms

Estimated time

beginner: 18m
intermediate: 45m
master: 85m