Normal Forms of Hyperelliptic Curves of Genus 3

Gerhard Frey^{1} & Ernst Kani ^{2,∗} 1 Institute for Experimental Mathematics University of Duisburg-Essen Essen, Germany e-mail: frey@exp-math.uni-essen.de 2 Department of Mathematics and Statistics Queen’s University Kingston, Ontario, K7L 3N6, Canada email: Kani@mast.queensu.ca ^{∗}

July 8, 2015

Dedicated to the Memory of Scott A. Vanstone

Abstract

The main motivation for the paper is to understand which hyperelliptic curves of genus 3 defined over a field K of characteristic ≠ 2 appear as the image of the Donagi-Livné-Smith construction. By results in [FK] this means that one has to determine the intersection W of a Hurwitz space defined by curves of genus 3 together with cover maps of degree 4 to ℙ_{K}^{1} and a certain ramification type with the hyperelliptic locus in the moduli space of curves of genus 3.

To achieve this aim we first study hyperelliptic curves of genus g as smooth curves C in ℙ_{K}^{1} × ℙ_{K}^{1} and prove that, under mild conditions on K, the curve C can be given by a “(g + 1, 2)-normal form”, namely by an affine equation in two variables of partial degrees g + 1 and 2 and hence of total degree ≤ g + 3, which is smaller than the degree of Weierstraß normal forms. Such curves are naturally parameterized by a Hurwitz space \overline{H}_{g,g+1}.

We then specialize to g = 3 and introduce Hurwitz spaces for 4-covers with special ramification types. The study of these spaces enables us to determine that W is irreducible of dimension 4. Moreover we find an explicitly given K-rational family of curves C in (4, 2)-normal form such that the isomorphism classes of its members are in W (K) and such that the image of the family in W is Zariski-dense. For these curves we describe the “inverse” of the Donagi-Livné-Smith construction. Keywords: hyperelliptic curves, normal forms, Hurwitz spaces attached to curves of genus 3 MSC 14H30, 14H45, 14Q05, 14G50

1 Introduction and Results

1.1 Motivation: A Relation to Public Key Cryptography

A well-known and very effective tool for public key cryptography is the use of discrete logarithms (DL) in divisor class groups Pic_{C}^{0} of degree 0 of curves C of genus g over finite fields 𝔽_{q} as crypto primitives for public key crypto systems. For details and following remarks see [CF].

∗Supported by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada (NSERC)

The security of these systems depend on the hardness of the computation of the DL, and one of the most dangerous attacks is based on very refined versions of index-calculus methods. As result one can bound the (probabilistic) complexity of the computation of the DL in Pic_{C}^{0}, up to logarithmic factors, by O(q^{(2−2/g)}) [DGTT]. Since, for fixed g, the asymptotic size of Pic_{C}^{0} is ∼ q^{g} one is forced to exclude curves C with genus ≥ 4.

Curves of genus 1 or 2 seem to be secure against this attack at least today and under certain precautions. Divisor class groups of curves of genus 3 lie on the border of security, if one looks at this estimate.

But it turns out that in addition to the genus the degree of plane equations for C is important.

Theorem 1 (Diem[D]) (a) If C is a non-nyperelliptic curve of genus 3 over 𝔽_{q} which is given by a plane equation of degree d = 4, then there exists an algorithm for computing the discrete logarithm in the group of divisor classes Pic^{0}(C) of degree 0 of C that has, up to logarithmic factors, complexity O(q).

(b) If C is a any curve over 𝔽_{q} which is given by a reflexive plane equation of degree d ≥ 4, then there exists an algorithm for computing the discrete logarithm that has, up to logarithmic factors, complexity O(q^{2− \frac{2}{d−2}} ).

By the theory of curves we know that non-hyperelliptic curves of genus 3 can be always be given by a plane equation of degree 4 and that plane equations of hyperelliptic curves of genus 3 cannot have a degree smaller than 5. Hence for hyperelliptic curves the above mentioned results yield complexity O(q^{4/3}) for the computation of the DL, which is not optimal but still acceptably near to the complexity of generic attacks.

But for non-hyperelliptic curves of genus 3 this complexity is, up to logarithmic factors, O(q) and hence very weak. We get the surprising

Consequence. Hyperelliptic curves of genus 3 are more secure than non-hyperelliptic (i.e., “generic”) curves of genus 3.

This result is exploited by Ben Smith [Sm] for an attack. He applies explicitly given isogenies η_{∗} of degree 8 of Jacobian varieties of hyperelliptic curves C′ of genus 3 whose image is again the Jacobian of a curve C of genus 3. This construction relies on the trigonal construction of Donagi-Livné and so we call it the Donagi-Livné-Smith construction. In [FK] we give an interpretation of this construction in terms of Hurwitz spaces and explain how η_{∗} is obtained by a correspondence η between C′ and C. As a consequence of this approach we get more information about C:

Fact. There is a cover map f : C → ℙ_{K}^{1} of degree 4 with monodromy group Gal(f) ≃ S_{4} such that exactly 4 points P_{1}, …, P_{4} ∈ ℙ^{1}(\overline{K}) of the ramification points of f are of type (2, 2), and the other 4 are of type (2, 1, 1).

Here, as usual, we mean by the monodromy group of f the Galois group of a Galois closure of the cover f. Moreover, we say that a point P_{i} ∈ ℙ^{1}(\overline{K}) has ramification type (2, 2) (respectively, (2, 1, 1)) with respect to f if f^{∗}(P_{i}) = 2(Q_{i,1} + Q_{i,2}), (respectively, f^{∗}(P_{i}) = 2Q_{i,1} + Q_{i,2} + Q_{i,3}), with Q_{i,j} ∈ C(\overline{K}) and Q_{i,j} ≠ Q_{i,k}, for j ≠ k.

Smith shows that there are curves C which are not hyperelliptic, and he assumes heuristically that the probability for C to be hyperelliptic should be ∼ 1/q. One of the main aims of the paper is to give an explanation of this heuristic in terms of Hurwitz spaces.^{1}

It turns out that for this the structure of the monodromy group and the ramification type of the cover f are crucial.

1We want to remark here that the arguments given in [FK], Subsection 5.2, for this claim are not sufficient. So part of the motivation for this paper is to repair this gap.

1.2 Results

This discussion motivates that we want to study the Hurwitz space \overline{H}_{3,4,4}(S_{4}) which “classifies” equivalence classes of morphisms f : C → ℙ_{K}^{1} of degree 4 with monodromy group S_{4} and ramification type (2, 2)^{4}(2, 1, 1)^{4}, where C is a hyperelliptic curve of genus 3. In order to understand this Hurwitz space, we first study more generally the related Hurwitz spaces \overline{H}_{3,4,k} which classify degree 4 covers f : C → ℙ_{K}^{1} (with C hyperelliptic of genus 3) ramified of type (2, 2) in at least k points (over \overline{K}), for k = 3, 4.

1.2.1 Hyperelliptic Curves in the Image of the Donagi-Livné-Smith construction

One of the main results of the paper is the following structure theorem.

Theorem 2 The Hurwitz space \overline{H}_{3,4,4}(S_{4}) is a unirational, irreducible variety of dimension 4, provided that char(K) > 5. Moreover, the natural forget map

\overline{μ}_{3} : \overline{H}_{3,4,4}(S_{4}) → M_{3}

to the moduli space M_{3} of genus 3 curves has finite fibres and so its image \overline{μ}_{3}(\overline{H}_{3,4,4}(S_{4})) is also irreducible of dimension 4.

This theorem is proved in Subsection 4.2.5. Moreover, we shall present explicit equations of a generic family of curve covers defined over K and parameterized by 4 rational parameters such that these define an non-empty open subscheme U′ of \overline{H}_{3,4,4}(S_{4}); cf. Subsection 4.2.4.

In Subsection 4.2.5 we also describe the inverse of the Donagi-Livné-Smith construction applied to curve covers in U′(K) (see Proposition 38) and give an explicit example.

Using this connection, it follows together with Theorem 2 that the isomorphism classes of hyperelliptic curves obtained by the Donagi-Livné-Smith Construction form a four-dimensional irreducible subspace W of the moduli space M_{3}^{h} of hyperelliptic curves of genus 3.

Now we are in a situation where we can use well-established results about the number of points on varieties over finite fields (see e.g. [GL]) and get that over finite fields 𝔽_{q} with q elements we have that |W (𝔽_{q})|/q^{4} ∼ 1. Since dim M_{3}^{h} = 5, this explains and proves the heuristics made in [Sm] and mentioned above.

1.2.2 Normal Forms

To prove Theorem 2 and other related results, we shall investigate normal forms for equations for curves C of genus g (mostly g = 3) which are different from the wellknown Weierstraß normal form.

In Section 3 we show that hyperelliptic curves C of arbitrary genus g with at least g(g − 1) + 1 K-rational points can be given by a smooth curve in ℙ_{K}^{1} × ℙ_{K}^{1} with an equation F (T_{0}, T_{1}; X_{0}, X_{1}) = 0 homogenous of degree 2 in T_{0}, T_{1} and homogenous of degree g+1 in X_{0}, X_{1} whose de-homogenization yields an affine plane curve with normal form equation

F (T, X) = ∑_{i=0}^{g+1} ∑_{j=0}^{2} r_{ij}X^{g+1−i}T^{j} with r_{ij} ∈ K

of total degree ≤ g + 3; ^{2} cf. formula (3) in Subsection 3.1.2. Since each such curve C comes equipped with two covers f_{i} : C → ℙ_{K}^{1}, we see that these equations naturally define a Hurwitz set H̃_{g,g+1}(K) which classifies isomorphism classes of triples (C, f_{1}, f_{2}), where f_{1} and f_{2} are cover of degree g + 1 and 2, respectively.

2By a linear change of variables that positions a K-rational point at (∞, ∞) ∈ ℙ_{K}^{1} × ℙ_{K}^{1}, one can even achieve degree g + 2, which is optimal ([CM]). Recall that the Weierstraß normal forms have degree ≥ 2g + 1.

1.2.3 Normal Forms with Given Ramification Types for g=3

In Section 4 we restrict attention to the case g = 3. For k = 3, 4 we consider the subsets H̃′_{3,4,k}(K) ⊂ H̃_{3,4}(K) which are defined by the condition that the degree 4 cover f_{1} : C → ℙ_{K}^{1} has at least k ramification points of type (2, 2). Moreover, by passing to equivalence classes of covers, we obtain the Hurwitz sets \overline{H}_{3,4,k}(K) ⊂ \overline{H}_{3,4}(K).

In this section we shall investigate the geometric structure behind these sets. More precisely, we will show that the set H̃′_{3,4,k}(K) is the the set of K-rational points of a subscheme H̃′_{3,4,k} of the Hurwitz space H̃_{3,4}, and that the elements of the set \overline{H}_{3,4,k}(K) give rise to K-rational points of a subscheme \overline{H}_{3,4,k} of the scheme \overline{H}_{3,4}.

The case k = 4 is obviously motivated by the Donagi-Livné-Smith construction, and k = 3 is remarkable because of the fact that the forget map \overline{μ}_{3} : \overline{H}_{3,4,3} → M_{3}^{h} which sends classes of triples (C, f_{1}, f_{2}) to the isomorphism class of C is generically finite; see Theorem 19.

The Role of Families. At this stage we want to say a word about the our strategy in this paper. We give arguments mostly for “generic” cases, i.e. we describe open nonempty subspaces of the varieties under investigation. This helps to avoid the discussion of special cases.

Moreover, the “generic” approach in our context has the big advantage that we can work with explicitly given rational families of curves defined over K which induce generically finite and dominant maps to the spaces under consideration and for which we find simple normal forms for the involved curves C as equations. But we have to pay a price for this. First of all, one misses a good part of the geometric picture that can be detected in a much more subtle and precise analysis and which is presented in the preprint [K2].

Moreover, in order to get simple curve equations we have to assume that ramification points of f_{1} are rational over K, and so rationality problems appear when K ≠ \overline{K}.^{3}

1.2.4 The Spaces H̃_{3,4,3} and \overline{H}_{3,4,3}

In order to obtain relatively simple equations for the curve covers in H̃′_{3,4,3}(K), we use Proposition 6 below to choose special representatives for the isomorphism classes of triples (C, f_{1}, f_{2}): We view C as an embedded curve on ℙ_{K}^{1} × ℙ_{K}^{1}, and let f_{1} = f_{C} := (pr_{1})_{|C} be the restriction to C of the first projection and let f_{2} = π_{C} := (pr_{2})_{|C} be the restriction to C of the second projection of ℙ_{K}^{1} × ℙ_{K}^{1}. Then we investigate triples (C, f_{C}, π_{C}) with the following additonal properties.

The cover f_{C} : C → ℙ_{K}^{1} has the special projective points P_{0} = (1 : 0), P_{1} = (1 : 1) and P_{−1} = (1 : −1) as ramification points of type (2, 2). In addition, we impose some extra conditions on the points on C in the fibres over P_{0}, P_{1} and P_{−1}; cf. equations (5)

– (7) in Subsection 4.1.

The isomorphism classes of triples (C, f_{C}, π_{C}) satisfying these conditions form a certain subset H̃_{3,4,3}(K) ⊂ H̃′_{3,4,3}(K) which is analyzed in detail in Subsection 4.1.

In particular, in Proposition 14 we determine a five-dimensional rational family defined over K with an explicitly given and simple affine normal form which defines an open subscheme H̃_{3,4,3}^{∗} of H̃_{3,4,3}.

3As an analogy, the reader might like to look at the Legendre family of elliptic curves E consisting of elliptic curves whose points of order 2 are K-rational.

This, together with more work done in [K2] yield that H̃_{3,4,3} is a smooth rational K-variety of dimension 5 (Proposition 16).

For arbitrary K (e.g., K = 𝔽_{q}), the natural map from H̃_{3,4,3}(K) to \overline{H}_{3,4,3}(K) is not surjective. But geometrically this is true (Lemma 11) and so for K = \overline{K} we get that there is a surjective map with finite fibres from H̃_{3,4,3}(\overline{K}) to \overline{H}_{3,4,3}(\overline{K}).

The affine normal form of curves in H̃_{3,4,3}(K) can be transformed to equations in Weierstraß normal form. By computational methods Hindry and Ritzenthaler [HR] show (cf. Theorem 18) that this family is generic, and so we get in Theorem 19 the following geometric description of \overline{H}_{3,4,3}: It is an irreducible unirational variety of dimension 5, and the map \overline{μ}_{3} from \overline{H}_{3,4,3} to the moduli space of hyperelliptic curves of genus 3 is generically finite and dominant. In particular, generic hyperelliptic curves of genus 3 over \overline{K} can be given by affine normal forms with coefficients depending on five parameters.

1.2.5 The Spaces H̃_{3,4,4}^{∗} and \overline{H}_{3,4,4}

Similarly, to obtain simple equations for the curves covers in H̃′_{3,4,4}(K), we restrict our attention to the set H̃_{3,4,4}^{∗}(K) := H̃′_{3,4,4}(K) ∩ H̃_{3,4,3}^{∗}(K). Thus, each curve cover f_{C} : C → ℙ_{K}^{1} in H̃_{3,4,4}^{∗}(K) is ramified of type (2, 2) at P_{0}, P_{1}, P_{−1} and at one further point P_{t} ∈ ℙ_{K}^{1}(K).

It turns out that the associated Hurwitz space H̃_{3,4,4}^{∗} is the union of precisely two irreducible components; cf. Corollary 25. Thus, using Lemma 26, it follows that the same is true for \overline{H}_{3,4,4}:

\overline{H}_{3,4,4} = V_{1} ∪ V_{2}.

We introduce in Subsection 4.2.2 two explicit families of curve covers by imposing polynomial inequalities (Condition U) which define non-empty open subschemes U_{1} and U_{2} of H̃_{3,4,4}^{∗} (cf. Theorem 23) with the property that U_{i} maps to a dense subset of V_{i} for i = 1, 2.

The nature of the two families of curve covers is (generically) quite different. Those in the first family (which defines U_{1}) are curve covers f_{C} : C → ℙ_{K}^{1} with the property that f_{C} factors over an elliptic curve (so the monodromy group of f_{C} is contained in the dihedral group D_{4}), whereas those in the second family have generically the symmetric group S_{4} as their monodromy group. In fact, it turns out that the desired Hurwitz space \overline{H}_{3,4,4}(S_{4}) of Theorem 2 is an open subscheme of V_{2}; cf. Subsection 4.2.5.

2 Covers of Hyperelliptic Curves: Definitions and Notation

Throughout, K is a field of characteristic ≠ 2, and \overline{K} is an algebraic closure of K. If not otherwise stated, geometric objects are defined over K.

As was motivated in the introduction, this paper deals with cover maps from hyperelliptic curves C to the projective line ℙ_{K}^{1} with various special properties. More precisely, we want to consider the set of all such cover maps (satisfying a fixed set of properties), modulo isomorphism or modulo equivalence of covers (cf. below).

For the convenience of the reader we give here an overview of the properties which we want to study.

It will be an important part of the paper to show that these sets have a “geometric interpretation” in terms of certain schemes, whose geometric properties are studied. Ideally, one would like to be able to identify these sets with the set of K-rational points of the associated scheme, but this is in general only possible when K = \overline{K} is algebraically closed.

2.1 Covers of Hyperelliptic Curves C of Genus g ≥ 2

M_{g}: the moduli scheme of curves of genus 3; its dimension is 3g − 3. M_{g}^{h}: the locus of hyperelliptic curves of genus g on M_{g}; its dimension is 2g − 1.

Typical notation

• C, C_{1}, C_{2}: hyperelliptic curves of fixed genus g ≥ 2,
• ω_{C}: the hyperelliptic involution of the hyperelliptic curve C,
• f : C → ℙ_{K}^{1} and f_{i} : C_{i} → ℙ_{K}^{1}: non-constant morphisms (“covers ”).

Isomorphism of covers. f_{1} ≃ f_{2} iff ∃ an isomorphism φ : C_{1} →^{∼} C_{2} with f_{2} ∘ φ = f_{1}.

Equivalence of covers. f_{1} ∼ f_{2} iff ∃φ : C_{1} →^{∼} C_{2} and α ∈ Aut(ℙ_{K}^{1}) with f_{2}∘φ = α∘f_{1}.

Example. If C = C_{1} = C_{2} and deg(f_{i}) = 2 for i = 1, 2, then the covers f_{i} are equivalent (with φ = id_{C}) and are called hyperelliptic covers of C.

Isomorphism of triples (C, f_{1}, f_{2}), where f_{i} : C → ℙ_{K}^{1}, are covers. (C, f_{1}, f_{2}) is isomorphic to (C′, f′_{1}, f′_{2}) iff ∃ an isomorphism φ : C →^{∼} C′ with f_{i} = f′_{i} ∘ φ for i = 1, 2.

Notation. The isomorphism class of the triple (C, f_{1}, f_{2}) is denoted by [C, f_{1}, f_{2}].

Convention. Let c be an isomorphism class of triples. The notation c = [C, f_{1}, f_{2}] means that the triple (C, f_{1}, f_{2}) is a representative of c.

Factorization of covers. A cover f_{1} : C → ℙ_{K}^{1} factors over a cover f_{2} : C → ℙ_{K}^{1} iff ∃ a cover φ : ℙ_{K}^{1} → ℙ_{K}^{1} such that f_{1} = φ ∘ f_{2}. If deg(f_{2}) = 2, then f_{1} factors over f_{2} iff f_{1} = f_{1} ∘ ω_{C}. Thus, f : C → ℙ_{K}^{1} does not factor over a hyperelliptic cover of C iff f ∘ ω_{C} ≠ f.

The basic Hurwitz sets H_{g,n}(K): Isomorphism classes of covers of degree n that do not factor over a hyper-

elliptic cover.

\overline{H}_{g,n}(K): Equivalence classes of covers of degree n that do not factor over a hyperel-

liptic cover.

H̃_{g,n}(K): Isomorphism classes of triples (C, f_{1}, f_{2}) where deg(f_{1}) = n, deg(f_{2}) = 2, and f_{1} does not factor over a hyperelliptic cover of C.

Group actions

The group Aut(ℙ_{K}^{1}) ≃ PGL_{2}(K) acts on H_{g,n}(K) via (α, f) ↦ α ∘ f, and the product group Aut(ℙ_{K}^{1}) × Aut(ℙ_{K}^{1}) acts on H̃_{g,n}(K) via

((α_{1}, α_{2}), (C, f_{1}, f_{2})) ↦ (C, α_{1} ∘ f_{1}, α_{2} ∘ f_{2}).

Via these actions, the first two Hurwitz sets are naturally the orbit sets of the third set as follows:

H_{g,n}(K) = (1 × Aut(ℙ_{K}^{1})\H̃_{g,n}(K),
\overline{H}_{g,n}(K) = (Aut(ℙ_{K}^{1}) × Aut(ℙ_{K}^{1}))\H̃_{g,n}(K)
= Aut(ℙ_{K}^{1})\H_{g,n}(K).

Results

• H̃_{g,g+1}(K) is the set of K-rational points of a non-empty open subscheme H̃_{g,g+1} of ℙ_{K}^{3g+5}, for every field K; cf. Proposition 6.

• H_{g,g+1}(\overline{K}) is the set of \overline{K}-rational points of a unirational scheme H_{g,g+1} of dimension 3g+2. More precisely, for any field K we have a map H_{g,g+1}(K) → H_{g,g+1}(K) which is compatible with base-change and which is a bijection when K = \overline{K}. We thus say that the scheme H_{g,g+1} classifies the Hurwitz problem H_{g,g+1}(·).

• \overline{H}_{g,g+1}(\overline{K}) is the set of \overline{K}-rational points of a unirational scheme \overline{H}_{g,g+1} of dimension 3g − 1. Moreover, the scheme \overline{H}_{g,g+1} is a good geometric quotient π_{g} : H̃_{g,g+1} → \overline{H}_{g,g+1} of H̃_{g,g+1}, and so \overline{H}_{g,g+1} classifies \overline{H}_{g,g+1}(·); cf. Subsection 3.1.5.

• There is a surjective morphism μ_{g} : H̃_{g,g+1} → M_{g}^{h}, which is obtained by sending [C, f_{1}, f_{2}] to the isomorphism class of C (over \overline{K}); cf. Corollary 7. Moreover, μ_{g} factors over π_{g}, and so we obtain a surjective morphism \overline{μ}_{g} : \overline{H}_{g,g+1} → M_{g}^{h} with μ_{g} = \overline{μ}_{g} ∘ π_{g}.

2.2 Covers of Hyperelliptic Curves C of Genus 3

In the following, C is a hyperelliptic curve of genus 3.

Special Hurwitz sets

\overline{H}_{3,4,k}(K): Equivalence classes of covers f : C → ℙ_{K}^{1} of degree 4 that do not factor over a hyperelliptic cover and with at least k ramification points (defined over \overline{K}) of type (2, 2); cf. Subsection 1.1.

\overline{H}_{3,4,4}(S_{4})(K): Equivalence classes of covers f : C → ℙ_{K}^{1} of degree 4 with monodromy group S_{4} and ramification type (2, 2)^{4}(2, 1, 1)^{4}. (See Subsection 1.1 for the definitions of these terms.)

H̃′_{3,4,k}(K): Isomorphism classes of triples [C, f_{1}, f_{2}] ∈ H̃_{3,4}(K) such that f_{1} has

at least k ramification points of type (2, 2);

H̃_{3,4,3}(K) Isomorphism classes of triples [C, f_{1}, f_{2}] ∈ H̃′_{3,4,3}(K) with f_{1} satisfy-

ing conditions (5), (6) and (7) in Subsection 4.1.

U_{1}(K) Isomorphism classes of triples [C, f_{1}, f_{2}] ∈ H̃′_{3,4,3}(K) for which C is

given by an equation F_{1}(T, X) = 0 with F_{1} as in (31).

U_{2}(K) Isomorphism classes of triples [C, f_{1}, f_{2}] ∈ H̃′_{3,4,3}(K) for which C is given by an equation F_{2}(T, X) = 0 with F_{2} as in (32).

U(K) = U_{1}(K) ∪ U_{2}(K)

U′(K) Isomorphism classes of triples [C, f_{1}, f_{2}] ∈ U(K) such that f_{1} has

monodromy group S_{4} and ramification type (2, 2)^{4}(2, 1, 1)^{4}.

Results

• The Hurwitz space \overline{H}_{3,4,3} which classifies \overline{H}_{3,4,3}(·) is an irreducible unirational variety of dimension 5; cf. Theorem 19.

• The Hurwitz space \overline{H}_{3,4,4} which classifies \overline{H}_{3,4,4}(·) is the union of two irreducible unirational varieties V_{1} and V_{2} of dimension 4; cf. Theorem 20.

• The Hurwitz space \overline{H}_{3,4,4}(S_{4}) which classifies \overline{H}_{3,4,4}(S_{4})(·) is irreducible of dimension 4 and is an open subscheme of one of the two components of \overline{H}_{3,4,4}; cf. Theorem 2 and Corollary 35.

• H̃_{3,4,3}(K) is the set of K-rational points of a smooth, rational scheme H̃_{3,4,3}, for every field K. Moreover, H̃_{3,4,3} is a locally closed subscheme of H̃_{3,4}; cf. Proposition 16.

• U_{i}(K) = ⋃_{t} U_{i,t}(K) is the set of K-rational points of a rational scheme U_{i} of dimension 4, for every field K. Moreover, U_{i} is locally closed in H̃_{3,4}; cf. Theorem 23 and the discussion after Theorem 24.

• U′(K) is the set of K-rational points of a non-empty open subscheme U′ of U_{2}, for every field K; cf. Corollary 35. Moreover, the image of U′ (and of U_{2}) in M_{3}^{h} is irreducible of dimension 4; cf. Proposition 41.

Spaces defined for technical reasons

By Proposition 6, the isomorphism classes in H̃_{3,4}(K) can be represented by triples (C, f_{C}, π_{C}) with C a smooth curve of genus 3 in ℙ_{K}^{1} × ℙ_{K}^{1} and f_{C} = pr_{1} is of degree 4, π_{C} = pr_{2} is of degree 2 where pr_{i} is the i-th projection of ℙ_{K}^{1} × ℙ_{K}^{1} restricted to C. With this notation, we define:

H̃_{3,4,3}^{∗}(K): [C, f_{C}, π_{C}] ∈ H̃_{3,4,3}(K) such that P_{1,∞} ∉ C (see Notation in Subsec-

tion 4.1);

H̃_{3,4,4,t}^{∗}(K): [C, f_{C}, π_{C}] ∈ H̃_{3,4,3}^{∗}(K) with f_{C} ramified of type (2, 2) at P_{t} = (1 : t), for t ∈ K ∪ {∞} \ {0, 1, −1}.

We thus have the inclusions H̃_{3,4,4,t}^{∗}(K) ⊂ H̃_{3,4,3}^{∗}(K) ⊂ H̃_{3,4,3}(K) ⊂ H̃_{3,4}(K). Moreover, put: H̃_{3,4,4}^{∗}(K) = ⋃_{t} H̃_{3,4,4,t}^{∗}(K).

3 Hyperelliptic Covers and Related Hurwitz Spaces

3.1 Hyperelliptic Curves in ℙ_{K}^{1} × ℙ_{K}^{1}

As was mentioned in the previous sections, we are interested in studing triples (C, f_{1}, f_{2}), where C is a hyperelliptic curve of genus g, f_{1} : C → ℙ_{K}^{1} a cover map of degree n with f_{1} ≠ f_{1} ∘ ω_{C} and f_{2} : C → ℙ_{K}^{1} is a hyperelliptic cover, i.e., deg(f_{2}) = 2. Note that Castelnovo’s Inequality ([St], [K1]) implies that n ≥ g + 1.

Each triple (C, f_{1}, f_{2}) as above defines (by the universal property of products) a unique K-morphism π = π_{f_{1},f_{2}} : C → ℙ_{K}^{1} × ℙ_{K}^{1} such that f_{i} = pr_{i} ∘ π, and the image C_{f_{1},f_{2}} = π_{f_{1},f_{2}}(C) is an irreducible curve on the surface ℙ_{K}^{1} × ℙ_{K}^{1}. Note that since f_{1} ≠ f_{1} ∘ ω_{C}, the curve C_{f_{1},f_{2}} is birationally equivalent to C (but C_{f_{1},f_{2}} may be singular). Moreover, C_{f_{1},f_{2}} is a divisor of type (2, n) on ℙ_{K}^{1} × ℙ_{K}^{1} i.e.,

C_{f_{1},f_{2}} ∼ D_{2,n} := 2(P × ℙ_{K}^{1}) + n(ℙ_{K}^{1} × P), for P ∈ ℙ_{K}^{1}(K).

In other words, C_{f_{1},f_{2}} ∈ |D_{2,n}|_{K} := {D ∈ Div(ℙ_{K}^{1} × ℙ_{K}^{1}) : D ≥ 0, D ∼ D_{2,n}}, the complete linear system of effective K-rational divisors defined by D_{2,n}; cf. [Ha], p. 157 (when K = \overline{K}). Note that |D_{2,n}|_{K} has the structure of a (naive) projective space of dimension 3n + 2 over K. Thus, if we choose (homogeneous) coordinates on ℙ_{K}^{1} such that P = (0 : 1), then C_{f_{1},f_{2}} is given by an equation F (T_{0}, T_{1}; X_{0}, X_{1}) = 0, where F (T_{0}, T_{1}; X_{0}, X_{1}) ∈ K[T_{0}, T_{1}, X_{0}, X_{1}] is a polynomial of the form

F (T_{0}, T_{1}; X_{0}, X_{1}) = ∑_{i=0}^{n} ∑_{j=0}^{2} r_{ij}X_{0}^{i}X_{1}^{n−i}T_{0}^{2−j}T_{1}^{j},  (1)

because the (bihomogeneous) monomials {X_{0}^{i}X_{1}^{n−i}T_{0}^{2−j}T_{1}^{j}}_{i,j} form a basis of the Kvector space of K-rational sections of the sheaf L(D_{2,n}). We note this result:

Proposition 3 Every element in H̃_{g,n}(K) can be represented by a triple (C, f_{C}, π_{C}) where C ⊂ ℙ_{K}^{1} × ℙ_{K}^{1} is an irreducible curve of genus g contained in |D_{2,n}|_{K}, and f_{C} = pr_{1}, π_{C} = pr_{2}. Moreover, C can be given by an equation of the form (1).

3.1.1 The Existence of g + 1-Covers

In the case that n = g + 1, much more can be said. For this, we first observe that a given hyperelliptic curve C/K of genus g always has subcover f_{1} : C → ℙ_{K}^{1} of degree g + 1 with f_{1} ≠ f_{1} ∘ ω_{C}, provided that C(K) is large enough. This follows from:

Lemma 4 Let C/K be a curve of genus g ≥ 2. If T ⊂ C(K) is a set consisting of at least g(g − 1) + 1 K-rational points, then there is an effective divisor D ∈ Div(C) of degree g + 1 with support in T such that dim |D| = 1 and |D| is base-point free. In particular, there exists a K-morphism f : C → ℙ_{K}^{1} of degree g + 1.

Proof. As is well-known (cf. [St], p. 35), there exists a non-special effective divisor D_{0} = P_{1} + … + P_{g} of degree g with support in T. Then for any P ∈ C(K) we have that dim |D_{0} + P | = 1 by the Theorem of Riemann-Roch. Now if Q is a base point of |D_{0} + P |, then Q = P_{i} for some i with 1 ≤ i ≤ g. Thus, if K_{C} denotes a canonical divisor, then by Riemann-Roch dim |K_{C} −D_{0} +P_{i} −P | = dim |D_{0} +P −P_{i}|−1 = 0, and so P ≤ D_{i}, where D_{i} ≥ 0 is the unique effective divisor such that D_{i} ∼ K_{C} − D_{0} + P_{i}. (Note that by Riemann-Roch, dim |K_{C} − D_{0} + P_{i}| = |D_{0} − P_{i}| = 0.) Thus, |D_{0} + P | is base point free whenever P ≰ ∑_{i=1}^{g} D_{i}. Since the latter divisor has degree g(g − 1), it is clear that we can choose a P ∈ T with this property.

Remark 5 Note that |C(K)| > g(g − 1) + 1 if K = \overline{K} or if K = 𝔽_{q} with q sufficiently large. More precisely, we have that q ≥ 6g^{2} is enough because the Hasse-Weil bound (cf. [St], p. 198) yields that |C(𝔽_{q})| ≥ (\sqrt{q} − g)^{2} − g^{2} + 1 ≥ (\sqrt{6} − 1)^{2}g^{2} − g^{2} + 1 > g^{2} ≥ g(g − 1) + 1. (For example, if g = 3, then it is enough to have q ≥ 59.)

The following result is fundamental for much of what follows.

Proposition 6 The rule (C, f_{1}, f_{2}) ↦ C_{f_{1},f_{2}} induces a bijection

κ_{g} = κ_{g,K} : H̃_{g,g+1}(K) →^{∼} |D_{2,g+1}|_{K}^{sm},

where |D_{2,g+1}|_{K}^{sm} ⊂ |D_{2,g+1}|_{K} denotes the subset of smooth divisors on ℙ_{K}^{1} × ℙ_{K}^{1} which are contained in |D_{2,g+1}|_{K}. Thus, H̃_{g,g+1}(K) is parameterized by the K-rational points of a non-empty open subscheme H̃_{g,g+1} of ℙ_{K}^{3g+5}.

Proof. Let [C, f_{1}, f_{2}] be in H̃_{g,g+1}(K). As was seen above, C_{f_{1},f_{2}} ∼ D_{2,g+1}. By the adjunction formula, the arithmetic genus of C_{f_{1},f_{2}} is p_{a}(C_{f_{1},f_{2}}) = g; cf. [Ha] or [K1]. Thus, C_{f_{1},f_{2}} is a smooth curve on ℙ_{K}^{1} × ℙ_{K}^{1}, and therefore C_{f_{1},f_{2}} ∈ |D_{2,g+1}|_{K}^{sm}. Thus, since C_{f_{1},f_{2}} depends only on the isomorphism class of (C, f_{1}, f_{2}), we obtain a map κ_{g} : H̃_{g,g+1}(K) → |D_{2,g+1}|_{K}^{sm}.

Now since D_{2,g+1} is (very) ample (cf. [Ha], II, 7.6.2), every divisor D ∈ |D_{2,g+1}|_{K} is connected ([Ha], III, 7.9.1) and so every D ∈ |D_{2,g+1}|_{K}^{sm} is a smooth, irreducible curve of genus g_{D} = p_{a}(D) = g. Thus, the rule D ↦ (D, pr_{1}_{|D}, pr_{2}_{|D}) induces a map κ′_{g} : |D_{2,g+1}|_{K}^{sm} → H̃_{g,g+1}(K) which is clearly inverse to κ_{g}, and so both are bijections.

To prove the last assertion, let N = 3g + 5, and let D := D_{2,g+1} be the subscheme of ℙ_{K}^{N} × (ℙ_{K}^{1} × ℙ_{K}^{1}) defined by equation (1) with n = g + 1, where we view the coefficients (r_{ij}) as variables of ℙ_{K}^{N}. Let p_{g,g+1} = (pr_{1})_{|D} : D → ℙ_{K}^{N} be the projection. It is immediate that the fibres of p_{g,g+1} at the K-rational points of ℙ_{K}^{N} give precisely the elements of |D_{2,g+1}|_{K}, and that this construction is compatible with base change. Thus, D defines an algebraic family of divisors on ℙ_{K}^{1} × ℙ_{K}^{1} parametrized by ℙ_{K}^{N} (in the sense of [Ha], p. 261). Thus, if H̃_{g,g+1} ⊂ ℙ_{K}^{N} denotes the open subscheme of ℙ_{K}^{N} where p_{g,g+1} is smooth, then the fibres of p_{g,g+1} at the points of H̃_{g,g+1}(K) are precisely the elements of |D_{2,g+1}|_{K}^{sm}. Since this construction is functorial in K, we have our desired parametrization.

Finally, to see that H̃_{g,g+1} is non-empty, we can use either Bertini’s Theorem (cf. [Ha],II.8.18) or Lemma 4 to show that |D_{2,g+1}|\overline{K} ≠ ∅, which implies that H̃_{g,g+1}(\overline{K}) ≠ ∅.

Corollary 7 The rule (C, f_{1}, f_{2}) ↦ C induces a surjective morphism

μ_{g} : H̃_{g,g+1} → M_{g}^{h}

with μ_{g}_{∗}(H̃_{g,g+1})(\overline{K}) = M_{g}^{h}(\overline{K}).

Proof. Since p_{g,g+1} : D_{2,g+1}^{sm} → H̃_{g,g+1} is a family of genus g curves, it induces (by the coarse moduli property of M_{g}) a morphism μ_{g} : H̃_{g,g+1} → M_{g} such that μ_{g}(x) = (isomorphism class of) p_{2,g+1}^{−1}(x) ∈ |D_{2,g+1}|sm ; \overline{K}, for x ∈ H̃_{g,g+1}(\overline{K}); cf. [Ha], p. 347. Thus, via the identifications of (the proof of) Proposition 6, this morphism is given by the rule (C, f_{1}, f_{2}) ↦ C. Moreover, since each C is hyperelliptic, it is clear that the image of μ_{g} is contained in the hyperelliptic locus M_{g}^{h}.

To see that μ_{g} is surjective, let C/\overline{K} be an arbitrary hyperelliptic curve of genus g. By Lemma 4 we get a morphism f := φ_{D} : C → ℙ1 ; \overline{K} of degree g + 1 attached to a divisor D ∈ Div(C) such that

dim |D| = 1, deg(D) = g + 1, and |D| is base-point free.

φ_{D} cannot factor over a hyperelliptic cover π_{C} because if φ_{D} = φ∘π_{C} for some φ : ℙ_{K}^{1} → ℙ_{K}^{1}, then deg(φ) = \frac{g+1}{2} (so g is odd) and then dim |φ_{D}^{∗}(\overline{P})| ≥ dim |φ^{∗}(\overline{P})| = \frac{g+1}{2} > 1, contradiction. Thus, (C, φ_{D}, π_{C}) ∈ H̃_{g,g+1}(\overline{K}), and hence μ_{g} is surjective.

3.1.2 Equations for Hyperelliptic Curves

We fix the genus g of curves.

In this subsection we shall assume that K has the following property: Every curve C of genus g over K has at least g(g − 1) + 1 K-rational points. As was mentioned in Remark 5, this condition is satisfied if K = \overline{K} or if K = 𝔽_{q} with q sufficiently large.

To get equations to represent hyperelliptic curves C of genus g by plane curves we fix coordinates on ℙ_{K}^{1}. Then as explained above, each divisor D in |D_{2,g+1}|_{K} can be represented by an equation F (T_{0}, T_{1}; X_{0}, X_{1}) = 0, where F has the form (1), i.e., F is homogeneous of degree 2 in T_{0}, T_{1} and of degree g + 1 in X_{0}, X_{1}, so

F (T_{0}, T_{1}; X_{0}, X_{1}) = ∑_{i=0}^{g+1} ∑_{j=0}^{2} r_{ij}X_{0}^{i}X_{1}^{g+1−i}T_{0}^{2−j}T_{1}^{j},  (2)

where r_{ij} ∈ K.

Since F is uniquely determined by D up to a multiplicative constant, we can view the coefficients of F as a point P_{D} = (r_{00} : r_{01} : r_{02} : r_{10} : …: r_{(g+1)2}) ∈ ℙ^{N}(K), where N = 3g + 5, and the rule D ↦ P_{D} gives the inverse of the above mentioned parametrization of |D_{2,g+1}|_{K}. This gives a concrete realization of the projective space structure on |D_{2,g+1}|_{K} over K.

Note that on the affine open subset U = {T_{0}X_{0} ≠ 0} of ℙ_{K}^{1} × ℙ_{K}^{1} we can represent each D ∈ |D_{2,g+1}|_{K} by an affine equation

F (T, X) = ∑_{i=0}^{g+1} ∑_{j=0}^{2} r_{ij}X^{g+1−i}T^{j},  (3)

with T = T_{1}/T_{0} and X = X_{1}/X_{0} and r_{ij} ∈ K.

It follows from Lemma 4 that if C/K is a hyperelliptic curve of genus g, then C/K can be described by an affine equation in 𝔸_{K}^{2} in two variables T, X, where the degree in T is 2 and the degree in X is g + 1, and so the total degree is ≤ g + 3, which is smaller than the degree of a Weierstraß equation (see also the footnote in Subsection 1.2.2).

3.1.3 Smoothness of Divisors

As was pointed out above, we are interested in smooth divisors. Having equations we can express this condition by discriminants and get the following explicit characterization of the divisors in |D_{2,g+1}|_{K}^{sm}.

Proposition 8 Let C ∈ |D_{2,g+1}|_{K} be given by F (T_{0}, T_{1}; X_{0}, X_{1}) as in (2), and let

D_{F}^{h} (X_{0}, X_{1}) = (∑_{i=0}^{g+1} r_{i1}X_{0}^{i}X_{1}^{g+1−i})^{2} −4 (∑_{i=0}^{g+1} r_{i0}X_{0}^{i}X_{1}^{g+1−i}) (∑_{i=0}^{g+1} r_{i2}X_{0}^{i}X_{1}^{g+1−i}) (4)

denote its homogeneous discriminant with respect to T. Then C ∈ |D_{2,g+1}|_{K}^{sm} if and only if D_{F}^{h} (X_{0}, X_{1}) is separable, i.e., D_{F}^{h} factors over \overline{K} into 2(g + 1) distinct linear factors.

Proof.^{4} From the proof of Proposition 4 we know that C ∈ |D_{2,g+1}|_{K}^{sm} if and only if (2) defines an irreducible curve. To examine this condition, write F as

F (T_{0}, T_{1}; X_{0}, X_{1}) = g_{0}(X_{0}, X_{1})T_{1}^{2} + g_{1}(X_{0}, X_{1})T_{0}T_{1} + g_{2}(X_{0}, X_{1})T_{0}^{2}.

Now if g_{0}, g_{1}, and g_{2} have common factor g, then clearly F is reducible, and then g^{2}|D_{F}^{h} = g_{1}^{2} − 4g_{0}g_{2}, so D_{F}^{h} is not separable, and hence the proposition holds in this case. Moreover, if g_{0} = 0, then F is again reducible (because T_{0}|F) and D_{F}^{h} = g_{1}^{2} is not separable. Thus, assume henceforth that gcd(g_{0}, g_{1}, g_{2}) = 1 and that g_{0} ≠ 0.

In this situation we see that F is irreducible if and only if its dehomogenization F̄(T, X) := F (1, T, 1, X) = ḡ_{0}(X)T^{2} + ḡ_{1}(X)T + ḡ_{2}(X) is irreducible. Moreover, since our hypotheses imply that gcd(ḡ_{0}, ḡ_{1}, ḡ_{2}) = 1 and ḡ_{0} ≠ 0, it follows from Gauss that F̄(T, X) is irreducible if and only if its discriminant ḡ_{1}(X)^{2} − 4ḡ_{0}(X)ḡ_{2}(X) = D_{F}^{h} (1, X) is not a square in K[X]. Thus, if D_{F}^{h} (X_{0}, X_{1}) is separable, then clearly D_{F}^{h} (1, X) cannot be a square and so F̄ and F are irreducible, so C is smooth.

Conversely, suppose that F̄ is irreducible, i.e., that D_{F}^{h} (1, X) is not a square in K[X]. Then the affine curves

C_{a} : F̄(T, X) = 0 and C′ : Y^{2} = D_{F}^{h} (1, X)

are both irreducible, and it is immediate that C′ is birationally equivalent to C_{a} because the substitution T = (Y − ḡ_{1})/2ḡ_{0} defines a birational equivalence between them. Thus, since C has genus g, the same is true for C′ (in the sense that its function field has genus g). Now since D_{F}^{h} (X_{0}, X_{1}) is homogeneous of degree 2g + 2, we see that deg D_{F}^{h} (1, X) ≤ 2g + 2. Thus, from the Hurwitz genus formula it follows that either deg(D_{F}^{h} (1, X)) = 2g + 2 and D_{F}^{h} (1, X) has 2g + 2 distinct roots (in \overline{K}) or that deg(D_{F}^{h} (1, X)) = 2g + 1 and D_{F}^{h} (1, X) has 2g + 1 distinct roots. Then in both cases D_{F}^{h} (X_{0}, X_{1}) = X_{0}^{2g+2}D_{h}(1, X_{1}/X_{0}) is separable, so the assertion follows.

Remark 9 If C ∈ |D_{2,g+1}|_{K}^{sm} is given by the equation (2), then the above proof shows implicitly that the hyperelliptic cover π_{C} : C → ℙ_{K}^{1} is ramified at the point P_{∞} = (0 : 1) if and only if X_{0} is a factor of D_{F}^{h} (X_{0}, X_{1}). This is equivalent to the condition that D_{F}^{h} (0, 1) = r_{01}^{2} − 4r_{00}r_{02} = 0.

4We want to thank the anonymous referee for the key idea of this proof which considerably shortens our original argument.

3.1.4 Weierstraß Normal Form

We state an immediate consequence of the proof of Proposition 8 and Remark 9:

Proposition 10 Assume that C is a curve of genus g given by

F (T_{0}, T_{1}; X_{0}, X_{1}) = ∑_{i=0}^{g+1} ∑_{j=0}^{2} r_{ij}X_{0}^{i}X_{1}^{g+1−i}T_{0}^{2−j}T_{1}^{j}

with r_{01}^{2} − 4r_{00}r_{02} ≠ 0. Then

Y^{2}X_{0}^{2g} = (∑_{i=0}^{g+1} r_{i1}X_{0}^{i}X_{1}^{g+1−i})^{2} − 4 (∑_{i=0}^{g+1} r_{i0}X_{0}^{i}X_{1}^{g+1−i}) (∑_{i=0}^{g+1} r_{i2}X_{0}^{i}X_{1}^{g+1−i})

is a Weierstraß equation for C.

It is an easy exercise to get an analogous result (with a homogenous equation of degree 2g + 1) in the case that (0 : 1) is a ramification point of f_{2}. Note that in the case that at least one of the Weierstraß points of C is K-rational the two cases can be transformed into each other by a K-rational projective transformation as usual.

3.1.5 The Hurwitz space \overline{H}_{g,g+1}

Let G = PGL_{2}. Then the G × G action on ℙ1 ; \overline{K} × ℙ1 ; \overline{K} permutes the elements of the complete linear system |D_{2,g+1}|\overline{K} and hence induces an action on ℙN ; \overline{K} (via p_{g,g+1}).

Since |D_{2,g+1}|sm ; \overline{K} is stable under this action, we have an induced action on H̃_{g,g+1}. By using the techniques of Mumford[Mu], one can show that the quotient scheme \overline{H}_{g,g+1} = (G × G)\H_{g,g+1} exists, and that the quotient map

π_{g} : H̃_{g,g+1} → \overline{H}_{g,g+1}

satisfies the properties of a good geometric quotient; cf. [K2]. Thus, since H̃_{g,g+1} is a rational variety of dimension 3g + 5, it follows that \overline{H}_{g,g+1} is a unirational variety of dimension 3g + 5 − 2 dim G = 3g − 1. Moreover, it follows from properties of good geometric quotients and from Proposition 6 and its proof that κ_{g,\overline{K}} induces a bijection

\overline{κ}_{g,\overline{K}} : \overline{H}_{g,g+1}(\overline{K}) →^{∼} \overline{H}_{g,g+1}(\overline{K}).

More generally, if L ⊂ \overline{K} is any subfield, then the bijection κ_{g,L} : H̃_{g,g+1}(L) →^{∼} H̃_{g,g+1}(L) of Proposition 6 induces a map

\overline{κ}_{g,L} : \overline{H}_{g,g+1}(L) = (G(L) × G(L))\H̃_{g,g+1}(L) → \overline{H}_{g,g+1}(L),

The maps \overline{κ}_{g,L} are compatible with field extensions. We thus see that the quotient scheme \overline{H}_{g,g+1} “classifies” the Hurwitz problem \overline{H}_{g,g+1}(·). Note, however, that for an arbitrary subfield L ⊂ \overline{K}, the map \overline{κ}_{g,L} is in general neither injective nor surjective.

3.2 Hurwitz Spaces with Given Ramification Type

We assume in this subsection that K = \overline{K}.

We have seen in Subsection 3.1.5 that the Hurwitz space \overline{H}_{g,g+1} is a unirational variety of dimension 3g − 1. Moreover, since the forget map μ_{g} : (C, f_{1}, f_{2}) ↦ C of Corollary 7 is G × G-invariant, it follows that μ_{g} factors over the quotient map π_{g}, and so there exists a unique morphism

\overline{μ}_{g} : \overline{H}_{g,g+1} → M_{g}^{h}

such that μ_{g} = \overline{μ}_{g} ∘ π_{g}. Note that M_{g}^{h} has dimension 2g − 1 and so \overline{μ}_{g} is obviously not finite.

But till now we have used only rudimentary parts of information that one can connect with Hurwitz spaces and we did not restrict the ramification type of the morphism f_{1}. (Since f_{2} is assumed to be a hyperelliptic cover, its ramification type has to be: 2g + 2 points are ramified of order 2.) Moreover, we did not prescribe the monodromy group of f_{1}. Since deg(f_{1}) = g + 1, the Riemann-Hurwitz genus formula shows that

4g = d_{f_{1}},

where d_{f_{1}} is the degree of the discriminant divisor of f_{1}.

The “generic” ramification type for covers of ℙ_{K}^{1} of degree g + 1 is expected to be as follows: Each ramified point on ℙ_{K}^{1} has one ramified extension of ramification order 2 and so there should be 4g ramified points. Using the action of the automorphism group of ℙ_{K}^{1} we should expect a “Hurwitz Space” of dimension 4g − 3. Speculating further that in our context we have to intersect this space with the space of covers attached to hyperelliptic curves, which is a subspace of the moduli space of curves of genus g of codimension g − 2, we can expect that as result of the “generic” ramification situation we would get a space of dimension 3g − 1 and so exactly of the dimension of \overline{H}_{g,g+1}.

If we change the ramification conditions by assuming that more than one point in the fibres of f_{1} can be ramified, then we will have fewer ramification points of ℙ_{K}^{1} for f_{1} and so the cover lies in a lower dimensional attached Hurwitz space. It is an interesting task to find out for which ramification type we get a finite map from the attached Hurwitz space to the moduli space of hyperelliptic curves, and to study the geometric properties of the resulting Hurwitz space.

To do this for g = 3 will be the content of the next section.

4 Hyperelliptic Curves of Genus 3

We assume in the whole section that K is a field with the property that every hyperelliptic curve of genus 3 has at least 7 K-rational points. For K = 𝔽_{q} it suffices that q ≥ 59; cf. Remark 5. Hence we know by Lemma 4 (and the proof of Corollary 7) that for every hyperelliptic curve C of genus 3 there is a K-rational 4-cover

f_{1} : C → ℙ_{K}^{1},

which does not factor over a hyperelliptic cover of C.

In fact we know more. In the isomorphism class of (C, f_{1}, f_{2}) we find a triple (C′, f_{C′}, π_{C′}) with C′ a smooth curve in ℙ_{K}^{1} × ℙ_{K}^{1} and f_{C′} = pr_{1}_{|C′}, π_{C′} = pr_{2}_{|C′}; cf. Proposition 6. Such triples are called “embedded”.

In the following we shall choose such embedded triples in isomorphism classes in H̃_{3,4}(K) and indicate this choice by the notation (C, f_{C}, π_{C}) respectively [C, f_{C}, π_{C}].

We recall that after this choice we have an affine equation for C:

C : ∑_{i=0}^{4} (∑_{j=0}^{2} r_{ij}T^{j}) X^{4−i} = 0

with r_{ij} ∈ K.

As was announced in Subsection 3.2, we introduce special ramification types for f_{C}. We only allow ramification orders ≤ 2. (This is generically no restriction). So the

ramified points Q ∈ ℙ_{K}^{1}(\overline{K}) of f_{C} are either of type (2, 1, 1) (i.e., there is exactly one ramified point P_{1} and two unramified points P_{2}, P_{3} in the fiber f_{C}^{−1}(Q)) or it is of type (2, 2) (the fiber of f_{C}^{−1}(Q) consists of two distinct ramified points P_{1}, P_{2}).

For k = 3, 4 we define the set H̃′_{3,4,k}(K) as the subset of H̃_{3,4}(K) consisting of those classes [C, f_{C}, π_{C}] such that f_{C} has at least k ramification points on ℙ_{K}^{1}(\overline{K}) that ramify of type (2, 2). As we shall see below (cf. Remark 4.1.2), the set H̃′_{3,4,k}(K) consists of the K-rational points of a locally closed subscheme H̃′_{3,4,k} of H̃_{3,4}.

Moreover, if we put \overline{H}_{3,4,k} := π_{3}(H̃′_{3,4,k}) ⊂ \overline{H}_{3,4}, then we see by the discussion in Subsection 3.1.5 that the elements of \overline{H}_{3,4,k}(\overline{K}) correspond (via \overline{κ}_{3}) to equivalence classes of pairs (C, f_{C}) ∈ \overline{H}_{3,4}(\overline{K}) such that f_{C} has at least k ramification points on ℙ_{K}^{1}(\overline{K}) of type (2, 2). Note that the set H̃′_{3,4,k}(K) is stable under the action of the group PGL_{2}(K)×PGL_{2}(K). In the next two subsections we will introduce subsets H̃_{3,4,k}(K) ⊂ H̃′_{3,4,k}(K) which serve (at least for K = \overline{K}) as a (partial) system of representatives for this action; cf. Lemmata 11 and 26. This allows us to find representatives of elements in \overline{H}_{3,4,k}(\overline{K}) with simple equations.

4.1 The Hurwitz Space \overline{H}_{3,4,3}

Notation. Fix coordinates on ℙ_{K}^{1}. Let P_{∞} = (0 : 1) ∈ ℙ_{K}^{1} be the point at infinity, and let P_{a} = (1 : a), for a ∈ K. Thus, any K-rational point on the product surface ℙ_{K}^{1} × ℙ_{K}^{1} has the form P_{a,b} := P_{a} × P_{b} ∈ ℙ_{K}^{1}(K) × ℙ_{K}^{1}(K), for a, b ∈ K ∪ {∞}.

Let H̃_{3,4,3}(K) denote the set of isomorphism classes in H̃_{3,4}(K) with representatives (C, f_{C}, π_{C}) satisfying the following conditions:

f_{C}^{∗}(P_{0}) = 2P_{0,∞} + 2P_{0,0}  (5)
f_{C}^{∗}(P_{1}) = 2P_{1,1} + 2P_{1,α}, for some α ∈ K, α ≠ 1  (6)
f_{C}^{∗}(P_{−1}) = 2D, for some D ∈ Div(C), D ≠ P_{−1,∞} + P_{−1,0}, D ≠ 2P, ∀P. (7)

Thus, f_{C} is ramified at P_{0}, P_{1}, P_{−1} of type (2, 2) and so (C, f_{C}, π_{C}) ∈ H̃′_{3,4,3}(K).

Although the above curves might seem to be rather special, they are general enough to represent all curves in \overline{H}_{3,4,3}(\overline{K}). More precisely (cf. [K2]):

Lemma 11 The map H̃_{3,4,3}(\overline{K}) → \overline{H}_{3,4,3}(\overline{K}), which sends isomorphism classes of triples (C, f_{C}, π_{C}) to equivalence classes of covers f_{C} : C → ℙ^{1}\overline{K}, is surjective and has finite fibres.

The strategy to find equations for embedded curves attached to elements in H̃_{3,4,3}(K) is obvious. We shall “plug in” the ramification conditions into the equation

C : ∑_{i=0}^{4} (∑_{j=0}^{2} r_{ij}T^{j}) X^{4−i}

and get normal forms for curves in H̃_{3,4,3}(K) and so, by using Lemma 11, for representatives for all classes in \overline{H}_{3,4,k}(\overline{K}).

Here are immediate consequences of our normalization for the coefficients r_{ij}:

First we use that P_{0,0} is a point on C ramified of order 2. It follows that r_{40} = 0 = r_{30}.

Similarly, since P_{0,∞} is ramified, we have r_{00} = 0 = r_{10}. Hence the equation for C has the form

C : F (X, T) = ∑_{i=0}^{4}(r_{i1}T + r_{i2}T^{2})X^{4−i} + r_{20}X^{2}.

Next we observe that P_{∞} ∈ ℙ_{K}^{1} is not a ramification point of the hyperelliptic cover π_{C}, for otherwise π_{C}^{∗}P_{∞} = 2P_{0,∞}, and then P_{0,∞} would be ramified under both projections of ℙ_{K}^{1} ×ℙ_{K}^{1}, which contradicts the smoothness of C ⊂ ℙ_{K}^{1} ×ℙ_{K}^{1}. This yields:

The discriminant of F (X, T) with respect to T has 8 distinct zeroes and so

D_{F} (X) = (∑_{i=0}^{4} r_{i1}X^{4−i})^{2} − 4r_{20}X^{2} (∑_{i=0}^{4} r_{i2}X^{4−i})

is a separable polynomial. Thus, by Proposition 10 we obtain that

W_{C} : Y^{2} = (∑_{i=0}^{4} r_{i1}X^{4−i})^{2} − 4r_{20}X^{2} (∑_{i=0}^{4} r_{i2}X^{4−i})

is a Weierstraß equation for C.

Recall that the fiber of P_{1} consists of the points P_{1,1} and P_{1,α}, for some α ≠ ∞. Thus, P_{1,∞} ∉ C, and hence it follows that

r_{01} + r_{02} ≠ 0.

The equation F (X, T) is uniquely determined up to a non-zero scalar factor, so we normalize it. Thus, we can and will assume that

r_{02} = 1 − r_{01}.

To continue, we will exploit the other ramification conditions. For this, it is useful to write down an elementary statement about points ramified of type (2, 2) under the map f_{C}.

4.1.1 A Criterion for Points of Type (2, 2)

Lemma 12 Let Q(X) = AX^{4} + BX^{3} + CX^{2} + DX + E ∈ K[X].

(a) If A ≠ 0, then Q(X) = Aq(X)^{2} for some monic quadratic polynomial q(X) = X^{2} + bX + c if and only if

u_{1} := BΔ − 8A^{2}D = 0 and u_{2} := 64EA^{3} − Δ^{2} = 0, where Δ = 4AC − B^{2}. (8)

Moreover, if this holds, then b = B/(2A) and c = Δ/(8A^{2}), and so q(X) has distinct roots in \overline{K} if and only if δ_{1} = B^{2} − 2Δ = 3B^{2} − 8AC ≠ 0. In addition, (8) implies that

g := AD^{2} − EB^{2} = 0.  (9)

(b) If E ≠ 0, then Q(X) = E(aX^{2} + bX + 1)^{2}, for some a, b ∈ K if and only if

u_{1}^{∗} := DΔ′ −8E^{2}B = 0 and u_{2}^{∗} := 64AE^{3} −(Δ′)^{2} = 0, where Δ′ = 4EC −D^{2}. (10)

Moreover, if this holds, then also (9) holds and b = D/(2E) and a = Δ′/(8E^{2}), and so b^{2} − 4a ≠ 0 if and only if δ_{2} := D^{2} − 2Δ′ = 3D^{2} − 8EC ≠ 0.

(c) Q(X) = λ(αX^{2} + βX + γ)^{2}, for some α, β, γ, λ ∈ K with λ ≠ 0 if and only if u_{1} = u_{2} = u_{1}^{∗} = u_{2}^{∗} = 0, and then also g = 0. Moreover, in that case the discriminant δ := λ(β^{2} − 4αγ) = 0 if and only if δ_{1} = δ_{2} = δ_{3} = 0, where δ_{3} := C^{2} − 36AE.

Proof. (a) Since q(X)^{2} = X^{4} +2bX^{3} +(b^{2} +2c)X^{2} +2bcX +c^{2}, we see that the equation Q(X) = Aq(X)^{2} implies that

B = 2Ab, C = A(b^{2} + 2c), D = 2Abc and E = Ac^{2}.  (11)

Thus Δ = 4AC −B^{2} = 8A^{2}c and so BΔ = 16A^{3}bc = 8A^{2}D and Δ^{2} = 64A^{4}c^{2} = 64A^{3}E. This shows that (8) holds and that b = B/(2A) and c = Δ/(8A^{2}). Moreover, since b^{2} −4c = (\frac{B}{2A})^{2} −4\frac{Δ}{8A^{2}} = \frac{B^{2}−2Δ}{4A^{2}} = \frac{3B^{2}−8AC}{4A^{2}} , we see that q(X) has distinct roots in \overline{K} if and only if B^{2} − 2Δ = 3B^{2} − 8AC ≠ 0. In addition, EB^{2} = (Ac^{2})(2Ab)^{2} = A(2Abc)^{2} = AD^{2}, which proves (9). Conversely, suppose that (8) holds. Put b = B/(2A) and c = Δ/(8A^{2}). Then Aq(X)^{2} = AX^{4} + BX^{3} + (\frac{B^{2}}{4A} + \frac{Δ}{4A})X^{2} + (B \frac{Δ}{8A^{2}} ) X + \frac{Δ^{2}}{64A^{3}} = Q(X) by (8).

(b) Apply part (a) to Q_{1}(X) = X^{4}Q(1/X) = A + BX + CX^{2} + DX^{3} + EX^{4}.

(c) Suppose first that u_{1} = u_{2} = u_{1}^{∗} = u_{2}^{∗} = 0. If A ≠ 0, then part (a) shows that Q(X) = λ(αX^{2} + βX + γ) with λ = A, α = 1, β = \frac{B}{2A} and γ = \frac{Δ}{8A^{2}} , and that g = 0. Similarly, if E ≠ 0, then part (b) shows that Q(X) = λ(αX^{2} + βX + γ) with λ = E, α = \frac{Δ′}{8E^{2}} , β = \frac{D}{2E} and γ = 1. Now suppose that A = E = 0. Then the condition u_{2} = 0 implies that B = 0, and u_{2}^{∗} = 0 shows that D = 0. Thus, Q(X) = CX^{2} = λ(βX)^{2}, for suitable λ ∈ K^{×}, β ∈ K. Clearly g = 0^{3} − 0^{3} = 0.

Conversely, suppose that Q(X) = λ(αX^{2} + βX + γ)^{2}. If AE ≠ 0, then by parts (a) and (b) we see that (8) and (10) hold. If A ≠ 0 and E = 0, then γ = 0, and then D = 2λγ = 0. Thus Δ′ = 0 and so u_{1}^{∗} = u_{2}^{∗} = 0. Moreover, u_{1} = u_{2} = 0 by part (a). Similarly, if A = 0 and E ≠ 0, then α = B = Δ = 0, so u_{1} = u_{2} = 0, and u_{1}^{∗} = u_{2}^{∗} = 0 by part(b). Finally, if A = E = 0, then α = γ = 0, so B = D = Δ = Δ′ = 0, and hence (8) and (10) hold trivially.

To prove the assertion about δ, note first that δ only depends on Q and not on the choice of λ, α, β, γ, as is easy to check. Now if A ≠ 0, then a short computation (using λ = A) shows that δ_{1} = 4Aδ, δ_{2} = 4Eδ and δ_{3} = δ(C + \frac{3Δ}{4A}), so δ = 0 ⇒ δ_{1} = δ_{2} = δ_{3} = 0 ⇒ δ_{1} = 0 ⇒ δ = 0, which proves the assertion in this case. Similarly, if E ≠ 0, then δ_{1} = 4Eδ, δ_{2} = 4Aδ and δ_{3} = δ(C + \frac{3Δ′}{4E}), and so the assertion follows. Finally, if A = E = 0, then δ = λβ^{2} = C and δ_{1} = δ_{2} = 0, δ_{3} = C^{2} = Cδ, so we see that here δ = 0 ⇔ δ_{3} = 0 ⇔ δ_{1} = δ_{2} = δ_{3} = 0, and so the assertion holds here well.

We will use the above Lemma 12 in the following way:

Proposition 13 Let C ⊂ ℙ_{K}^{1} ×ℙ_{K}^{1} be a genus 3 curve given by an equation F (T, X) = 0 of the form (3) with n = 4, and let f_{C} = (pr_{1})_{|C} : C → ℙ_{K}^{1} be the associated cover of degree 4. Moreover, let t ∈ K, and let u_{1}, u_{2}, u_{1}^{∗}u_{2}^{∗} and δ_{i} be the elements associated to Q(X) = F (t, X). Then f_{C}^{∗}(P_{t}) has type (2, 2) if and only if u_{1} = u_{2} = u_{1}^{∗} = u_{2}^{∗} = 0 and some δ_{i} ≠ 0, for i = 1, 2, 3.

Proof. Suppose first that deg Q = 4. Then Q(X) = A(X − α_{1})^{m_{1}} ⋯ (X − α_{s})^{m_{s}} with distinct α_{1}, …, α_{s} ∈ \overline{K}, and then f_{C}^{∗}(P_{t}) = ∑ m_{i}P_{t,α_{i}}. It is thus clear that f_{C}^{∗}(P_{t}) has type (2, 2) if and only if Q(X)/A is a square with distinct roots, and so the assertion follows from Lemma 12(c).

Next, suppose that deg Q = 4 − m, where m ≥ 1. Then P_{t,∞} has multiplicity m in f_{C}^{∗}(P_{t}), and so f_{C}^{∗}(P_{t}) has type (2, 2) ⇔ f_{C}^{∗}(P_{t}) = 2P_{t,∞} + 2P_{t,α}, for some α ∈ K ⇔ Q(X) = λ(X − α)^{2}, for some α ∈ K, λ ∈ K^{×} ⇔ Q(X)/λ is a square and δ ≠ 0. Thus, the assertion follows from Lemma 12(c).

Application to the ramification at P_{1}. We apply this to the fiber over P_{1} and recall that f_{C}^{∗}(P_{1}) = 2(P_{1,1} + P_{1,α}). So

F (1, X) = (X − 1)^{2}(X − α)^{2}.

Thus, from (11) it follows that

\begin{aligned}r_{11} + r_{12} & = & −2(α + 1), & r_{20} + r_{21} + r_{22} & = & α^{2} + 4α + 1, \\ r_{31} + r_{32} & = & −2α(α + 1), & r_{41} + r_{42} & = & α^{2}. \end{aligned}  (12)

Note that the first relation shows that α = −\frac{1}{2}(r_{11} + r_{12} + 2).

Application to the ramification at P_{−1}. We have still to exploit that P_{−1} is ramified of type (2, 2), and we have to use elimination to find a system of independent parameters for H̃_{3,4,3}(K). It is not surprising that we have to make a case discussion. This leads two subsets H̃_{3,4,3}^{∗}(K) and H̃_{3,4,3}^{∗∗}(K) which cover H̃_{3,4,3}(K). Following our strategy expressed in the introduction, we shall discuss only H̃_{3,4,3}^{∗}(K) in the sequel; cf.

[K2] for H̃_{3,4,3}^{∗∗}(K). The set H̃_{3,4,3}^{∗}(K) is defined by

H̃_{3,4,3}^{∗}(K) = {[C, f_{C}, π_{C}] ∈ H̃_{3,4,3}(K) with P_{−1,∞} ∉ C}.

The assumption that P_{−1,∞} ∉ C is equivalent to the condition that a_{0} := 1 − 2r_{01} ≠ 0 because if F^{h} is the bi-homogenization of F, then F^{h}(1, −1; 0, 1) = r_{02} − r_{01} = 1 − 2r_{01}.

We now use that P_{−1} is ramified of type (2, 2). Since the leading coefficient of F (−1, X) is a_{0} ≠ 0, condition (7) is equivalent to the assertion that

F (−1, X) = a_{0}(X^{2} + bX + c)^{2},

for some b, c ∈ K, and so by Lemma 12(a) we obtain the relations (8) which yield

r_{32} − r_{31} = \frac{1}{8}(r_{12} − r_{11})Δ and r_{42} − r_{41} = \frac{1}{64}a_{0}Δ^{2},  (13)

with Δ = (4a_{0}(r_{20} − r_{21} + r_{22}) − (r_{12} − r_{11})^{2})/a_{0}^{2}. Thus, by using the first relation of (13) and the third relation of (12) we can solve for r_{31} and r_{32} and so we obtain

r_{31} = −α^{2} − α − \frac{1}{16}(r_{12} − r_{11})Δ

and

r_{32} = −α^{2} − α + \frac{1}{16}(r_{12} − r_{11})Δ.

Similarly, we can solve for r_{41} and r_{42} by using the second relation of (13) and the fourth relation of (12), and so we obtain

r_{41} = \frac{1}{2}α^{2} − \frac{1}{128}(1 − 2r_{01})Δ^{2}

and

r_{42} = \frac{1}{2}α^{2} + \frac{1}{128}(1 − 2r_{01})Δ^{2}

with

Δ = \frac{4(1 − 2r_{01})(α^{2} + 4α + 1 − 2r_{21}) − (r_{12} − r_{11})^{2}}{(1 − 2r_{01})^{2}} .

Moreover, since X^{2} + bX + c has distinct roots, it follows from Lemma 12(a) that

(r_{12} − r_{11})^{2} ≠ 2(1 − 2r_{01})^{2}Δ.

We summarize our results as follows:

Proposition 14 Every element in H̃_{3,4,3}^{∗}(K) has a representative (C, f_{C}, π_{C}) with a curve C given by an equation F (T, X) = 0, where

F (T, X) = ∑_{i=0}^{4}(r_{i1}T + r_{i2}T^{2})X^{4−i} + r_{20}X^{2}  (14)

and the r_{ij} ∈ K are such that the polynomial

D_{F} (X) := (∑_{i=0}^{4} r_{i1}X^{4−i})^{2} − 4r_{20}X^{2} (∑_{i=0}^{4} r_{i2}X^{4−i})  (15)

is separable of degree 8 and such that the following relations hold:

r_{02} = 1 − r_{01}  (16)
r_{22} = α^{2} + 4α + 1 − r_{20} − r_{21}  (17)
r_{31} = −α^{2} − α − \frac{1}{16}a_{1}Δ  (18)
r_{32} = −α^{2} − α + \frac{1}{16}a_{1}Δ  (19)
r_{41} = \frac{1}{2}α^{2} − \frac{1}{128}a_{0}Δ^{2}  (20)
r_{42} = \frac{1}{2}α^{2} + \frac{1}{128}a_{0}Δ^{2}  (21)

in which α = −\frac{1}{2}(r_{11} + r_{12} + 2) ≠ 1, a_{0} = 1 − 2r_{01} ≠ 0, a_{1} = r_{12} − r_{11} and

Δ = Δ(r_{01}, r_{11}, r_{12}, r_{21}) := (4a_{0}(α^{2} + 4α + 1 − 2r_{21}) − a_{1}^{2})/a_{0}^{2}.  (22)

In addition, we have that

a_{1}^{2} ≠ 2a_{0}^{2}Δ.  (23)

A Weierstraß Normal Form for C is given by

Y^{2} = (r_{01}X^{4} + r_{11}X^{3} + r_{21}X^{2} − (α^{2} + α + \frac{1}{16}a_{1}Δ)X + \frac{1}{2}α^{2} − \frac{1}{128}a_{0}Δ^{2})^{2}
−4r_{20}X^{2}((1 − r_{01})X^{4} + r_{12}X^{3} + (α^{2} + 4α + 1 − r_{20} − r_{21})X^{2}
+(−α^{2} − α + \frac{1}{16}a_{1}Δ)X + \frac{1}{2}α^{2} + \frac{1}{128}a_{0}Δ^{2}).

Conversely, if C ⊂ ℙ_{K}^{1}×ℙ_{K}^{1} is a curve given by an equation satisfying the above relations, then the isomorphism class of (C, f_{C}, π_{C}) is in H̃_{3,4,3}^{∗}(K).

It remains to prove the “converse” part of the proposition.

If F (T, X) ∈ K[T, X] satisfies the conditions above, then D_{F}^{h} (X_{0}, X_{1}) = X_{0}^{8}D_{F} (X_{1}/X_{0}) has 8 distinct linear factors, and so by Proposition 8 we see that the equation F (T, X) = 0 defines a smooth curve C ∈ |D_{2,4}|^{sm}. Moreover, we know that P_{−1,∞} ∉ C because r_{02} − r_{01} = 1 − 2r_{01} = a_{0} ≠ 0.

It remains to show that C satisfies conditions (5) – (7). Now if we substitute T = 0, 1, −1 in F (T, X), then we obtain that

F (0, X) = r_{20}X^{2}  (24)
F (1, X) = (X − 1)^{2}(X − α)^{2}  (25)
F (−1, X) = a_{0} (X^{2} + \frac{a_{1}}{2a_{0}} X + \frac{Δ}{8})^{2} ,  (26)

as a quick computation (using MAPLE) shows. Since a_{0} ≠ 0, a_{1}^{2} ≠ 2a_{0}^{2}Δ and r_{20} ≠ 0 (else D_{F} would be a square), it follows that (5) – (7) hold, and so (C, f_{C}, π_{C}) lies in a class of H̃_{3,4,3}^{∗}(K).

In the following Example 15, we give, for all fields K of odd characteristic, examples of curves C/K for which (C, f_{C}, π_{C}) is a representative of a class in H̃_{3,4,3}^{∗}(K). In particular, it follows that H̃_{3,4,3}^{∗}(K) ≠ ∅.

Example 15 (a) Choose r_{01} = r_{20} = 1 and r_{11} = r_{12} = r_{21} = 0. Then a_{0} = α = −1, and so we can define r_{02}, r_{22}, …, r_{42} by (16) – (22). This gives the polynomial

F_{01}(T, X) = X^{4}T − X^{2}(3T^{2} − 1) + T

Here D_{F_{01}}(X) = X^{8} + 14X^{4} + 1 and Δ = 8. Using a computer algebra program (such as MAPLE) we see that the discriminant of D_{F_{01}} is 2^{40}3^{4}, and so it follows that D_{F_{01}} is separable of degree 8 whenever char(K) ≠ 3. Thus F_{01}(T, X) defines a smooth genus 3 curve C ∈ H̃_{3,4,3}(K) whenever char(K) ≠ 3.

(b) Choose r_{01} = r_{21} = 1, r_{20} = 2 and r_{11} = r_{12} = 0. Then as before a_{0} = α = −1, and so we can define r_{02}, r_{22}, …, r_{42} by (16) – (22). This gives the polynomial

F_{02}(T, X) = X^{4}T − X^{2}(5T^{2} − T − 2) − \frac{3}{2}T^{2} + \frac{5}{2}T

Here Δ = 16 and D_{F_{02}}(X) = X^{8} + 2X^{6} + 46X^{4} + 17X^{2} + \frac{25}{4}, which (by MAPLE) has discriminant 2^{20}5^{10}13^{2}17^{4}. Thus F_{02}(T, X) defines a smooth genus 3 curve C ∈ H̃_{3,4,3}(K) whenever char(K) ≠ 5, 13, 17.

4.1.2 Scheme Structures

By using Proposition 13, one can show that H̃′_{3,4,k}(K) can be identified (via κ_{3,K}) with the set of K-rational points of a locally closed subset H̃′_{3,4,k} of H̃_{3,4}; cf. [K2]. More precisely, one can show (by considering (ℙ_{K}^{1})^{k} × H̃_{3,4}) that H̃′_{3,4,k} = V ∩ U, where V (respectively, U) is a closed (respectively, open) subscheme of H̃_{3,4} which is invariant under the G × G-action. It thus follows that \overline{H}_{3,4,k} = π_{3}(H̃′_{3,4,k}) is also locally closed. Thus, both H̃′_{3,4,k} and \overline{H}_{3,4,k} have an induced scheme structure.

In a similar way one can interpret H̃_{3,4,3}(K), H̃_{3,4,3}^{∗}(K), and H̃_{3,4,3}^{∗∗}(K) as the sets of K-rational points of locally closed subschemes H̃_{3,4,3}, H̃_{3,4,3}^{∗}, and H̃_{3,4,3}^{∗∗} of H′_{3,4,3}, respectively. It is then immediate that H̃_{3,4,3}^{∗} and H̃_{3,4,3}^{∗∗} are open subschemes of H̃_{3,4,3}.

With the explicit description for H̃_{3,4,3}^{∗} one is not far away from a geometrical description of H̃_{3,4,3}, though some work still has to be done; cf. [K2] for the details. One gets:

Proposition 16 The Hurwitz space H̃_{3,4,3} is a smooth rational variety of dimension 5 containing H̃_{3,4,3}^{∗} as open subscheme which is isomorphic to an open subscheme of 𝔸_{K}^{5}.

An explicit parametrization of H̃_{3,4,3}^{∗} is given by Proposition 14.

Remark 17 The equations F (X, T) for the curves in H̃_{3,4,3}^{∗}(K) are of degree 5 if r_{01} = 1 (with the normalization made above). This is the case in our examples.

Hence we find a hyperplane of H̃_{3,4,3}^{∗} for which the corresponding hyperelliptic curves C are described by plane equations of degree 5.

4.1.3 From Hurwitz Spaces to Moduli Spaces

Recall from Subsection 3.2 that we have the morphism

\overline{μ}_{3} : \overline{H}_{3,4} → M_{3}^{h},

which is not (generically) finite, as we have already seen. But by Lemma 11 and Proposition 16 we know that we have a surjective map from the 5-dimensional irreducible scheme H̃_{3,4,3} to \overline{H}_{3,4,3} and from Proposition 14 we get a family of Weierstraß equations for the hyperelliptic curves which define points in this space.

We repeat their definition: Take s_{1}, s_{2}, s_{3}, t, u as algebraically independent elements over K. Put

α := −\frac{1}{2}(s_{2} + t + 2),

Δ := (4(1 − 2s_{1})(α^{2} + 4α + 1 − 2s_{3}) − (t − s_{2})^{2})/(1 − 2s_{1})^{2}, and define the hyperelliptic curve C(s_{1}, s_{2}, s_{3}, t, u) =: C by the Weierstraß equation

Y^{2} = (s_{1}X^{4} + s_{2}X^{3} + s_{3}X^{2} − (α^{2} + α + \frac{1}{16}(t − s_{2})Δ)X + \frac{1}{2}α^{2} − \frac{1}{128}(1 − 2s_{1})Δ^{2})^{2}
−4uX^{2}((1 − s_{1})X^{4} + tX^{3} + (α^{2} + 4α + 1 − u − s_{3})X^{2}
−(α^{2} + α + \frac{1}{16}(s_{2} − t)Δ)X + \frac{1}{2}α^{2} + \frac{1}{128}(1 − 2s_{1})Δ^{2}).

Theorem 18 (Hindry-Ritzenthaler) The family of curves C(s_{1}, s_{2}, s_{3}, t, u) is generic of dimension 5.

The proof of this theorem ([HR]) is based on computational methods as developed in [LR] and crucially enhanced by ideas of Marc Hindry that enable to compute the dimension of the tangent space around an arbitrary point of the family.^{5} We use Lemma 11 and get from Proposition 16 and Theorem 18:

Theorem 19 The Hurwitz space \overline{H}_{3,4,3} is an irreducible unirational variety of dimension 5 which is covered by the irreducible and rational variety H̃_{3,4,3} of dimension 5.

The restriction of the forget map \overline{μ}_{3} to \overline{H}_{3,4,3} is generically finite and dominant in the moduli space M_{3}^{h}.

4.2 The Hurwitz space \overline{H}_{3,4,4}

The aim of this subsection is to describe \overline{H}_{3,4,4}. From the geometrical point of view, the main result is the following statement:

Theorem 20 The Hurwitz space \overline{H}_{3,4,4} is a scheme which has two irreducible components V_{1} and V_{2}, both unirational of dimension 4.

The proof of this theorem uses, in addition to the information we get in this paper, rather complicated constructions from algebraic geometry; cf. [K2] for the details.

In this paper we shall find equations for two 4-dimensional families of hyperelliptic curves C which define subsets U_{1}(K), U_{2}(K) of H̃_{3,4,3}(K). The cover maps f_{C} of these curves C have at least 4 ramification points of type (2, 2), so U_{i}(K) ⊂ H̃′_{3,4,4}. Moreover, each U_{i}(K) is the set of K-rational points of a subscheme U_{i} ⊂ H̃′_{3,4,4} whose image in \overline{H}_{3,4,4} is dense in V_{i}, for i = 1, 2.

4.2.1 Two Natural Subspaces of \overline{H}_{3,4,4}

Curves with elliptic differentials. First look at hyperelliptic curves C with cover map

π_{E} : C → E

where E is an elliptic curve and deg(π_{E}) = 2, i.e., C has an elliptic differential of degree 2. This implies that the Jacobian variety J_{C} of C has an elliptic subvariety E^{∗} = π_{E}^{∗}(E) and a two-dimensional abelian subvariety A with A ∩ E^{∗} = E^{∗}[2]. Such hyperelliptic curves are well-studied and it is known that over \overline{K} they generate a 3-dimensional subspace of the moduli space of hyperelliptic curves of genus 3 ([GS]).

Assume that

α : E → ℙ_{K}^{1}

is a cover of degree 2, not induced by the hyperelliptic cover. Then

f_{C} : C ⟶^{α∘π_{E}} ℙ_{K}^{1}

5We are very thankful to Marc Hindry and Christophe Ritzenthaler for answering our question concerning the dimension of the above family very quickly.

is a cover of degree 4 which does not factor over the hyperelliptic cover, and hence the class [C, f_{C}, π_{C}] satisfies the conditions imposed on elements in H̃_{3,4}(K).

We examine the ramification structure of f_{C}. By the Hurwitz genus formula there are 4 points Q_{1}, …, Q_{4} on E(\overline{K}) that are ramified under π_{E}, and α has four ramification points P_{1}, …, P_{4} on ℙ_{K}^{1}(\overline{K}).

Assume that α can be chosen such that the associated involution σ_{α} (with Aut(α) = ⟨σ_{α}⟩) has no fixed points in common with Q_{1}, …, Q_{4}. Then the points α^{−1}P_{1}, … α^{−1}P_{4} are unramified under π_{E} and hence have ramification type (2, 2) with respect to f_{C}. Thus, (C, f_{C}, π_{C}) represents an element in H̃′_{3,4,4}(K), and the equivalence class of f_{C} defines an element in \overline{H}_{3,4,4}(K).

Remark 21 (a) The assumptions made for α can be satisfied for K large enough (e.g., for K = \overline{K}).

(b) “ Generically” the monodromy group of f_{C} is the dihedral group D_{4}.

(c) Again “generically” we have that α(Q_{i}) ≠ α(Q_{j}) for i ≠ j and so the ramification type of α(Q_{i}) is (2, 1, 1).

Remark 22 The above covers f_{C} = α ∘ π_{E} are clearly imprimitive in the sense that they factor over a nontrivial subcover. Now it turns out that every imprimitive cover f_{C} : C → ℙ_{K}^{1} in H_{3,4}(K) is of the above form.

Indeed, suppose that f_{C} = α ∘ π, for some subcover π : C → C′ of degree 2. By Riemann-Hurwitz we have that g_{C′} = 0, 1, or 2. But if g_{C′} = 0, then π is equivalent to the hyperelliptic cover π_{C}, and so f_{C} factors over π_{C}, contradiction. Next, if g_{C′} = 2, then α = π_{C′} is (equivalent to) the hyperelliptic cover of C′. But then α is induced by the hyperelliptic cover of C, and so f_{C} factors over π_{C}, contradiction. Thus, we must have g_{C′} = 1, so C′ = E is an elliptic curve.

Covers with monodromy group equal to S_{4}. For more details of the following discussion see [FK].

Assume that

f_{C} : C → ℙ_{K}^{1}

is primitive, and that there are exactly 4 points P_{1}, …, P_{4} of ramification type (2, 2) and 4 ramification points Q_{1}, …, Q_{4} of ramification type (2, 1, 1). It follows that the discriminant divisor of f_{C} is equal to Δ_{C} = 2P_{1} + ⋯ + 2P_{4} + Q_{1} + ⋯ + Q_{4} and that the monodromy group of f_{C} is equal to S_{4}. Let C̃ be the cover curve obtained as the Galois closure of f_{C}.

Let G_{2} be a non-cyclic subgroup of S_{4} of order 4 and different from the Klein group in A_{4}. Then G_{2} = N_{S_{4}}(τ) is the normalizer of a transposition τ, and hence any two such groups are conjugate in S_{4}. By analyzing the ramification groups, we obtain that C′ := C̃/G_{2} is a curve of genus 3. Moreover, if P_{2} denotes the unique 2-Sylow subgroup of S_{4} which contains G_{2}, then C̃/P_{2} has genus 0. Thus, C′ has the following property:

There is a degree-2 cover u_{2} : C′ → ℙ_{K}^{1}, and so C′ is hyperelliptic, and a degree-3 cover f_{3} : ℙ_{K}^{1} → ℙ_{K}^{1} such that the Galois closure of f_{6} := f_{3} ∘ u_{2} is C̃ and the monodromy group of f_{6} is the symmetric group S_{4}. Hence f_{6} : C′ → ℙ^{1} is a trigonal cover in the sense of Donagi-Livné. The cover maps φ_{1} : C̃ → C′ and φ_{2} : C̃ → C induce (via conorm respectively norm maps) a correspondence φ_{2}_{∗} ∘ φ_{1}^{∗} and so a morphism from Pic_{C′}^{0} to Pic_{C}^{0} which is (essentially) as constructed by Smith [Sm].

In other words, C is the curve obtained from C′ by Smith’s construction. The special property here is that C is hyperelliptic.

4.2.2 Equations for H̃_{3,4,4}^{∗}

We use the assumptions and results from Subsection 4.1 and add the condition that for fixed t ∈ K ∪ {∞} \ {0, 1, −1}, the point P_{t} ∈ ℙ_{K}^{1} is ramified under f_{C} with ramification type (2, 2), i.e. f_{C}^{∗}(P_{t}) = 2D_{t}, for some D_{t} ∈ Div(C), D_{t} ≠ 2P for any P ∈ C(K).

The subset of isomorphism classes [C, f_{C}, π_{C}] ∈ H̃_{3,4,3}^{∗}(K) for which C satisfies this additional condition is denoted by H̃_{3,4,4,t}^{∗}(K).

Moreover, define H̃_{3,4,4}^{∗}(K) = ⋃_{t} H̃_{3,4,4,t}^{∗}(K), where the union is over all t ∈ K ∪ \{∞\}\\{0, 1, −1\}. It is not difficult to see (cf. [K2]) that H̃_{3,4,4}^{∗}(\overline{K}) is the set of \overline{K}-rational points of the scheme H̃_{3,4,4}^{∗} := H̃_{3,4,3}^{∗} ∩ H̃′_{3,4,4}; cf. Subsection 4.1.2.

We will find equations for two rational families of triples (C, f_{C}, π_{C}) with [C, f_{C}, π_{C}] ∈ H̃_{3,4,4}^{∗}; these will then also give equations for two open subschemes U_{i} of H̃_{3,4,4}^{∗} whose union U = U_{1} ∪ U_{2} is dense in H̃_{3,4,4}^{∗}.

We use the notation introduced in Subsection 4.1. Our aim is to show that we can find (Zariski-)open conditions for the parameter set r_{01}, r_{11}, r_{12}, t such that for the resulting subset U(K) ⊂ 𝔸_{K}^{4}(K) = K^{4} we get: There is exactly one triple (C, f_{C}, π_{C}) with [C, f_{C}, π_{C}] ∈ H̃_{3,4,4,t}(K) with coefficients r_{01}, r_{11}, r_{12} in its normal form, and the other coefficients in the normal form are given by rational expressions in these parameters.

It is not surprising that the formulas become a bit involved, and to write them down it is useful to introduce some additional notation.

Notation. For r_{01}, r_{11}, r_{12}, t ∈ K, put

a_{0} = 1 − 2r_{01}  a_{3} = r_{01}r_{11} + r_{01}r_{12} − r_{11}  a_{5} = (1 − r_{01})t + r_{01}
a_{1} = r_{12} − r_{11}  α = −\frac{1}{2}(r_{11} + r_{12} + 2)  a_{6} = r_{12}t + r_{11}.
a_{2} = r_{12} + r_{11}

A(T) = r_{01}T + (1 − r_{01})T^{2},  (27)
B(T) = r_{11}T + r_{12}T^{2},  (28)
C(T) = r_{20} + r_{21}T + (α^{2} + 4α + 1 − r_{20} − r_{21})T^{2},  (29)

where r_{20} and r_{21} have to be determined as functions of r_{01}, r_{11}, r_{12}, t. For later use define

r_{201} := \frac{ta_{3}^{2}}{4a_{0}a_{5}} and r_{211} := \frac{4a_{0}(4αr_{01} + (α + 1)^{2}) − a_{1}^{2}}{8a_{0}} .  (30)

We add open conditions for the parameter set:

Condition U. We assume that r_{01}, r_{11}, r_{12}, t ∈ K are such that

• a_{0} ≠ 0, i.e., r_{01} ≠ \frac{1}{2},

• a_{5} ≠ 0, i.e., either r_{01} = 1 or r_{01} ≠ 1 and t ≠ \frac{r_{01}}{−1+r_{01}}

• a_{6} ≠ 0,

• αa_{3} ≠ 0 and so d := 4αa_{0}a_{3} ≠ 0.

• Moreover, assume that q ≠ 0, where

q := (a_{2}(r_{11}t + r_{12} − 3a_{6})a_{5} + 2a_{6}^{2})/(t − 1).

Claim. If C ⊂ ℙ^{1} × ℙ^{1} is a curve with cover maps f_{C} and π_{C} such that the isomorphism class of (C, f_{C}, π_{C}) is in H̃_{3,4,4,t}^{∗}(K) and the parameters r_{01}, r_{11}, r_{12}, t ∈ K satisfy Condition U, then C can be described by one of the following two types of equations (31) and (32), as will be explained presently. Define

F_{1}(T, X) = A(T)X^{4} + B(T)X^{3} + C(T)X^{2} + αB(T)X + α^{2}A(T),  (31)

where A(T), B(T) and C(T) are as in (27) – (29) with r_{20} = r_{201} and r_{21} = r_{211}, and

F_{2}(T, X) = F_{1}(T, X) + \frac{d}{q}G(T, X),  (32)

with

G(T, X) = (c_{2}(1 − T^{2}) + a_{6}T (1 − T))X^{2} + c_{3}T (1 − T)X + c_{4}T (1 − T),

and

c_{2} = \frac{ta_{3}}{a_{0}} , c_{3} = \frac{a_{1}a_{6}}{2a_{0}} , c_{4} = −\frac{αa_{1}a_{2}a_{5}a_{6}}{q} .

Proof of the claim. C is a curve satisfying the hypotheses of Condition U, so C belongs to a triple with isomorphism class in H̃_{3,4,3}^{∗}(K), and hence by Proposition 14 we know that C is given by an equation F (T, X) = 0 of the form (14), where the coefficients r_{ij} satisfy the conditions (16) – (21) and are uniquely determined by C.

We thus have that F (T, X) = ∑_{i=0}^{4} A_{i}(T)X^{4−i}, with A_{0}(T) = A(T), A_{1}(T) = B(T) and A_{2} = C(T), where A, B, C are as in (27) – (29). Moreover, by Proposition 14 we also know that D_{F} is separable of degree 8.

Since f_{C}^{∗}(P_{t}) = 2D_{t} and P_{t,∞} ∉ C by hypothesis, we see that

F (t, X) = a(X^{2} + bX + c)^{2},

for some b, c ∈ K, where a = A_{0}(t) = ta_{5} ≠ 0 is the leading coefficient of F (t, X). Thus, by Lemma 12(a) we obtain the relations (8) and (9), which can be written in the form u_{1} = u_{2} = g = 0, where

u_{1} = 4a_{5}a_{6}c − 8ta_{5}^{2}a_{7} − ta_{6}^{3}, u_{2} = 64t^{2}a_{5}^{3}a_{8} − (4a_{5}c − ta_{6}^{2})^{2}, g = a_{6}^{2}a_{8} − a_{5}a_{7}^{2},

with

c = C(t) = (1 − t^{2})r_{20} + (t − t^{2})r_{21} + t^{2}(α^{2} + 4α + 1),

a_{7} = A_{4}(t)/t = r_{32}t + r_{31} = −(t + 1)(α^{2} + α) + \frac{1}{16}(t − 1)a_{1}Δ(r_{01}, r_{11}, r_{12}, r_{21}),

and

a_{8} = A_{5}(t)/t = r_{42}t + r_{41} = \frac{1}{2}(t + 1)α^{2} + \frac{1}{128}(t − 1)a_{0}Δ(r_{01}, r_{11}, r_{12}, r_{21})^{2}

with Δ(r_{01}, r_{11}, r_{12}, r_{21}) as in (22). (Here we’ve used the relations (17) – (21).) To analyze these relations, we first observe (by using MAPLE) that g factors as

64a_{0}^{4}g = (1 − t^{2})(4a_{0}r_{21} − c_{211})(4a_{0}qr_{21} − c_{212}) = (1 − t^{2})g_{1}g_{2},  (33)

in which g_{1} = 4a_{0}(r_{21}−r_{211}) = 4a_{0}r_{21}−c_{211} and g_{2} = 4a_{0}qr_{21}−c_{212} with c_{211} := 4a_{0}r_{211} and c_{212} := qc_{211} + 4a_{0}a_{6}d.

Since (1 − t^{2}) ≠ 0 we have to discuss two cases. Case 1. g_{1} = 0. This implies that

r_{21} = r_{211} = \frac{4a_{0}(4αr_{01} + (α + 1)^{2}) − a_{1}^{2}}{8a_{0}} .

Substituting this into (22) and simplifying (with MAPLE’s help) yields that

Δ(r_{01}, r_{11}, r_{12}, r_{21}) = Δ(r_{01}, r_{11}, r_{12}, \frac{4a_{0}(4αr_{01} + (α + 1)^{2}) − a_{1}^{2}}{8a_{0}} ) = 8α.

Thus, by (18) we obtain that r_{31} = αr_{11} and similarly (19) yields r_{32} = αr_{12}. Moreover, from (20) and (21) we obtain that r_{41} = r_{01}α^{2} and r_{42} = (1 − r_{01})α^{2}. It remains to determine r_{20}.

Substituting the values for r_{21}, r_{31}, r_{32} and r_{42} into the equation u_{1} = 0, we obtain

(1 − t^{2})a_{6}h_{1}/a_{0} = 0

with

h_{1} = 4a_{0}a_{5}r_{20} − ta_{3}^{2}.

Hence by our assumptions we get that h_{1} = 0 and so

r_{20} = r_{201} = \frac{ta_{3}^{2}}{4a_{0}a_{5}} .

By (23) we have that a_{1}^{2} ≠ 2a_{0}^{2}Δ_{1} = 16a_{0}^{2}α. Moreover, since

F_{1}(t, X) = a_{5}t (X^{2} + \frac{a_{6}}{2a_{5}} X + α)^{2} ,

we see that F_{1}(t, X) has two distinct roots if and only if a_{6}^{2} ≠ 16a_{5}^{2}α. We note this inequality and will find it in Theorem 23 below as a part of Inequality (35). Case 2. g_{2} = 0. Since a_{0} ≠ 0 and q ≠ 0 it is clear that g_{2} = 0 is equivalent to

r_{21} = r_{212} := \frac{c_{212}}{4a_{0}q} = r_{211} + \frac{da_{6}}{q} .

From the relation between r_{211} and r_{212} we see immediately from (22) that

Δ(r_{01}, r_{11}, r_{12}, r_{21}) = Δ(r_{01}, r_{11}, r_{12}, r_{212}) = Δ(r_{01}, r_{11}, r_{12}, r_{211}) − (\frac{da_{6}}{q} ) (\frac{8}{a_{0}} ) .

Define

\begin{aligned}r_{311} & := & −α^{2} − α − \frac{1}{16}a_{1}Δ(r_{01}, r_{11}, r_{12}, r_{211}), \\ r_{321} & := & −α^{2} − α + \frac{1}{16}a_{1}Δ(r_{01}, r_{11}, r_{12}, r_{211}), \\ r_{411} & := & \frac{1}{2}α^{2} − \frac{1}{128}a_{0}Δ^{2}(r_{01}, r_{11}, r_{12}, r_{211}), \\ r_{421} & := & \frac{1}{2}α^{2} + \frac{1}{128}a_{0}Δ^{2}(r_{01}, r_{11}, r_{12}, r_{211}). \end{aligned}

By (18) – (19) it follows that r_{31} = r_{311} + (\frac{d}{q})c_{3} and r_{32} = r_{321} − (\frac{d}{q})c_{3}, with c_{3} = \frac{a_{1}a_{6}}{2a_{0}} . Similarly, from (20) – (21) we see that r_{41} = r_{411} + \frac{d}{q}c_{4}, and r_{42} = r_{421} − \frac{d}{q}c_{4}, with c_{4} = −\frac{αa_{1}a_{2}a_{5}a_{6}}{q}. In addition, since the condition u_{1} = 0 means that r_{20} = r_{202} := \frac{c_{202}}{4a_{0}a_{5}q} = r_{201} + \frac{d}{q}c_{2}, where c_{2} = \frac{ta_{3}}{a_{0}} , we obtain that

F (T, X) = F_{1}(T, X) + \frac{d}{q}G(T, X) = F_{2}(T, X).

Moreover, by (23) we have that a_{1}^{2} ≠ 16a_{0}^{2} \frac{Δ(r_{01},r_{11},r_{12},r_{21})}{8} = 16a_{0}^{2}(α− \frac{da_{6}}{a_{0}q}), and since

F_{2}(t, X) = a_{5}t (X^{2} + \frac{a_{6}}{2a_{5}} X + (α − \frac{(t − 1)da_{1}}{2a_{0}q}))^{2} ,  (34)

we see that F_{2}(t, X) has two distinct roots if and only if a_{6}^{2} ≠ 16a_{5}^{2}(α + (1 − t)\frac{da_{1}}{2a_{0}q}). We summarize and get the following result:

Theorem 23 Fix t ∈ K \{0, ±1}, and let U_{3,4,4,t} denote the open subscheme of H̃_{3,4,4,t}^{∗} where Condition U holds.

Then U_{3,4,4,t} is the union of the two three-dimensional varieties U_{1,t} and U_{2,t} which are given as follows: There exist unique elements r_{01}, r_{11}, r_{12} ∈ K such that a curve C representing an element in U_{i,t}(K) is given by the equation

F_{i}(T, X) = 0,

where F_{i}(T, X) is defined as above by equation (31) or (32).

Moreover, we have that D_{F_{i}}(X) is separable of degree 8 and that

a_{1}^{2} ≠ 16a_{0}^{2} Δ′_{i} and a_{6}^{2} ≠ 16a_{5}^{2}Δ′′_{i}  (35)

where Δ′_{1} = Δ′′_{1} = α, Δ′_{2} = α − \frac{da_{6}}{a_{0}q} , Δ′′_{2} = α + (1 − t)\frac{da_{1}}{2a_{0}q}.

Conversely, every equation of the above form with αa_{0}a_{3}a_{5}q ≠ 0 defines a curve C which corresponds to a point in U_{3,4,4,t}(K).

After our discussion, one only has to prove the converse part of the theorem and this is, in principle, an easy check (with the help of MAPLE).

Let us emphasize that a lot more work has to be invested if one wants to remove the restrictions given by Condition U; cf. [K2] for the details. The result is the following:

Theorem 24 ([K2]) For each t ∈ K ∪ {∞}, t ≠ 0, ±1, the Hurwitz space H̃_{3,4,4,t}^{∗} consists of two irreducible, rational components H̃_{3,4,4,t,1}^{∗} and H̃_{3,4,4,t,2}^{∗} of dimension 3.

We now use this result to study the Hurwitz space H̃_{3,4,4}^{∗}. Taking t = τ as a variable over K and replacing K by K(τ), we obtain the 3-dimensional rational scheme H̃_{3,4,4,τ}^{∗} over K(τ). We would like to interpret H̃_{3,4,4,τ}^{∗} as the generic fibre of H̃_{3,4,4}^{∗} with respect to the “morphism” from H̃_{3,4,4}^{∗} to ℙ_{K}^{1} which takes a point corresponding to C to t.

But this does not lead to a morphism because the (2, 2)-ramification point t is not uniquely determined by (C, f_{C}, π_{C}) since f_{C} may have more than 4 ramification points of type (2, 2).

However, we can cover H̃_{3,4,4}^{∗} by a locally closed subscheme H̃_{3,4,4}^{†} ⊂ ℙ_{K}^{1} × H̃_{3,4,3}^{∗} such that the fibre at t ∈ K of p_{1} := (pr_{1})_{|H̃_{3,4,4}^{†}} → ℙ_{K}^{1} can be identified with H̃_{3,4,4,t}^{∗}

and such that its generic fibre is H̃_{3,4,4,τ}^{∗}. From this it is not difficult to deduce the following result (cf. [K2]):

Corollary 25 The Hurwitz space H̃_{3,4,4}^{∗} consists of two irreducible, rational components of dimension 4.

From this corollary it is easy to deduce Theorem 20 by using the following fact (cf. [K2]) which is a partial analogue of Lemma 11:

Lemma 26 The map H̃_{3,4,4}^{∗}(\overline{K}) → \overline{H}_{3,4,4}(\overline{K}) is surjective and has finite fibres.

The following examples (as well as other examples) are used in the proof of Theorems 20 and 23 to show that the Hurwitz spaces H̃_{3,4,4}^{∗} and hence \overline{H}_{3,4,4} are non-empty.

Example 27 (a) Substituting r_{01} = 1, r_{11} = 0, and r_{12} = 4 in F_{1}(T, X) yields the polynomial

F_{11}(T, X) = T X^{4} + 4T^{2}X^{3} + (4tT^{2} − 2T − 4t)X^{2} − 12T^{2}X + 9T.

By MAPLE we find that the discriminant of F_{11}(T, X) with respect to T is

D_{F_{11}}(X) = X^{8} − 4X^{6} + 64tX^{5} + (64t^{2} + 22)X^{4} − 192tX^{3} − 36X^{2} + 81,

which in turn has discriminant d_{F_{11}}(t) = 2^{64} · 3^{12} · (t^{4} + 14t^{2} + 1)t^{4}(t − 1)^{4}(t + 1)^{4}. It thus follows from Proposition 8 that the equation F_{11}(T, X) = 0 defines a smooth genus 3 curve C_{11} on ℙ_{K}^{1} × ℙ_{K}^{1} whenever char(K) ∤ d_{F_{11}}^{∗} = 3(t^{4} + 14t^{2} + 1). Moreover, since a_{0} = −a_{5} = −1, α − 1 = −4 and a_{1}^{2} − 16a_{0}^{2}Δ′_{1} = 784 = 2^{4}7^{2} and a_{6}^{2} − 16a_{5}^{2}Δ′′_{1} = 784t^{2}, it follows that C_{11} ∈ H̃_{3,4,3,t} whenever char(K) ∤ f_{11}(t) := 7d_{F_{11}}^{∗} = 21(t^{4} + 14t^{2} + 1). To check the ramification behavior of the associated 4-cover f_{11} : C_{11} → ℙ_{K}^{1}, we compute its discriminant divisor which is given by the discriminant disc(F_{11}) of F_{11}(T, X) with respect to X. By MAPLE we find that

disc(F_{11}) = 2^{12}3^{2}T^{2}(T − 1)^{2}(T + 1)^{2}(T − t)^{2}disc^{∗}(F_{11}),

where

disc^{∗}(F_{11}) := (t^{2} + 12)T^{4} − 4tT^{3} + (4 − 2t^{2})T^{2} + 4tT + t^{2}.

Since the discriminant of disc^{∗}(F_{11}) is (by MAPLE) 2^{12}3^{2}t^{4}(t^{4} + 14t^{2} + 1), we see that disc^{∗}(F_{11}) has four distinct roots (in \overline{K}) whenever char(K) ∤ 2^{12}3^{2}t^{4}(t^{4} + 14t^{2} + 1). Since this condition follows from the previous one involving f_{11}(t), we see that whenever char(K) ∤ f_{11}(t) = 21(t^{4} + 14t^{2} + 1), then F_{11} is ramified of type (2, 2) at T = 0, 1, −1, t and of type (2, 1, 1) at the 4 distinct roots of disc^{∗}(F_{11}).

For example, if we specialize to the case t = 2, then f_{11}(2) = 3^{2} · 5 · 7, so we see that whenever char(K) > 7, then the equation

F_{112}(T, X) = T X^{4} + 4T^{2}X^{3} + (8T^{2} − 2T − 8)X^{2} − 12T^{2}X + 9T

defines a smooth curve C_{112} of genus 3 on ℙ_{K}^{1} × ℙ_{K}^{1} whose associated 4-cover f_{112} : C_{112} → ℙ_{K}^{1} is ramified of type (2, 2) at T = 0, 1, −1, 2 and of type (2, 1, 1) at the roots of 4T^{4} − 2T^{3} − T^{2} + 2T + 1.

(b) Substituting r_{01} = 1, r_{11} = −1, r_{12} = 2 and t = 2 in F_{2}(T, X) yields the polynomial

F_{21}(T, X) = T X^{4} + (2T^{2} − T)X^{3} + (\frac{7}{3}T^{2} + \frac{9}{4}T − \frac{22}{3})X^{2} + (3T^{2} − \frac{9}{2}T)X + \frac{17}{4}T − 2T^{2}.

By MAPLE, the discriminant D_{F_{21}} ∈ K[X] has degree 8, and its discriminant is

d_{F_{21}} = (2)^{12}(5)^{2}(11)^{4}(13)^{2}(17)^{6}(19)^{2}(47)^{2}(191)^{2}(3)^{−14},

so F_{21} defines a smooth curve whenever char(K) > 19 and ≠ 47, 191. The discriminant of F_{21} (and of the cover) is

disc(F_{21}) = 18^{2}T^{2}(T − 1)^{2}(T + 1)^{2}(T − 2)^{2}disc^{∗}(F_{21}),

where disc^{∗}(F_{21}) = 15929408−7986000T −7592871T^{2} +6397864T^{3} −1297776T^{4}. Since the discriminant of disc^{∗}(F_{21}) is −(2)^{22}(3)^{17}(7)^{6}(11)^{12}(379)^{3}, we see that the polynomial disc^{∗}(F_{21}) has distinct roots if, in addition, char(K) ≠ 379. Thus, for char(K) > 19 and ≠ 47, 191, 379, we obtain a cover which is ramified of type (2, 2) at T = 0, 1, −1, 2 and of type (2, 1, 1) at the four roots of disc^{∗}(F_{21}).

4.2.3 The Lagrange Resolvent

If we compare the results of Theorem 23 (and/or Theorem 24) with the motivating discussion of Subsection 4.2.1, then the natural question arises: Which of the two families U_{1} and U_{2} corresponds to the curves with elliptic differentials, and which corresponds to the S_{4}-covers?

To decide this, we have to look at the monodromy group of covers in more detail. A convenient tool for this is the associated Lagrange Resolvent, as we shall see.

Definition. The Lagrange resolvent of a general quartic polynomial

f(x) = ax^{4} + bx^{3} + cx^{2} + dx + e  (36)

is the monic cubic polynomial r_{f}(x) which is defined by

r_{f}(x) = x^{3} − cx^{2} + (bd − 4ae)x + a(4ce − d^{2}) − b^{2}e.

Remark 28 If f is monic, then this definition of r_{f} agrees with the usual definition; cf. Hungerford[Hu], p. 272. In general, however, we have that r_{f}(ax) = a^{3}rf̃(x), where f̃(x) = f(x)/a is the associated monic polynomial (when a ≠ 0).

We can use the Lagrange resolvent to detect primitivity because of the following basic fact.

Lemma 29 Let x be a root of an irreducible quartic f(X) ∈ k[X]. Then k(x)/k is primitive if and only if r_{f}(X) is irreducible.

Proof. This follows from [Hu], Proposition V.4.11 (p. 273) because in the list of groups given there, k(x)/k is primitive if and only if Gal_{f} ≃ A_{4} or S_{4}, as is easy to see.

The following type of polynomials will play an important role in analyzing the curves represented by the points in U_{1}.

Lemma 30 Let f(X) ∈ k[X] be an irreducible quartic of the form

f(X) = aX^{4} + bX^{3} + cX^{2} + αbX + α^{2}a.  (37)

(a) We have that Gal_{f} ≃ D_{4} or Gal_{f} ≃ ℤ/4ℤ or Gal_{f} ≃ ℤ/2ℤ × ℤ/2ℤ. Moreover, the latter case occurs if and only if the discriminant disc(f) of f is a square in k.

(b) If x is a root of f(X), and if we put y = (2ax^{2} + bx + 2aα)/x, then y^{2} = b^{2} − 4ac + 8a^{2}α, and so k(y) is a quadratic subfield of k(x).

Proof. (a) It is easy to see that the Lagrange resolvent of f(x) factors as

r_{f}(x) = (x − 2aα)(x^{2} + (2aα − c)x + α(b^{2} − 2ac)).

Thus, r_{f}(x) is reducible over k and so the first assertion follows from Proposition 4.11 of Hungerford[Hu], p. 273. Moreover, since

disc(f) = α^{2}(b^{2} + 8a^{2}α − 4ac)^{2}((2aα + c)^{2} − 4αb^{2}),

we see that disc(f) is a square in k if and only if (2aα+c)^{2} −4αb^{2} = (2aα−c)^{2} −4α(b^{2} − 2ac) is a square in k. Since the latter is the discriminant of x^{2} +(2aα−c)x+α(b^{2} −2ac), it follows that disc(f) is a square in k if and only if r_{f} splits in k. By Hungerford[Hu], Proposition 4.11, this is equivalent to the condition that Gal_{f} ≃ Z/2ℤ × ℤ/2ℤ.

(b) The first assertion holds because

y^{2} − (b^{2} − 4ac + 8a^{2}α) = 4a(ax^{4} + bx^{3} + cx^{2} + αbx + α^{2}a)/x^{2} = 0.

Thus, [k(y) : k] ≤ 2. On the other hand, since x is a root of the polynomial 2aX^{2} + (b − y)X + 2aα ∈ k(y)[X], we see that [k(x) : k(y)] ≤ 2, and so [k(x) : k(y)] = [k(y) : k] = 2 because [k(x) : k] = 4.

4.2.4 Application to the Monodromy Group of Covers

Let the isomorphism class of (C, f_{C}, π_{C}) belong to H̃_{3,4,4}^{∗}(K). We want to determine the monodromy group of the 4-cover f_{C} : C → ℙ_{K}^{1}, which is the same as the Galois group of the associated polynomial F (T, X) over K(T).

Curves with Elliptic Differentials. Recall the definition of the open subscheme U_{1} of H̃_{3,4,4}^{∗} given in Theorem 23: The curves representing elements in this set satisfy the equation (31), which has the form

F_{1}(T, X) = A(T)X^{4} + B(T)X^{3} + C(T)X^{2} + αB(T)X + α^{2}A(T).

We look at this polynomial over K(T) and see that it satisfies the conditions of Lemma 30. Hence its resolvent is reducible and f_{C} factors through a quadratic subcover

g_{2} : C → E,

where E is given by the equation

Y^{2} = B(T)^{2} − 4A(T)C(T) + 8αA(T)^{2}

with

A(T) = r_{01}T + (1 − r_{01})T^{2},
B(T) = r_{11}T + r_{12}T^{2},
C(T) = r_{201} + r_{211}T + (α^{2} + 4α + 1 − r_{201} − r_{211})T^{2}

where α = −\frac{1}{2}(r_{11} + r_{12} + 2) and r_{201} and r_{211} are defined in (30) and satisfy the relations described in Theorem 23. In particular, E is an elliptic curve.

Hence we get:

Theorem 31 The members of the family U_{1} in H̃_{3,4,4}^{∗} are attached to hyperelliptic curves C of genus 3 with an elliptic differential of degree 2.

The elliptic curve covered by such curves C is given by the Weierstraß equation

Y^{2} = B(T)^{2} − 4A(T)C(T) + 8αA(T)^{2},

which is uniquely and explicitly determined by the parameters r_{01}, r_{11}, r_{12}, t.

Corollary 32 The monodromy group Gal_{f_{C}} of f_{C} for (C, f_{C}, π_{C}) belonging to an isomorphism class in U_{1}(K) is either the dihedral group D_{4} or the Klein 4-group V ≃ ℤ/2ℤ × ℤ/2ℤ. Moreover, Gal_{f_{C}} ≃ D_{4} if and only if the discriminant disc(F_{1}) of its associated equation F_{1}(T, X) is not a square in K(T). Thus, the isomorphism classes of triples (C, f_{C}, π_{C}) that lie in U_{1}(K) with Gal_{f_{C}} ≃ D_{4} are K-rational points of an open non-empty subscheme of U_{1} (the generic case).

Proof. Since f_{C} is ramified of type (2, 2) at some point, Gal_{f_{C}} contains a (2, 2)-cycle and hence cannot be cyclic. In view of this, the first and second assertions follow from Lemma 30(a). Since over \overline{K} the condition of a being a square can be described by polynomial conditions (cf. Lemma 12), the last assertion follows.

Remark 33 It follows from Corollary 32 that the examples presented in Example 27(a) all have D_{4} as their monodromy group. Indeed, it seems difficult to find examples in U_{1} with monodromy group V.

The Primitive Case. We now come to the second family U_{2} in H̃_{3,4,4}^{∗}. Each curve C ⊂ ℙ^{1} × ℙ^{1} attached to elements in U_{2}(K) is given by an equation of the form

F_{2}(T, X) = F_{1}(T, X) + \frac{d}{q}G(T, X).

By the previous discussion we would expect that at least generically such curves should give primitive 4-covers f_{C} : C → ℙ_{K}^{1} with monodromy group S_{4}. The following result shows that this expectation is correct:

Theorem 34 Take (C, f_{C}, π_{C}) such that its isomorphism class lies in U_{2}(K), and suppose that f_{C} has ramification type (2, 2)^{4}(2, 1, 1)^{4} or, equivalently, that the discriminant of its associated polynomial F_{2}(T, X) has the form

disc(F_{2}) = T^{2}(X^{2} − 1)^{2}(X − t)^{2}disc^{∗}(F_{2}),

and disc^{∗}(F_{2}) ∈ K[T] is squarefree. If r_{11} ≠ r_{12}, then f_{C} is a primitive cover and hence its monodromy group is S_{4}.

Proof. (Sketch) To show that f_{C} is primitive, it is enough to show that the Lagrange resolvent r_{F_{2}} is irreducible over K(T); Lemma 29. Suppose the contrary, i.e., that

r_{F_{2}}(X) = (X − y)(X^{2} + aX + b)

for some y, a, b ∈ K(T). Since r_{f}(X) ∈ K[T, X], it follows from Gauss that y, a, b ∈ K[T]. Furthermore, a careful degree count shows that deg(y) ≤ 2; cf. [K2].

By specializing f(X) modulo T − 1, T + 1 and T − t and using (25), (26) and (34) and suitable properties of r_{f}, it follows that y satisfies the congruences

y ≡ 2α (mod T − 1)
y ≡ 2a_{0}α (mod T + 1)
y ≡ 2a_{5}tβ (mod T − t),

where β = α − \frac{da_{1}(t−1)}{2a_{0}q}; cf. [K2] for the details. Since deg(y) ≤ 2, the Lagrange Interpolation Formula shows that

y = α(T − t) (\frac{T + 1}{1 − t} + a_{0} \frac{T − 1}{t + 1}) + 2a_{5}tβ \frac{T^{2} − 1}{t^{2} − 1}.

On the other hand, the fact (24) that F_{2}(0, X) = r_{20}X^{2} implies that

r_{F_{2}}(X) ≡ X^{3} − r_{20}X^{2} (mod T),

and from this one obtains that either y ≡ 0 (mod T) or that y ≡ r_{20} (mod T). Now if y ≡ r_{20} (mod T), then b^{2} − 4c ≡ 0 (mod T), and this contradicts the fact that disc^{∗}(F_{2}) is squarefree; cf. [K2]. We thus have that y(0) = 0. But from the above expression fro y we have (after simplification) that

y(0) = \frac{2ta_{5}}{t^{2} − 1}(α − β).

We thus have that α = β, which is equivalent to da_{1} = 0. Since d ≠ 0 by the definition of U_{2} and a_{1} ≠ 0 by hypothesis, it follows that no such y exists, and hence r_{F_{2}} is irreducible, as desired.

Thus, f_{C} is primitive and hence Gal_{f_{C}} ≃ S_{4} because Gal_{f_{C}} contains a transposition since we have points which are ramified of type (2, 1, 1).

Notation. Let U′ ⊂ U denote the subfamily determined by the condition that the monodromy group of covers f_{C} attached to isomorphism classes of triples (C, f_{C}, π_{C}) corresponding to points in U′(\overline{K}) is equal to S_{4} and that the ramification type is (2, 2)^{4}(2, 1, 1)^{4}.

Corollary 35 The family U′ is open in U_{2} and its K-rational elements coincide with the set of isomorphism classes of triples (C, f_{C}, π_{C}) satisfying the hypotheses of Theorem 34. Moreover, U′(K) is non-empty provided that char(K) > 5.

Proof. The fact that U′ is a subscheme of U_{2} follows from Corollary 32. Now if the class of (C, f_{C}, π_{C}) corresponds to a point in U′(K), then by Proposition 36 below we see that C must satisfy the hypotheses of Theorem 34. Since triples attached to these curves clearly define an open subscheme of U_{2}, it follows that U′ is open in U_{2}.

Since Example 27(b) satisfies the hypotheses of Theorem 34 whenever char(K) > 19 and ≠ 47, 191, 379, it follows that U′(K) ≠ ∅ in these cases. By considering other curves (cf. [K2]) one concludes that U′(K) ≠ ∅ whenever char(K) > 5.

Note that we do have imprimitive covers in U_{2}, as the following simple criterion shows.

Proposition 36 Let C ∈ U_{2}(K) be a curve defined by F_{2}(T, X) = 0 with parameters (r_{01}, r_{11}, r_{12}, t). If r_{11} = r_{12}, or if r_{11} = −tr_{12}, then Gal_{f_{C}} ≃ D_{4} or V. Moreover, the latter case occurs if and only if disc(F_{2}) is a square in K(T).

Proof. The condition r_{11} = r_{12} (respectively, r_{11} = −tr_{12}) means that a_{1} = 0 (respectively, a_{6} = 0), and so it follows that c_{3} = c_{4} = 0 in formula (32). Thus, G(T, X) = C_{1}(T)X^{2}, with C_{1}(T) = c_{2}(1 − T^{2}) + a_{6}T (1 − T), and so F_{2}(T, X) has the form of Lemma 30 (with c = C(T) + \frac{d}{q}C_{1}(T)). Thus, the result follows by the same argument as in the proof of Corollary 32.

4.2.5 The “Inverse” of the Donagi-Livné-Smith Construction

The basic strategy was already explained in Subsection 4.2.1:

Given a 4-cover f : C → ℙ_{K}^{1} with monodromy S_{4} and ramification type (2, 2)^{4}(2, 1, 1)^{4}, we want to construct a hyperelliptic genus 3 curve C′ with a degree 6 cover f_{6} : C′ → ℙ_{K}^{1} which factors over the hyperelliptic cover of C′ in such a way that the Galois hull f̃: C̃ → ℙ_{K}^{1} of f factors over f_{6}.

The following lemma makes this construction more explicit.

Lemma 37 Let f(X) ∈ k[X] be an irreducible quartic with Gal_{f} ≃ S_{4}, and let L/k be a splitting field of f. Then the field L also splits the Lagrange resolvent r_{f}(X). Let x ∈ L be a root of f and ξ ∈ L a root of r_{f}. Then:

(a) Gal(L/k(ξ)) is a 2-Sylow subgroup of G = Gal_{f}.

(b) [L : k(x, ξ)] = 2.

(c) There is an element y ∈ k(x, ξ) such that y^{2} = ξ^{2} − 4ae, where a and e are the coefficients of f as in (36), and then [k(ξ, y) : k(ξ)] = 2.

Proof. If f(X) = a(X − x_{1})(X − x_{2})(X − x_{3})(X − x_{4}) is the factorization of f(X) in L, then

r_{f}(X) = (X − ξ_{1})(X − ξ_{2})(X − ξ_{3}),

where ξ_{1} = a(x_{1}x_{2} + x_{3}x_{4}), ξ_{2} = a(x_{1}x_{3} + x_{2}x_{4}) and ξ_{3} = a(x_{1}x_{4} + x_{2}x_{3}); cf. [Hu], Lemma V.4.10 (together with Remark 28). Thus L also splits r_{f}(X).

To prove the rest of the assertions, we may assume (after renumbering, if necessary) that x = x_{1} and ξ = ξ_{1}. We identify an element σ ∈ S_{4} with the elements of Gal(L/k) via the natural relation σ(x_{i}) = x_{σ(i)}.

(a) Since r_{f} is irreducible, we have [k(ξ_{i}) : k] = 3 and so |Gal(L/k(ξ_{i}))| = [L : k(ξ)] = \frac{24}{3} = 8, which means that Gal(L/k(ξ)) is a 2-Sylow subgroup of Gal(L/k).

(b) We have Gal(L/k(x_{1})) = ⟨(34), (234)⟩ =: H and Gal(L/k(ξ_{1})) = ⟨(34), (1324)⟩ =: P_{2}. Thus, Gal(L/k(x_{1}, ξ_{1})) = H ∩ P_{2} = ⟨(34)⟩, and so [L : k(x_{1}, ξ_{1})] = |⟨(34)⟩| = 2.

(c) Let y_{1} = ax_{1}x_{2} and y_{2} = ax_{3}x_{4}. Since Gal(L/k(x_{1}, ξ_{1})) = ⟨(34)⟩ by part (b), we see that y_{1}, y_{2} ∈ k(x_{1}, ξ_{1}). Clearly y_{1} + y_{2} = ξ_{1} and y_{1}y_{2} = a^{2}x_{1}x_{2}x_{3}x_{4} = ae, so

(X − y_{1})(X − y_{2}) = X^{2} − ξ_{1}X + ae.

Put y = y_{1} − y_{2}. By the quadratic formula we have that y^{2} = ξ_{1}^{2} − 4ae ≠ 0. Since (1324)y_{1} = y_{2} and (1324)y_{2} = y_{1}, we see that (1324)y = −y, and so [k(ξ_{1}, y) : k(ξ_{1})] = 2, as desired.

By combining this lemma with what was said in Subsection 4.2.1, we obtain:

Proposition 38 Let F (T, X) ∈ K[T, X] be a polynomial with deg_{X}(F) = 4 such that the associated quartic cover f : C → ℙ_{K}^{1} has monodromy group S_{4} and ramification type (2, 2)^{4}(2, 1, 1)^{4}. Then the Lagrange resolvent r_{F} defines a degree 3 cover f_{3} : C_{0} → ℙ_{K}^{1}, and the curve C_{0} : r_{F} = 0 has genus 0. Furthermore, if (p(Z), q(Z)) is a parametrization of C_{0}, and if A(T) and E(T) are the leading and constant terms of F (T, X), respectively, then

C′ : Y^{2} = q(Z)^{2} − 4A(p(Z))E(p(Z))

defines a hyperelliptic curve of genus 3, and the Galois hull f̃: C̃ → ℙ_{K}^{1} of f factors over f_{3} ∘ f_{2}, where f_{2} : C′ → C_{0} ≃ ℙ_{K}^{1} is the hyperelliptic subcover of C′.

We now show how this works in an example.

Example 39 Let C = C_{21} be the curve defined by F_{21}(T, X) = 0, where F_{21} is as in Example 27(b). Thus, if char(K) > 19 and ≠ 47, 191, 379, then C is a smooth curve and f_{C} : C → ℙ_{K}^{1} is a cover of type (2, 2)^{4}(2, 1, 1)^{4}. Since here r_{11} = −1 ≠ 2 = r_{12}, we conclude from Theorem 34 that f_{C} has monodromy group S_{4}. We can thus apply the above proposition.

The Lagrange resolvent of F_{21} is

R(T, X) = X^{3} − (\frac{7}{3}T^{2} + \frac{9}{4}T − \frac{22}{3})X^{2} + (6T^{4} − 4T^{3} − \frac{25}{2}T^{2})X
+8T^{6} − \frac{158}{3}T^{5} + \frac{203}{3}T^{4} + \frac{869}{12}T^{3} − \frac{374}{3}T^{2}

By Proposition 38, R(T, X) = 0 defines a curve C_{0} of genus 0, MAPLE finds the following parametrization of C_{0}:

T = −484p_{1}(Z)/q(Z), and X = p_{2}(Z)/q(Z)^{2},

where

p_{1}(Z) = 162Z^{2} − 99Z − 20812
p_{2}(Z) = −\frac{11}{3}p_{1}(Z)(26244Z^{4} + 416988Z^{3} − 21434787Z^{2} − 298325016Z + 186702032)
q(Z) = 9(162Z^{3} − 99Z^{2} + 18392Z + 574992).

Note that since 484 = (2)^{2}(11)^{2}, 162 = (2)(3)^{4} and 26244 = (2)^{3}(3)^{8}, the indicated top coefficients of the numerator and denominator of T (Z) and X(Z) are non-zero in K. Following the recipe of Proposition 38, put h(Z) = X(Z)^{2} − 4A(T (Z))E(T (Z)). Here A(T) = T and E(T) = \frac{17}{4}T − 2T^{2}, and by MAPLE we find that h(Z) = h_{1}(Z)h_{2}(Z)^{2}, where

h_{1}(Z) = (162Z^{2} − 99Z − 20812)(162Z^{2} − 5643Z − 86636)
·(162Z^{2} + 3861Z + 6292)(162Z^{2} + 7029Z − 55660)
h_{2}(Z) = \frac{11(162Z^{2} − 99Z − 20812)}{243(162Z^{3} − 99Z^{2} + 18392Z + 574992)^{2}}

It thus follows from Proposition 38 that C′ : Y^{2} = h_{1}(Z) is the desired hyperelliptic curve of genus 3.

Note that since disc(h_{1}) = (2)^{152}(3)^{165}(5)(11)^{62}(13)(17)^{3}(19)(47)(191) ≠ 0, the roots of h_{1}(Z) are distinct over \overline{K}, and so C′ is indeed a curve of genus 3.

Remark 40 One might be tempted to use the curves C attached to elements in U_{2}(𝔽_{q}) to avoid Smith’s attack. In fact, let C′ be the curve obtained from C by the inverse of the Donagi-Livné-Smith construction. If there is only one 𝔽_{q}-rational possibility for a correspondence η for C′ (with resulting curve C necessarily), then the Smith attack cannot be applied to C′ over 𝔽_{q}. Whether C′ satisfies this condition can be decided by looking at the Galois structure of the Weierstraß points of C′ (see [Sm]) and this structure can be rediscovered by the Galois structure of the ramification points of type (2, 2) of f_{C}. For instance a “good” case is that these points form one Galois orbit. (This should be the “generic” case).

But then necessarily [C, f_{C}, π_{C}] ∉ U_{2}(𝔽_{q}) and so over 𝔽_{q} we cannot use the simple equations defining this family to find C. (Recall that a random choice of C will be not in U′(𝔽_{q}).)

Of course, one could try to go to extensions 𝔽_{q^{d}} (d ≤ 4 will be enough), find [C, f_{C}, π_{C}] ∈ U_{2}(𝔽_{q^{d}}) with the extra condition that C is defined over 𝔽_{q} and then try Galois descent to find a 4-cover of C over 𝔽_{q}. But it is very doubtful that this strategy will lead to a computationally effective algorithm.

The image of U in M_{3}^{h}. We now consider the map

ϕ := (μ_{3})_{|U} : U = U_{1} ∪ U_{2} → M_{3}^{h}

which is defined by the rule ϕ([C, f_{C}, π_{C}]) = isomorphism class of C. On the component U_{1} of U this map cannot be generically finite because all fibres are infinite. However, we do have the following result:

Proposition 41 The restriction of ϕ to the subscheme U′ is quasi-finite, and hence the restriction of μ_{3} to U_{2} is generically finite. Thus, the image of U_{2} in M_{3}^{h} is a 4-dimensional variety.

Proof. To verify this, let H_{3,6}^{h}(S_{4}) denote the Hurwitz space which classifies 6-covers f_{6} : C′ → ℙ_{K}^{1} with the property that C′ is a hyperelliptic curve, f_{6} factors over the hyperelliptic involution and that f_{6} has monodromy group S_{4} (with a special ramification structure as was explained in [FK], Theorem 3). Then the construction of Proposition 38 induces a (set) map from U′(\overline{K}) to H_{3,6}^{h}(S_{4})(\overline{K}) with finite fibres.

Moreover, the existence of a correspondence between C and C′ which induces a (polarized) isogeny of degree 8 between the Jacobians J_{C} and J_{C′} implies (by Torelli) that the isomorphism class of C produces only finitely many isomorphism classes of curves C′.

Now since the forget map H_{3,6}(S_{4}) → M_{3}^{h} is quasi-finite (because for a given C′ there are, up to equivalence, only finitely many trigonal maps f_{3} : ℙ_{K}^{1} → ℙ_{K}^{1} such that f_{6} = f_{3} ∘ π_{C′} satisfies the required ramification conditions), it follows that the forget map ϕ : U′ → M_{3}^{h} is also quasi-finite.

Corollary 42 The restriction of \overline{μ}_{3} to V_{2} is generically finite.

Finally, we are ready to present at least a sketch of the proof of the main result announced in Subsection 1.2.1.

Proof of Theorem 2 (sketch). We have that \overline{H}_{3,4,4}(S_{4}) ⊂ \overline{H}_{3,4,4} = V_{1} ∪ V_{2}, the latter by Theorem 20. By a similar argument as that of the proof of Theorem 31 we see that \overline{H}_{3,4,4}(S_{4}) ⊂ V_{2}; cf. [K2].

Next, by Theorem 34 and Corollary 35 we see that \overline{H}_{3,4,4}(S_{4}) contains the non-empty open subscheme π_{3}(U′) when char(K) > 5. Moreover, by an extension of the argument of Theorem 34 it follows that \overline{H}_{3,4,4}(S_{4}) is open in V_{2} (cf. [K2]), and so it follows from Theorem 20 that \overline{H}_{3,4,4}(S_{4}) is an irreducible, unirational variety of Dimension 4.

The assertions about \overline{μ}_{3} follow by a similar argument as that of the proof of Proposition 41.

Acknowledgment. We would like to thank the referees for their extremely careful reading of the paper, for their numerous useful comments which greatly improved the exposition of this paper, and for providing a long list of typos, some of which required considerable ingenuity and insight to find.

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