ON PLANE QUARTICS WITH A GALOIS INVARIANT CAYLEY OCTAD

ANDREAS-STEPHAN ELSENHANS AND JÖRG JAHNEL

Abstract. We describe a construction of plane quartics with prescribed Galois operation on the 28 bitangents, in the particular case of a Galois invariant Cayley octad. As an application, we solve the inverse Galois problem for degree two del Pezzo surfaces in the corresponding particular case.

1. Introduction

It is well known [Pl] that a nonsingular plane quartic curve C over an algebraically closed field of characteristic 2 has exactly 28 bitangents. The same is still true if the base field is only separably closed, as is easily deduced from [Va, Theorem 1.6]. If C is defined over a separably non-closed field k then the bitangents are defined over a finite extension field l of k, which is normal and separable, and permuted by the Galois group Gal l k .

By far not every permutation in S_{28} may occur. It is well-known (cf., e.g., [PSV]) that the subgroup G S_{28} of all admissible permutations is independent of the choice of C and isomorphic to Sp_{6} 𝔽_{2} . A natural question arising is thus the following.

Question. Given a field k and a subgroup g G, does there exist a nonsingular plane quartic C over k, for which the group homomorphism Gal k^{sep} k G S_{28}, given by the Galois operation on the 28 bitangents, has image g? This question depends only on the conjugacy class of the subgroup g G.

Remark 1.1. At least when k is a global field, the Galois operation on the 28 bitangents and, in particular, the subgroup of G being its image, form important invariants concerning the arithmetic of the curve C. For interesting applications in the closely related situations of Del Pezzo surfaces of low degree, we advise the reader to consult Yu. I. Manin’s book [Ma] or the article [Shi] of T. Shioda.

The group G Sp_{6} 𝔽_{2} is the simple group of order 1 451 520. It has 1369 conjugacy classes of subgroups. Among these, there are eight maximal subgroups, which are of indices 28, 36, 63, 120, 135, 315, 336, and 960, respectively.

Date: September 6, 2018.

2010 Mathematics Subject Classification. Primary 14H25; Secondary 14J20, 14J45, 11G35.

Key words and phrases. plane quartic, Cayley octad, degree two del Pezzo surface.

Acknowledgement. We wish to thank the anonymous referee, whose remarks helped us to significantly improve this article.

An example of a nonsingular plane quartic over ℚ such that Gal \overline{ℚ} ℚ operates on the bitangents via the full Sp_{6} 𝔽_{2} has been constructed by T. Shioda [Shi, Section 3] in 1993 and, almost at the same time, by R. Erné [Er, Corollary 3]. Moreover, there is an obvious approach to construct examples for the groups contained in the index 28 subgroup. Indeed, in this case, there is a rational bitangent. One may start with a cubic surface with the right Galois operation [EJ15], blow-up a rational point, and use the connection between degree two del Pezzo surfaces and plane quartics [Ko96, Theorem 3.3.5], cf. the application discussed below (and mainly in Section 3). For the group U_{63} of index 63 and its subgroups, we gave a complete answer in our previous work [EJ17].

In this article, we deal with the subgroup U_{36} G of index 36 and the groups contained within. More precisely, we show the following result, which answers a more refined question than the one asked above.

Theorem 1.2. Let an infinite field k of characteristic not 2, a normal and separable extension field l, and an injective group homomorphism

i: Gal l k U_{36}

be given. Then there exists a nonsingular quartic curve C over k such that l is the field of definition of the 28 bitangents and each σ Gal l k permutes the bitangents as described by i σ G S_{28}.

Among the 1369 conjugacy classes of subgroups of G Sp_{6} 𝔽_{2} , 296 are contained in U_{36}. However, 262 of these are also contained in U_{63}, such that they are covered by [EJ17], already, while 34 are new.

The subgroup U_{36} has a geometric meaning. The assumption that the image of the natural homomorphism i: Gal k^{sep} k G is contained in U_{36} expresses the fact that there exists a Galois invariant Cayley octad (cf. [Sen, §5], [PSV, §3], or [Do, Definition 6.3.4]) defining the quartic C. Moreover, one has an isomorphism of groups U_{36} S_{8} and U_{36} permutes the eight points forming the octad accordingly. We use Cayley octads for our proof of existence, which is completely constructive.

Remarks 1.3. i) Theorem 1.2 is clearly not true, in general, when k is a finite field. For example, there cannot be a nonsingular quartic curve over 𝔽_{3}, all whose bitangents are 𝔽_{3}-rational , simply because the projective plane contains only 13 𝔽_{3}-ratio-nal lines. ii) We ignore about characteristic 2 in this article, as this case happens to be very different. Even over an algebraically closed field, a plane quartic cannot have more than seven bitangents [SV, p. 60].

As an application, one may answer the analogous question for degree two del Pezzo surfaces. The double cover of P^{2}, ramified at a nonsingular quartic curve C is a del Pezzo surface of degree two. Here, considerations can be made that are very similar to the ones above. First of all, it is well known [Ma, Theorem 26.2.(iii)] that a del Pezzo surface S of degree two over an algebraically closed field contains exactly 56 exceptional curves, i.e. such of self-intersection number 1 . Again, the same is true when the base field is only separably closed. If S is defined over a separably non-closed field k then the exceptional curves are defined over a normal and separable finite extension field l of k and permuted by Gal l k . Once again, not every permutation in S_{56} may occur. The maximal subgroup G S_{56} that respects the intersection pairing is isomorphic to the Weyl group W E_{7} [Ma, Theorem 23.9].

Every bitangent of C is covered by exactly two of the exceptional curves of S. Thus, for the operation of Gal k^{sep} k on the 56 exceptional curves on S, there seem to be two independent conditions. On one hand, Gal k^{sep} k must operate via a subgroup of W E_{7} G S_{56} . On the other hand, the induced operation on the blocks of size two must take place via a subgroup of Sp_{6} 𝔽_{2} G S_{28}. In turns out, however, that there is an isomorphism W E_{7} Z ^{}Sp_{6} 𝔽_{2} , for Z W E_{7} the centre, that makes the two conditions equivalent. Cf. [EJ17, Corollary 2.17].

The group G W E_{7} already has 8074 conjugacy classes of subgroups. Two subgroups with the same image under the quotient map p: G ↠ G Z ^{}G correspond to del Pezzo surfaces of degree two that are quadratic twists of each other. Theorem 1.2 therefore extends word-by-word to del Pezzo surfaces of degree two and homomorphisms Gal l k G with image contained in p^{1} U_{36} .

Remark 1.4 (Conjugacy classes in G versus conjugacy classes in U_{36}). As noticed above, 296 of the conjugacy classes of subgroups of G Sp_{6} 𝔽_{2} are contained in S_{8} U_{36} G. On the other hand, one may readily check that S_{8} itself has precisely 296 conjugacy classes of subgroups. In other words, two subgroups U_{1}, U_{2} U_{36} that are conjugate in G must be conjugate in U_{36}, already.

This result came to us originally as an experimental finding, and as quite a surprise. As it is pure group theory and thus somewhat off the main topic of this article, we provide a proof in an appendix.

Convention 1.5. In this article, by a field, we mean a field of characteristic 2. For the convenience of the reader, the assumption on the characteristic will be repeated in the formulations of our final results, but not during the intermediate steps.

Computations. All computations are done using magma [BCP].

2. Cayley octads with prescribed Galois operation

All space quadrics over a field k form a P^{9}, the generic quadric being

T_{0} T_{1} T_{2} T_{3} ; ; ; \begin{matrix}a_{00} & a_{01} & a_{02} & a_{03} \\ a_{01} & a_{11} & a_{12} & a_{13} \\ a_{02} & a_{12} & a_{22} & a_{23} \\ a_{03} & a_{13} & a_{23} & a_{33} \end{matrix} ; ; ; ; ; ; \begin{matrix}T_{0} \\ T_{1} \\ T_{2} \\ T_{3} \end{matrix} ; ; ; 0 .

Among these, the singular quadrics are parametrised by the quartic Q_{s} P^{9}, given by D_{s} 0, for

D_{s} : det ; ; ; \begin{matrix}a_{00} & a_{01} & a_{02} & a_{03} \\ a_{01} & a_{11} & a_{12} & a_{13} \\ a_{02} & a_{12} & a_{22} & a_{23} \\ a_{03} & a_{13} & a_{23} & a_{33} \end{matrix} ; ; ; .

Lemma 2.1. The quartic Q_{s} is singular, the singular points corresponding exactly to the quadrics of rank 2. Proof. This may be tested after base extension to the algebraic closure. Thus, let us suppose that k \overline{k}. The quartic Q_{s} parametrises all singular quadrics, i.e. those of rank 3. If the rank of a quadric q is exactly three then, after a linear transformation of coordinates, we may assume that q T_{0}^{2} T_{1}^{2} T_{2}^{2}. At the point where a_{00} a_{11} a_{22} 1 and all other coordinates vanish, the gradient of D_{s} is a_{33}. Thus, q corresponds to a regular point.

On the other hand, if the rank of q is 2 then one may assume without restriction that q T_{0}^{2} cT_{1}^{2}, for c 1 or 0. At the point where a_{00} 1, a_{11} c, and all other coordinates vanish, the gradient of D_{s} is zero. The point is singular. □

Let now q_{1}, q_{2}, and q_{3} be three linearly independent quadratic forms, defined by the symmetric matrices M_{1}, M_{2}, and M_{3}. These span a net

Λ : Z u_{1}q_{1} u_{2}q_{2} u_{3}q_{3} _{u_{1}:u_{2}:u_{3} P^{2}}

of quadrics, which contains its singular members at the locus

C_{Λ} : Z det u_{1}M_{1} u_{2}M_{2} u_{3}M_{3} P^{2} .

This is a plane quartic.

Lemma 2.2. Assume that the base locus of the net Λ consists of eight distinct points, no four of which are coplanar. Then the quartic C_{Λ} is nonsingular. Proof. Again, this may be tested after base change to the algebraic closure. The quartic C_{Λ} is then the intersection of Q_{s} with a two-dimensional linear subspace E P^{9}. Thus, there are two ways, in which C_{Λ} may become singular. Either, the linear subspace intersects the singular locus of Q_{s} or it meets Q_{s} somewhere tangentially.

By Lemma 2.1, the first option would mean that Λ q_{1}, q_{2}, q_{3} contains a quadric of rank 2. Such a quadric, however, splits into two linear forms, so the base locus of Λ, given in P^{3} by q_{1} q_{2} q_{3} 0, is necessarily contained in the union of two planes. This contradicts our assumptions.

Thus, let us assume that E meets Q_{s} somewhere tangentially. Without restriction, the point of tangency corresponds to the space quadric q T_{0}^{2} T_{1}^{2} T_{2}^{2}. The tangent hyperplane at the corresponding point on Q_{s} P^{9} is given by a_{33} 0, so that our assumption means that Λ q, q_{1}, q_{2} and both q_{1} and q_{2} have coefficient zero at T_{3}^{2}. But then the base locus of Λ contains 0 : 0 : 0 : 1 at least as a double point, which is a contradiction, too. □

Definition 2.3. A set X P^{3} of eight points is called a Cayley octad if it is the base locus of a unique net of quadrics. In this case, we write Λ_{X} for the net of quadrics through X.

Remarks 2.4. i) Observe that eight points in general position in P^{3} define only a pencil of quadrics. Thus, Cayley octads are point sets in non-general position, although three general quadrics have eight points of intersection. ii) Let k be any field and X P_{k^{sep}}^{3} be a Cayley octad that is invariant under Gal k^{sep} k . Then the net Λ_{X} is spanned by k-rational quadrics, so that C_{Λ_{X}} is a plane quartic defined over k.

Proposition 2.5. Let k be any field and X P_{k^{sep}}^{3} be a Cayley octad that is invariant under Gal k^{sep} k . Assume that the quartic C_{Λ_{X}} is nonsingular. Then the operation of Gal k^{sep} k on the 28 bitangents of C_{Λ_{X}} coincides with that on pairs of the eight points of X. In particular, if each of the eight points of X is defined over an extension field l of k then the 28 bitangents of C are defined over l. Proof. As is classically known (cf. [Sen, Proposition 5.4] or [PSV, Proposition 3.3]), each pair of points of X defines a bitangent of C_{Λ_{X}}, which may be rationally computed from the pair. The assertion immediately follows from this. □

Proposition 2.6. Let k be a field and f k T a separable polynomial of degree 8, whose coefficient at T^{7} vanishes. Denote by l_{0} : k T f the étale algebra defined by f, and let α : T mod f be its natural generator. Furthermore, write l for the splitting field of f, and let σ_{1}, …, σ_{8} : l_{0} l be the eight homomorphisms, which are given by the roots of f. The operation of Gal l k on the homomorphisms then yields an injection ι: Gal l k S_{8}.

a) Then the eight points 1 : σ_{i} α : σ_{i} α ^{2} : σ_{i} α ^{4} P^{3} l , for i 1, …, 8, form a Cayley octad X.

b) Moreover, X is Gal l k -invariant, the eight points being acted upon by Gal l k as described by ι.

c) Suppose that, for each subset i_{1}, …, i_{4} 1, …, 8 of size four,

_{j 1}^{4} σ_{i_{j}} α 0 .

Then the quartic C_{Λ_{X}} associated with X is nonsingular.

Proof. a) The 2-uple embedding maps 1:σ_{i} α :σ_{i} α ^{2} :σ_{i} α ^{4} to

1:σ_{i} α ^{2} :σ_{i} α ^{4} :σ_{i} α ^{8} :σ_{i} α :σ_{i} α ^{2} :σ_{i} α ^{4} :σ_{i} α ^{3} :σ_{i} α ^{5} :σ_{i} α ^{6} . (1)

We first claim that the 8 10-matrix formed by the eight rows of the form (1) is of rank 7. For this, we may reorder the columns, thereby omitting the repeated ones.

The result is the 8 8-matrix

; ; ; ; ; ; ; ; ; ; \begin{matrix}1 & σ_{1} α & σ_{1} α ^{2} & σ_{1} α ^{3} & σ_{1} α ^{4} & σ_{1} α ^{5} & σ_{1} α ^{6} & σ_{1} α ^{8} \\ 1 & σ_{2} α & σ_{2} α ^{2} & σ_{2} α ^{3} & σ_{2} α ^{4} & σ_{2} α ^{5} & σ_{2} α ^{6} & σ_{2} α ^{8} \\ 1 & σ_{3} α & σ_{3} α ^{2} & σ_{3} α ^{3} & σ_{3} α ^{4} & σ_{3} α ^{5} & σ_{3} α ^{6} & σ_{3} α ^{8} \\ 1 & σ_{4} α & σ_{4} α ^{2} & σ_{4} α ^{3} & σ_{4} α ^{4} & σ_{4} α ^{5} & σ_{4} α ^{6} & σ_{4} α ^{8} \\ 1 & σ_{5} α & σ_{5} α ^{2} & σ_{5} α ^{3} & σ_{5} α ^{4} & σ_{5} α ^{5} & σ_{5} α ^{6} & σ_{5} α ^{8} \\ 1 & σ_{6} α & σ_{6} α ^{2} & σ_{6} α ^{3} & σ_{6} α ^{4} & σ_{6} α ^{5} & σ_{6} α ^{6} & σ_{6} α ^{8} \\ 1 & σ_{7} α & σ_{7} α ^{2} & σ_{7} α ^{3} & σ_{7} α ^{4} & σ_{7} α ^{5} & σ_{7} α ^{6} & σ_{7} α ^{8} \\ 1 & σ_{8} α & σ_{8} α ^{2} & σ_{8} α ^{3} & σ_{8} α ^{4} & σ_{8} α ^{5} & σ_{8} α ^{6} & σ_{8} α ^{8} \end{matrix} ; ; ; ; ; ; ; ; ; ; ,

which is clearly of rank 7, as the rightmost column is a linear combination of the other seven. On the other hand, the upper left 7 7-minor is Vandermonde and thus of value

_{1 i j 7} σ_{i} α σ_{j} α 0 ,

which implies our claim. Consequently, the quadrics through X indeed form a net Λ_{X}.

In order to prove that X is indeed a Cayley octad, it now suffices to verify that the base locus of Λ_{X} is zero dimensional. In fact, in this case, according to Bezout, it cannot consist of more than the eight points given. To show zero dimensionality, we first observe that the net Λ_{X} contains the pencil L spanned by Z q_{1} and Z q_{2} , for q_{1} : T_{1}^{2} T_{0}T_{2} and q_{2} : T_{2}^{2} T_{0}T_{3}. The pencil L contains the nonsingular quadric Q : Z q_{1} q_{2} and has

C: T_{1}^{2} T_{0}T_{2} T_{2}^{2} T_{0}T_{3} 0

as its base locus, which is an irreducible curve. Indeed, on the affine chart “T_{0} 1”, one has the system of equations T_{1}^{2} T_{2} and T_{2}^{2} T_{3} that yields the rational parametrisation P^{1} C, s:t s^{4} : s^{3}t : s^{2}t^{2} : t^{4} , for C. Consequently, the base locus of Λ_{X} could be of positive dimension only if the third generator of Λ_{X} contained the curve C entirely.

However, left exactness of the global section functor, applied to the standard exact sequences

0 𝒪_{P}^{3} ^{q_{1} q_{2}} 𝒪_{P}^{3} 2 𝒪_{Q} 2 0

and

0 𝒪_{Q} ^{q_{1}} 𝒪_{Q} 2 𝒪_{C} 2 0

shows that no quadratic form vanishes on C, except for those linearly spanned by q_{1} and q_{2}.

b) This assertion is clear from the construction of X.

c) Since the eight points given are clearly distinct, all that is left to verify for C_{Λ_{X}} being nonsingular is that no four of the points are coplanar. But this is clear, too.

Indeed, the determinants

det ; ; ; \begin{matrix}1 & σ_{i_{1}} α & σ_{i_{1}} α ^{2} & σ_{i_{1}} α ^{4} \\ 1 & σ_{i_{2}} α & σ_{i_{2}} α ^{2} & σ_{i_{2}} α ^{4} \\ 1 & σ_{i_{3}} α & σ_{i_{3}} α ^{2} & σ_{i_{3}} α ^{4} \\ 1 & σ_{i_{4}} α & σ_{i_{4}} α ^{2} & σ_{i_{4}} α ^{4} \end{matrix} ; ; ; ,

for 1 i_{1} … i_{4} 8, are easily calculated to be

σ_{i_{1}} α σ_{i_{2}} α σ_{i_{3}} α σ_{i_{4}} α _{1 j_{1} j_{2} 4} σ_{j_{1}} α σ_{j_{2}} α ,

an expression, in which the factor on the left is nonzero by assumption while that on the right is nonzero by construction. □

Remark 2.7. The more obvious choice of the eight points 1 : σ_{i} α : σ_{i} α ^{2} : σ_{i} α ^{3} does not lead to a Cayley octad. There is actually a net of quadrics through these eight points, spanned by Z T_{1}^{2} T_{0}T_{2} , Z T_{2}^{2} T_{1}T_{3} , and Z T_{1}T_{2} T_{0}T_{3} , but the base locus of this net is the twisted cubic curve.

We may thus conclude the following result.

Theorem 2.8. Let an infinite field k of characteristic not 2, a normal and separable extension field l, and an injective group homomorphism

i: Gal l k U_{36} G Sp_{6} 𝔽_{2} S_{28}

be given. Then there exists a nonsingular quartic curve C over k such that l is the field of definition of the 28 bitangents and each σ Gal l k permutes the bitangents as described by i σ G S_{28}. Proof. The natural operation of S_{8} on the 28 \frac{8 7}{2} 2-sets yields an injective group homomorphism S_{8} S_{28}, the image of which is exactly U_{36}. Thus, the homomorphism i given induces an injection ı^{}: Gal l k S_{8}.

This immediately yields an étale algebra l_{0} of degree eight. In the case of a transitive subgroup, l_{0} is the field corresponding under the Galois correspondence to the point stabiliser. In general, one has to take the direct product of the fields, corresponding to the point stabilisers of the orbits.

Moreover, by the primitive element theorem, every element of l_{0} is a generator, except for the union of finitely many k-linear subspaces of lower dimension. The same is still true for the trace zero subspace

h : α l_{0} Tr_{l_{0} k} α 0 l_{0} .

Indeed, with α l_{0}, the element α^{}: α \frac{1}{8} Tr_{l_{0} k} α of trace zero is a generator, too.

Furthermore, the eight homomorphisms σ_{i}: l_{0} l are k-linearly independent. Hence, the subset

h_{c} : α h _{j 1}^{4} σ_{i_{j}} α 0 for some 1 i_{1} … i_{4} 8 h ,

too, is a union of finitely many lower-dimensional k-linear subspaces.

Consequently, as k is infinite, there exists some element α h h_{c} that is a generator of the étale algebra l_{0}. We take f k T to be the minimal polynomial of α. Then, one has l_{0} k T f and the generator T mod f is mapped to α under this isomorphism. The coefficient of f at T^{7} is zero, because of Tr_{l_{0} k} α 0. Moreover, f is separable, as, for α a generator, σ_{1} α , …, σ_{8} α are automatically distinct. Thus, all the assumptions of Proposition 2.6 are fulfilled.

Therefore, we are given a Galois invariant Cayley octad X delivering a nonsingular plane quartic C_{Λ_{X}}. Proposition 2.5 shows, finally, that the Galois operation on the bitangents of C_{Λ_{X}} is exactly the one desired. □

Corollary 2.9 (The case that k is a number field). Let k be a number field and g a subgroup of G Sp_{6} 𝔽_{2} that is contained in U_{36}. Then there exists a nonsingular quartic curve C over k such that the natural permutation representation

i: Gal \overline{k} k G S_{28}

on the 28 bitangents of C has the subgroup g as its image. Proof. According to Theorem 2.8, it suffices to show that, for every number field k and each subgroup g U_{36}, there exists a normal extension field l such that Gal l k is isomorphic to g. This is a particular instance of the inverse Galois problem for number fields and thus potentially hard. But, fortunately, the groups occurring are easy enough.

In fact, 296 of the 1369 conjugacy classes of subgroups of G are contained in U_{36}. These are exactly the conjugacy classes of subgroups of U_{36} S_{8}. It does not happen that two subgroups of U_{36} not being conjugate to each other become conjugate in G. Cf. Corollary A.4.

Among these 296 conjugacy classes, 268 consist of solvable groups. In this case, the inverse Galois problem has been solved by I. R. Shafarevich [Sha], cf. [NSW, Theorem 9.5.1]. The groups in the remaining 28 conjugacy classes turn out to be i) isomorphic to a symmetric group S_{n} or alternating group A_{n}, for n 5, 6, 7, or 8, in 13 cases, and ii) isomorphic to a direct product S_{m} S_{n}, S_{m} A_{n}, or A_{m} A_{n}, in ten further cases. iii) The remaining five cases are the simple group of order 168, occurring transitively and intransitively, the transitive subgroups of the types PGL_{2} 𝔽_{7} and AGL_{3} 𝔽_{2} , having the orders 336 and 1344, respectively, and the subdirect product of S_{5} and S_{3}, of order 360. The groups of types i) and ii) are classically known to occur as Galois groups over an arbitrary number field k [MM, Theorem 9.4 and 9.5]. Moreover, the first three groups of type iii) are realised over ℚ t , still over \overline{ℚ} t , and consequently over k t , too, by the polynomials f_{7,3}, f_{8,43}, and f_{8,48} from the tables in [MM, Appendix]. Finally, for the subdirect product, the polynomial

T^{3} 3 3t 1 T 2 3t 1 T^{5} 5 5t 1 T 4 5t 1 ℚ t T

enjoys the same property. According to the Hilbert irreducibility theorem [Se, Section 9.2, Proposition 2], in each case, there are infinitely many specialisations of t to an element of k, such that the resulting polynomial over k has the same Galois group.

□

Example 2.10. For the polynomial

f T : T^{8} 42T^{4} 168T^{2} 1152T 1197 ℚ T ,

we find the nonsingular plane quartic over ℚ, given by

2T_{0}^{3}T_{2} 5T_{0}^{2}T_{2}^{2} 3T_{0}T_{1}^{3} 6T_{0}T_{1}^{2}T_{2} 24T_{0}T_{1}T_{2}^{2} 8T_{0}T_{2}^{3} 6T_{1}^{3}T_{2} 12T_{1}^{2}T_{2}^{2}

24T_{1}T_{2}^{3} 16T_{2}^{4} 0 .

Here, the Galois group of f is the simple group of order 168, realised as a transitive subgroup of S_{8}. This group is in fact doubly transitive, so that the operation on the 28 bitangents is transitive, too.

Example 2.11. Over the function field 𝔽_{3} t , the polynomial

f T : T^{8} 2tT^{6} 2t^{2}T^{5} t^{3} 2t^{2} t 2 T^{4} 2t^{3}T^{3} 2t^{3} t 2 T^{2}

t^{5} t^{4} t^{3} 2t^{2} T t^{6} t^{4} 2t^{3} t^{2} 1  𝔽_{3} t T

provides the same Galois group. We obtain the nonsingular plane quartic over 𝔽_{3} t , given by

t^{13} t^{12} 2t^{10} t^{8} 2t^{6} 2t^{4} t^{3} t^{2} t 1 T_{0}^{4}
2t^{9} 2t^{8} t^{7} t^{6} 2t^{5} t^{4} 2t 1 T_{0}^{3}T_{1}
t^{10} t^{9} t^{8} 2t^{7} t^{6} 2t^{3} t^{2} 2t 1 T_{0}^{3}T_{2} t^{4} t^{3} 2t 1 T_{0}^{2}T_{1}^{2}
t^{6} t^{5} 2t^{4} 2t^{3} t 1 T_{0}^{2}T_{1}T_{2} t^{7} 2t^{4} t^{3} t^{2} 2t 2 T_{0}^{2}T_{2}^{2} T_{0}T_{1}^{3}
tT_{0}T_{1}^{2}T_{2} t^{3} 2t^{2} t 2 T_{0}T_{1}T_{2}^{2} 2t^{3} t 2 T_{0}T_{2}^{3} 2T_{1}T_{2}^{3} 0 .

Example 2.12. For the polynomial

f T : T^{8} 5T^{6} T^{5} 7T^{4} T^{3} 4T^{2} 1 ℚ T ,

the nonsingular plane quartic over ℚ, given by

T_{0}^{4} 2T_{0}^{3}T_{1} 5T_{0}^{3}T_{2} 6T_{0}^{2}T_{1}T_{2} 7T_{0}^{2}T_{2}^{2} 8T_{0}T_{1}^{2}T_{2} 6T_{0}T_{1}T_{2}^{2} 5T_{0}T_{2}^{3}

8T_{1}^{3}T_{2} T_{1}^{2}T_{2}^{2} 10T_{1}T_{2}^{3} 2T_{2}^{4} 0

results. Here, the Galois group of f is A_{8}. The operation on the 28 bitangents is transitive.

Example 2.13. Over the function field 𝔽_{3} t , the polynomial

f T : T^{8} tT^{5} 2tT^{2} 1 𝔽_{3} t T

has Galois group A_{8}. We find the nonsingular plane quartic over 𝔽_{3} t , given by

t^{2} t T_{0}^{2}T_{1}^{2} T_{0}T_{1}^{3} t^{2} t T_{0}^{3}T_{2} T_{0}^{2}T_{1}T_{2} 2tT_{0}T_{2}^{3} 2T_{1}T_{2}^{3} 0 .

Remark 2.14. The examples above were chosen from the huge supply in the hope that they are of some particular interest. For instance, in both cases the groups occurring as Galois groups are simple. The corresponding degree two del Pezzo surfaces are, independently of the twist chosen, all of Picard rank 1.

Remark 2.15 (The situation of an algebraically closed base field—Moduli schemes). There is a coarse moduli scheme of nonsingular quartic curves, provided by Geometric Invariant Theory. It is nothing but the quotient ℳ : 𝒱 PGL_{3}, for 𝒱 P Sym^{4} k^{3} ^{}P^{14} the open subscheme parametrising nonsingular plane quartics. Observe here that, by [Mu, Remark 1.13], every nonsingular plane quartic corresponds to a PGL_{3}-stable point on P Sym^{4} k^{3} ^{}.

The PGL_{3}-invariants have been determined by J. Dixmier [Di] and T. Ohno, Ohno in his unpublished paper [Oh] proving completeness. There is a more recent treatment due to A.-S. Elsenhans [El], who also provided an implementation into magma. The Dixmier–Ohno invariants yield an embedding

𝒱 PGL_{3} ℳ P 1, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 9

into a weighted projective space.

On the other hand, there is the moduli space ℳ^{ev} of projective equivalence classes of Cayley octads, cf. [Do, Corollary 6.3.12]. As is classically known, every nonsingular plane quartic may be obtained from a Cayley octad in 36 non-equivalent ways [He, §14]. Accordingly, ℳ^{ev} naturally provides a 36:1 étale covering

π: ℳ^{ev} ℳ

of the moduli scheme ℳ of nonsingular plane quartics [Do, Theorem 6.3.19].

Our construction, from the geometric point of view, describes a particular kind of Cayley octads, namely those of the type

1:x_{i}:x_{i}^{2} :x_{i}^{4} P^{3} k i 1, …, 8 ,

for x_{1}, …, x_{8} k a subset consisting of eight elements such that x_{1} x_{8} 0. It therefore yields a rational map

ι: A^{7} −−→ ℳ^{ev} .

Moreover, a simple experiment in magma reveals the fact that ι is dominant. Indeed, we verified dominance of the composition π ι: A^{7} −−→ ℳ by showing that the tangent map T π_{x}: T_{x}A^{7} T_{ι x}ℳ at a single point x A^{7} is surjective. Here, we used the first author’s code [El] for calculating the Dixmier–Ohno invariants of a ternary quartic form.

As a consequence, we find that both, ℳ^{ev} and ℳ are unirational varieties. This result, however, is not new. In fact, ℳ^{ev} and ℳ are even known to be rational. For ℳ, this is due to P. Katsylo [Ka, Theorem 0.1], while the rationality of ℳ^{ev} follows from this in view of [Do, Theorem 6.3.19].

Anyway, our approach to construct quartic curves with an arbitrarily given Galois operation on the 28 bitangents, having the image of Galois contained in U_{36} G Sp_{6} 𝔽_{2} , seemed to run quite smoothly. It is our distinct impression that the rationality of the moduli scheme ℳ^{ev} is the proper reason for this.

3. Application to del Pezzo surfaces of degree two–Twisting

There is the double cover p: G G of finite groups, for W E_{7} G S_{56} and Sp_{6} 𝔽_{2} G S_{28}, which is given by the operation on the size two blocks. The kernel of p is exactly the centre Z G . For a subgroup H G , one therefore has two options.

i) Either p _{H}: H p H is two-to-one. Then H p^{1} h , for h : p H . In this case, H contains the centre of G and, as abstract groups, one has an isomorphism H p H ℤ 2ℤ. ii) Or p _{H}: H p H is bijective. In our geometric setting, the first case is the generic one. More precisely, let C: q 0 be a nonsingular plane quartic such that the 28 bitangents are acted upon by the group h G. Then, for λ an indeterminate, the 56 exceptional curves on S_{λ}: λw^{2} q are operated upon by h ℤ 2ℤ. For particular choices of λ, every subgroup of G may be realised that has image h under the projection p.

Lemma 3.1. Let a field k of characteristic not 2, a normal and separable extension field l, and an injective group homomorphism i: Gal l k G be given. Write l^{} for the subfield corresponding to i^{1} Z under the Galois correspondence.

a) Then there is a commutative diagram

\xymatrix{Gal l k \ar@{^{(}->}[r]^{i} \ar@{->>}[d]_{res} & G \ar@{->>}[d]^{p} \\ Gal l^{}k \ar@{^{(}->}[r]^{\overline{ı}} & G ,}

the downward arrow on the left being the restriction.

b) Let C: q 0 be a nonsingular plane quartic over k, the 28 bitangents of which are defined over l^{}and acted upon by Gal l^{}k as described by \overline{ı}. Then there exists some λ k^{}such that the 56 exceptional curves of the degree two del Pezzo surface

S_{λ}: λw^{2} q

are defined over l and each automorphism σ Gal l k permutes them as described by i σ G S_{56} .

Proof. Cf. [EJ17, Theorem 4.2]. □

Thus, from our main result on plane quartics, we may draw the following conclusion.

Theorem 3.2. Let an infinite field k of characteristic not 2, a normal and separable extension field l, and an injective group homomorphism

i: Gal l k p^{1} U_{36}

be given. Then there exists a degree two del Pezzo surface S over k such that l is the field of definition of the 56 exceptional curves and each σ Gal l k permutes them as described by i σ G S_{56} . Proof. This follows from Lemma 3.1 together with Theorem 2.8. □

Remark 3.3 (2-torsion Brauer classes). There is a unique subgroup

H_{Br} p^{1} U_{36} G S_{56}

of index 2 that is intransitive of orbit type 28, 28 . Clearly, one has H_{Br} U_{36} S_{8}. The subgroup H_{Br} is of index 72 in G.

Assume that S is a degree two del Pezzo surface over a field k such that the Galois group operating on the 56 exceptional curves is a subgroup of H_{Br}. In this case, S carries a global Brauer class α Br S_{λ} _{2}. In fact, the 2-torsion Brauer classes on degree two del Pezzo surfaces have been systematically studied by P. Corn in [Co]. It turns out that the conjugacy classes of subgroups of W E_{7} that lead to such a Brauer class form a partially ordered set with exactly two maximal elements. The subgroup H_{Br} described above is one of them. It yields the Brauer classes of the second type in Corn’s terminology.

There can be no doubt that this type is more interesting than the other. A short experiment shows the following facts.

i) For 176 out of the 296 conjugacy classes H of subgroups of H_{Br}, the Brauer class α is nontrivial. ii) For 87 of these, the Brauer class α is non-cyclic. I.e., there is no normal subgroup H^{}H with cyclic quotient annihilating the class. Moreover, somewhat surprisingly, in each of these 87 cases, the restriction to the 2-Sylow subgroup of H is still non-cyclic. This shows that, in the situation that k is a number field and ν one of its places, the evaluation of such a Brauer class at a k_{ν}-rational point cannot be computed in the usual, class field theoretic way (cf., e.g., [EJ10, §6]).

Remark 3.4. Again, two subgroups U_{1}, U_{2} H_{Br} not being conjugate to each other do not become conjugate when considered as subgroups of G. Proof. Indeed, observe the commutative diagram below,

\xymatrix{H_{Br} \ar@{^{(}->}[r] \ar[d]_{p _{H_{Br}}}^{} & G \ar@{->>}[d]^{p} \\ U_{36} \ar@{^{(}->}[r] & G .}

From this, one sees that if U_{1}, U_{2} H_{Br} are conjugate in G then p U_{1} , p U_{2} U_{36} are conjugate in G. But then Corollary A.4 shows that p U_{1} , p U_{2} U_{36} are already conjugate in U_{36}. And, finally, since p _{H_{Br}} is an isomorphism, U_{1}, U_{2} must be conjugate to each other in H_{Br}. □

4. Another application: Cubic surfaces with a Galois invariant double-six

A nonsingular cubic surface over an algebraically closed field contains exactly 27 lines. The maximal subgroup G_{max} S_{27} that respects the intersection pairing is isomorphic to the Weyl group W E_{6} [Ma, Theorem 23.9] of order 51 840.

A double-six (cf. [Ha, Remark V.4.9.1] or [Do, Subsection 9.1.1]) is a configuration of twelve lines E_{1}, …, E_{6}, E_{1}^{}, …, E_{6}^{}such that

i) E_{1}, …, E_{6} are mutually skew, ii) E_{1}^{}, …, E_{6}^{}are mutually skew, and iii) E_{i} E_{j}^{}1, for i j, 1 i, j 6, and E_{i} E_{i}^{}0 for i 1, …, 6. Every cubic surface contains exactly 36 double-sixes, which are transitively acted upon by G_{max} [Do, Theorem 9.1.3]. Thus, there is an index-36 subgroup U_{ds} G_{max} stabilising a double-six. This is one of the maximal subgroups of G_{max} W E_{6} . Up to conjugation, W E_{6} has maximal subgroups of indices 2, 27, 36, 40, 40, and 45.

As an application of Theorem 2.8 and Lemma 3.1, we now have a simpler proof for the following result, which was shown in [EJ17, Theorem 5.1].

Theorem 4.1. Let an infinite field k of characteristic not 2, a normal and separable extension field l, and an injective group homomorphism

i: Gal l k U_{ds} G_{max} W E_{6} S_{27}

be given. Then there exists a nonsingular cubic surface S over k such that

i) the 27 lines on S are defined over l and each σ Gal l k permutes them as described by i σ U_{ds} S_{27}. ii) S is k-unirational.

Proof. There is an injective homomorphism ι: W E_{6} W E_{7} that corresponds to the blow-up of a point on a cubic surface. The subgroup ι U_{ds} W E_{7} is of index 36 56 2016. It is sufficient to show that the image \overline{ι} U_{ds} under the composition

\overline{ι}: W E_{6} ^{ι} W E_{7} ↠^{p} W E_{7} Z G

is contained in U_{36}. Indeed, then Theorem 3.2 yields a degree two del Pezzo surface S^{}of degree two with a k-rational line L and the proposed Galois operation on the 27 lines that do not meet L. Blowing down L, we obtain a nonsingular cubic surface S satisfying i). As there is a k-rational point on S, the blow down of L, S is k-unirational [Ko02, Theorem 2].

To show the group-theoretic claim, we first observe that the homomorphism \overline{ι} is still an injection. Furthermore, the subgroup \overline{ι} W E_{6} G of index 28 is exactly the point stabiliser of the transitive group G S_{28}. The subgroup U_{36} S_{28} is still transitive, as U_{36} S_{8}, operating on 2-sets, and S_{8} is doubly transitive. Thus, U_{36} \overline{ι} W E_{6} is of index 28 in U_{36}. Moreover, \overline{ι} yields an isomorphism

\overline{ι} _{\overline{ι}^{1} U_{36}}: \overline{ι}^{1} U_{36} ^{}U_{36} \overline{ι} W E_{6} .

In particular, \overline{ι}^{1} U_{36} is of the same order as U_{36} \overline{ι} W E_{6} and hence of index 36 in the group W E_{6} . Therefore, up to conjugation, \overline{ι}^{1} U_{36} U_{ds}, which implies

\overline{ι} U_{ds}  \overline{ι} \overline{ι}^{1} U_{36} U_{36}, as required.  □

Appendix A. A group-theoretic observation

Proposition A.1. Let G be a finite group and H G a subgroup. Consider the permutation representation

ι: G  Sym G H ,
g  aH gaH

on left cosets modulo H, and assume that, for all ω, ω^{}G H, ω^{}ω, the following two conditions are satisfied.

i) There is some g G such that ι g Sym G H interchanges ω and ω^{}. I.e., such that the permutation ι g contains the 2-cycle ωω^{}in its canonical decomposition into disjoint cycles. ii) The twofold point stabiliser G_{ω} _{ω^{}}is a group having only inner automorphisms. Then the following is true. If two subgroups U_{1}, U_{2} H are conjugate in G then they are conjugate already in H. Proof. Consider, at first, two elements ω, ω^{}G H such that ω^{}ω. Then, according to assumption i), there is some g G such that ι g ωω^{}. Conjugation by g then clearly provides an automorphism of G_{ω} _{ω^{}}. Moreover, by assumption ii), that must be inner. I.e., there exists some h G_{ω} _{ω^{}}satisfying hxh^{1} gxg^{1} for every x G_{ω} _{ω^{}}, and therefore xh^{1} g h^{1} gx. In other words, we found an element

c_{ω,ω^{}}: h^{1} g C_{G} G_{ω} _{ω^{}}

in the centraliser of G_{ω} _{ω^{}}that is non-trivial, because of i c_{ω,ω^{}}ωω^{}.

Now suppose that

uU_{1}u^{1} U_{2}  (2)

for some u G. We let ω : H, i.e. the trivial coset. Because of U_{1}, U_{2} H, ω is a fixed point of U_{1}, as well as of U_{2}. Moreover, assumption (2) shows that ω^{}: u^{1} ω must be a fixed point of U_{1}, too.

If ω^{}ω then u H and the assertion is trivially true. Otherwise, U_{1} G_{ω} _{ω^{}} and one has

U_{2} uU_{1}u^{1}
uc_{ω,ω^{}}U_{1}c_{ω,ω^{}}^{1}u^{1} ,

since c_{ω,ω^{}}is a central element. Here, uc_{ω,ω^{}}fixes ω, which shows that uc_{ω,ω^{}}H. □

Remarks A.2. i) The permutation representation ι is always transitive. Thus, in the assumptions i) and ii), it suffices to restrict considerations to the case that ω : H. ii) It is, of course, possible to find a purely group-theoretic formulation, avoiding permutations. For instance, the twofold point stabiliser of ω H and ω^{}xH, for x H, is H xHx^{1} .

Remark A.3. A relevant particular case occurs when ι doubly transitive. Indeed, in this case, assumption i) is automatically fulfilled. Furthermore, all twofold point stabilisers G_{ω} _{ω^{}}are isomorphic to each other.

Corollary A.4. If two subgroups U_{1}, U_{2} U_{36} are conjugate in G Sp_{6} 𝔽_{2} then they are conjugate already in U_{36}. Proof. In our case, G Sp_{6} 𝔽_{2} and H U_{36} S_{8}. As this is a maximal subgroup, the permutation representation ι is primitive of degree 36. It is wellknown that Sp_{6} 𝔽_{2} allows only one primitive permutation representation in this degree, and that this is doubly transitive [DM, Theorem 7.7.A, for n 3].

Moreover, G_{ω} S_{8} allows only one conjugacy class of subgroups of index 35, and that contains the imprimitive wreath product S_{4} ≀ S_{2}. Thus, all the twofold point stabilisers of ι are isomorphic to S_{4} ≀ S_{2}. Finally, this group allows only one faithful permutation representation in degree 8 and, hence, has only inner automorphisms. Thus, Proposition A.1 applies and yields the assertion. □

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Institut für Mathematik, Universität Würzburg, Emil-Fischer-Straße 30, D-97074 Würzburg, Germany

E-mail address: stephan.elsenhans@mathematik.uni-wuerzburg.de
URL: https://math.uni-paderborn.de/ag/ca/elsenhans/

Department Mathematik, Univ. Siegen, Walter-Flex-Str. 3, D-57068 Siegen, Germany

E-mail address: ~jjahnel@mathematik.uni-siegen.de
URL: http://wwwuser.gwdg.de/~jjahnel