On cubic surfaces with a rational line

Andreas-Stephan Elsenhans^{∗} and Jörg Jahnel^{‡}

Abstract

We report on our project to construct non-singular cubic surfaces over ℚ with a rational line. Our method is to start with degree 4 Del Pezzo surfaces in diagonal form. For these, we develop an explicit version of Galois descent.

1 Introduction

1.1. –––– The configuration of the 27 lines upon a smooth cubic surface is highly symmetric. The group of all permutations respecting the canonical class as well as the intersection pairing is isomorphic to the Weyl group W (E_{6}) of order 51 840.

When S is a cubic surface over ℚ, the absolute Galois group Gal(\overline{ℚ} /ℚ) operates on the 27 lines. This yields a subgroup G ⊆ W (E_{6}). It is an open problem whether each of the 350 conjugacy classes of subgroups of W (E_{6}) may be realized by a cubic surface over ℚ.

Exactly 172 of the 350 conjugacy classes fix a line. We constructed examples of cubic surfaces over ℚ realizing each of these subgroups. The goal of this note is to report on our investigations.

1.2. Remark. –––– The analogous question for Del Pezzo surfaces of degree 4 is somewhat easier as it leads to subgroups of W (D_{5}). B. È. Kunyavskij, A. N. Skorobogatov, and M. A. Tsfasman [KST] showed that every subgroup of W (D_{5}) may be realized by a surface defined over ℚ.

∗Mathematisches Institut, Universität Bayreuth, Univ’straße 30, D-95440 Bayreuth, Germany, Stephan.Elsenhans@uni-bayreuth.de, Website: http://www.staff.uni-bayreuth.de/∼btm216

‡Département Mathematik, Universität Siegen, Walter-Flex-Str. 3, D-57068 Siegen, Germany, ~jjahnel@mathematik.uni-siegen.de, Website: http://wwwuser.gwdg.de/~jjahnel

∗The first author was supported in part by the Deutsche Forschungsgemeinschaft (DFG) through a funded research project.

Key words and phrases. Cubic surface, Degree 4 Del Pezzo surface, Explicit Galois descent

2 Constructions

2.1. –––– Cubic surfaces with a rational line are closely related to Del Pezzo surfaces of degree 4. Indeed, blowing down the line leads to a degree 4 Del Pezzo surface having a rational point. On the other hand, blowing up a rational point on a degree 4 Del Pezzo surface yields a cubic surface with a rational line. These two constructions may easily be made explicit.

2.2. Constructions (Cubic surfaces versus Del Pezzo surfaces of degree 4). —– Let a base field K be fixed once and for all.

i) For two linear forms l_{0}, l_{1}, suppose that the line l_{0} = l_{1} = 0 is contained in the cubic surface S given by F (x_{0}, …, x_{3}) = 0. Then, F may be written as F = l_{0}q_{0} + l_{1}q_{1} for quadratic forms q_{0} and q_{1}. The corresponding degree 4 Del Pezzo surface V is given by q_{0} + l_{1}x_{4} = q_{1} − l_{0}x_{4} = 0. ii) On the other hand, let a Del Pezzo surface V of degree 4 be given by Q_{0}(x_{0}, …, x_{4}) = Q_{1}(x_{0}, …, x_{4}) = 0. If (0 : 0 : 0 : 0 : 1) ∈ V then Q_{0} and Q_{1} may be written as Q_{0} = q_{0} + l_{0}x_{4} and Q_{1} = q_{1} + l_{1}x_{4} for q_{0}, q_{1} quadratic forms and l_{0}, l_{1} linear forms in x_{0}, …, x_{3}, only. The corresponding cubic surface S is given by q_{0}l_{1} − q_{1}l_{0} = 0.

2.3. Remarks. –––– a) These two constructions are inverse to each other.

b) One may start construction ii) as well with arbitrary generators of the pencil spanned by Q_{0} and Q_{1}.

2.4. Fact. –––– Let A be a symmetric matrix representing the quadratic form q_{0}|_{l_{0}=0}. If the eigenvalues of A are z_{1}, z_{2}, z_{3} then there is a symmetric matrix representing Q_{0} with eigenvalues (−1), 1, z_{1}, z_{2}, z_{3}.

2.5. Corollary. –––– i) In particular, Q_{0} is of rank < 5 if and only if q_{0}|_{l_{0}=0} is of rank <3. Hence, the five degenerate quadratic forms in the pencil [Q_{0}, Q_{1}] are in one-to-one correspondence with the five tritangent planes through the line considered. ii) If the eigenvalues of a symmetric matrix representing Q_{0} are 0, Z_{1}, …, Z_{4} then “l_{0} = 0” is a tritangent plane on S. The conic, defined by S on this plane, splits into two lines over the field K(\sqrt{Z_{1}Z_{2}Z_{3}Z_{4}}).

2.6. Example. –––– Consider the case that Q_{0} := a_{0}x_{0}^{2} + … + a_{4}x_{4}^{2} and Q_{1} := b_{0}x_{0}^{2} +…+b_{4}x_{4}^{2} are diagonal forms over the field K. Then, the five tritangent planes correspond to the points ((−b_{i}) : a_{i}) ∈ P^{1} as (−b_{i}Q_{0} + a_{i}Q_{1}) is degenerate. The conics split over the fields

K(\sqrt{∏_{j≠i}(−b_{i}a_{j} + a_{i}b_{j})})

for i = 0, …, 4. Observe that the product of the five radicands is a perfect square.

On the corresponding cubic surface, all 27 lines are defined over

L = K(\sqrt{∏_{j≠0}(−b_{0}a_{j} + a_{0}b_{j})}, …, \sqrt{∏_{j≠4}(−b_{4}a_{j} + a_{4}b_{j})}) .

Indeed, the subgroup of W (E_{6}) stabilizing a line is clearly of order 51840/27 = 1920. It is actually the semi-direct product T ⋊ S_{5}, where T ⊂ (ℤ/2ℤ)^{5} is the subgroup of order 16 formed by the elements having an even number of components equal to 1. As Gal(\overline{ℚ} /L) stabilizes not only the five tritangent planes but also the lines on them, it must act through the trivial subgroup of T ⋊ S_{5}.

2.7. Construction (Explicit Galois descent). —– Let A be a commutative étale algebra of degree 5 over ℚ and ι_{0}, …, ι_{4} : A → ℂ be the five embeddings.

i) For general a,b ∈A, the equations ι_{0}(a)x_{0}^{2}+⋯+ι_{4}(a)x_{4}^{2} = ι_{0}(b)x_{0}^{2}+⋯+ι_{4}(b)x_{4}^{2} = 0 define a Del Pezzo surface V of degree 4 over \overline{ℚ} . ii) Let l be a linear form in five variables with coefficients in A. Then, by symmetry, the quadratic forms ι_{0}(a)(l^{ι_{0}})^{2} +⋯+ι_{4}(a)(l^{ι_{4}})^{2} and ι_{0}(b)(l^{ι_{0}})^{2} +⋯+ι_{4}(b)(l^{ι_{4}})^{2} have rational coefficients. If l^{ι_{0}}, …, l^{ι_{4}} are linearly independent then we have a Del Pezzo surface V_{0} of degree 4 over ℚ such that its base change to \overline{ℚ} is isomorphic to V.

2.8. Remarks. –––– a) This construction is analogous to [EJ1, Theorem 6.1].

b) The five tritangent planes on V_{0} correspond to the points ((−ι_{i}(b)) : ι_{i}(a)) ∈ P^{1}. Hence, the Galois operation on them is the same as that on the embeddings ι_{i}.

c) When a ≠ 0, the conic on the tritangent plane corresponding to ((−ι_{i}(b)) : ι_{i}(a)) splits into two lines over the field

ℚ(ι_{i}(−b/a), \sqrt{∏_{j≠i}(−ι_{i}(b)ι_{j}(a) + ι_{i}(a)ι_{j}(b))}) .  (1)

The radicand may be rewritten as N(a) ι_{i}(a^{3} δ_{A/ℚ} (−b/a)), where δ_{A/ℚ} denotes the different of an element of A.

2.9. –––– Thus, given a subgroup G ⊆ T ⋊ S_{5}, there is the following strategy to construct a cubic surface S over ℚ such that Gal(\overline{ℚ} /ℚ) operates via G on the 27 lines.

Strategy. i) Find a number field K, normal over ℚ, such that Gal(K/ℚ) ≅ G. Identify the normal subextension K′ ⊆ K such that Gal(K′/ℚ) is the image G′ of G in S_{5} [MM]. ii) Find five elements r_{0}, …, r_{4} ∈ K′ with the properties below. r_{0}, …, r_{4} are permuted by G′ exactly via the embedding G′ ⊆ S_{5}. Further, the square roots ±\sqrt{r_{0}}, …, ±\sqrt{r_{4}} are elements of K and acted upon by G according to the embedding G ⊆ T ⋊S_{5}. Put p(T) := (T − r_{0}) · . . . · (T − r_{4}) and A := ℚ[T]/(p). This is a commutative étale algebra of degree 5 over ℚ with a distinguished element r := (T mod (p)). iii) Choose x ∈ A and put d := δ_{A/ℚ} (x). Set a := dr and b := −xa. iv) Execute Construction 2.7 for a, b ∈ A. On the Del Pezzo surface V_{0} found, search for a ℚ-rational point. If none is found then go back to step iii). Otherwise, determine the cubic surface S.

2.10. Remarks. –––– a) The properties required in ii) imply \sqrt{r_{0}} · . . . · \sqrt{r_{4}} ∈ ℚ. I.e., N(r) is a perfect square.

b) The construction yields N(a)a^{3} δ_{A/ℚ} (−b/a) = N(a)(d^{2}r)^{2}r. As the product of the five radicands in (1) is a square, the norm of a is a perfect square automatically.

3 Examples

3.1. –––– There are 172 conjugacy classes of subgroups of W (E_{6}) that fix a line. We constructed examples for each such group.

Actually, 81 of the 172 classes also stabilize a double-six and 49 of the 172 classes stabilize a pair of Steiner trihedra. 34 classes do both. Thus, examples for 96 of the 172 conjugacy classes had been constructed before [EJ1, EJ2]. The remaining 76 classes were of interest.

After naive trials and an extensive search through surfaces with small coefficients, only six of the 76 classes remained open. For these, we applied Strategy 2.9.

3.2. Remark. –––– In Strategy 2.9, we regularly run into reiteration, because there were no ℚ-rational points on the Del Pezzo surfaces of degree 4.

3.3. –––– The list containing our examples of cubic surfaces is available

on the second author’s web page at http://wwwuser.gwdg.de/~jjahnel/

Arbeiten/Kub Fl/list rat ger.txt. The numbering of the subgroups is that created by GAP, version 4.4.12.

3.4. Example. –––– As a conclusion, let us show how Strategy 2.9 works on a particular example. We consider the subgroup of number 107.

Abstractly, this is a group G of order 16. Its center is isomorphic to the Klein four-group. The operation on the 27 lines causes orbits of lengths 1, 2, 4, 4, and 16. On the two orbits of size four, G acts via two different quotients, both isomorphic to the dihedral group D_{4} of order eight. The operation on the five tritangent planes through the rational line is via a quotient G′ of order four. The orbits are of sizes 1, 2, and 2.

i) An example of a field with Galois group G is the composite K := K_{1}K_{2} of K_{1} := ℚ(\sqrt{3 ± \sqrt{3}}) and K_{2} := ℚ(\sqrt{−9 ± \sqrt{6}}) . Then, the subfield corresponding to G′ is K′ = ℚ(\sqrt{3}, \sqrt{6}) = ℚ(\sqrt{2}, \sqrt{3}). Observe that both fields K_{1} and K_{2} contain K′. Further, both are extensions of ℚ of type D_{4}. ii) Thus, we chose r_{0}, …, r_{4} to be 2, 3 ± \sqrt{3}, and −9 ± \sqrt{6}. This yields

p(T) = (T − 2)[(T − 3)^{2} − 3][(T + 9)^{2} − 6] .

iii) We worked with x := r = (T mod (p)). iv) The biggest coefficient of the resulting del Pezzo surface V_{0} is 524 391 211 895 464. An isomorphic surface is given by the equations

4x_{0}^{2} + 10x_{0}x_{1} + 20x_{0}x_{2} − 112x_{0}x_{3} − 134x_{0}x_{4} + 7x_{1}^{2} − 26x_{1}x_{2} − 134x_{1}x_{3}
− 148x_{1}x_{4} − 2x_{2}^{2} + 140x_{2}x_{3} − 2x_{2}x_{4} + 10x_{3}^{2} − 38x_{3}x_{4} − 323x_{4}^{2}
= 47x_{0}^{2} − 18x_{0}x_{1} + 10x_{0}x_{2} − 188x_{0}x_{3} − 178x_{0}x_{4} + 63x_{1}^{2} − 22x_{1}x_{2} + 376x_{1}x_{3}
− 86x_{1}x_{4} + 71x_{2}^{2} − 580x_{2}x_{3} + 146x_{2}x_{4} − 364x_{3}^{2} − 296x_{3}x_{4} − 21x_{4}^{2} = 0 .

Here, a point search in magma with an initial height limit of 100 shows 14 rational points. Blowing up (8 : −13 : 4 : 2 : −3) leads to a cubic surface with coefficients up to 3 838 320. Reembedding gives us the final result, the cubic surface V with the equation

2x^{2}y + 6x^{2}z − 4xy^{2} + 6xyz + 4xyw − 10xz^{2} − 4xzw − 7xw^{2} + 2y^{3} − 9y^{2}z − 4y^{2}w + 4yz^{2} − 26yzw + 6yw^{2} + z^{3} + 10z^{2}w − 7zw^{2} − 5w^{3} = 0 .

3.5. Remark. –––– The rational line on V connects (5 : 0 : 0 : −7) with (0 : 5 : 10 : 2).

3.6. Remark. –––– There are actually a few more particularities characterizing the subgroup of number 107.

a) First of all, the two D_{4} extensions K_{1} und K_{2} become cyclic over the same quadratic field ℚ(\sqrt{2}).

b) On the other hand, over ℚ(\sqrt{3}) and ℚ(\sqrt{6}), they are of Kleinian type. However, there is yet another oddity. While Gal(K_{1}/ℚ(\sqrt{3})) operates on the corresponding four lines via two disjoint two-cycles, Gal(K_{2}/ℚ(\sqrt{3})) acts on its orbit by double-transpositions. Over ℚ(\sqrt{3}) instead of ℚ(\sqrt{6}), the situation is vice versa. To realize such a behaviour, it was essential to choose r_{1} in ℚ(\sqrt{3}) fulfilling N(r_{1}) ∈ 6(ℚ^{∗})^{2} and r_{3} in ℚ(\sqrt{6}) such that N(r_{3}) ∈ 3(ℚ^{∗})^{2}.

References

[E] Elsenhans, A.-S.: Good models for cubic surfaces, Preprint

[EJ1] Elsenhans, A.-S. and Jahnel, J.: Cubic surfaces with a Galois invariant double-six, Central European Journal of Mathematics 8 (2010), 646-661 [EJ2] Elsenhans, A.-S. and Jahnel, J.: Cubic surfaces with a Galois invariant pair

of Steiner trihedra, to appear in: International Journal of Number Theory

[KST] Kunyavskij, B. È., Skorobogatov, A. N., and Tsfasman, M. A.: Del Pezzo surfaces of degree four, Mém. Soc. Math. France 37 (1989), 1–113

[MM] Malle, G. and Matzat, B. H.: Inverse Galois theory, Springer, Berlin 1999

[Ma] Manin, Yu. I.: Cubic forms, algebra, geometry, arithmetic, North-Holland Publishing Co. and American Elsevier Publishing Co., Amsterdam, London, and New York 1974