ON THE ARITHMETIC OF THE DISCRIMINANT FOR CUBIC SURFACES

ANDREAS-STEPHAN ELSENHANS AND JÖRG JAHNEL

Abstract. The 27 lines on a smooth cubic surface over ℚ are acted upon by a finite quotient of Gal(\overline{ℚ} /ℚ). We construct explicit examples such that the operation is via the index two subgroup of the maximal possible group. This is the simple group of order 25 920. Our examples are given in pentahedral normal form with rational coefficients. On the corresponding parameter space, we search for rational points, discuss their asymptotic, and construct an accumulating subvariety.

1. Introduction

1.1. Let 𝒮 ⊂ P^{3} be a smooth cubic surface over an algebraically closed field. It is well known that there are exactly 27 lines on 𝒮. The intersection matrix of these lines is essentially the same for every smooth cubic surface. The group of all permutations of the 27 lines respecting the intersection matrix is isomorphic to the Weyl group W(E_{6}).

For a smooth cubic surface S ⊂ P^{3} over ℚ, the 27 lines are, in general, not defined over ℚ, but over an algebraic field extension L. The Galois group Gal(L/ℚ) is a subgroup of W(E_{6}).

1.2. In a previous article [4], we described a strategy how to find explicit examples of cubic surfaces over ℚ such that the Galois group Gal(L/ℚ) is exactly the index two subgroup D^{1}W(E_{6}) ⊂ W(E_{6}). This is the simple group of order 25 920.

Our approach was as follows. We considered cubic surfaces in pentahedral normal form with rational coefficients. For these, we studied the discriminant Δ. We showed that Gal(L/ℚ) is contained in the index two subgroup if and only if (−3)Δ is a perfect square.

1.3. This leads to a point search on the double covering of P^{4} ramified at the degree 32 discriminantal variety. A generalized Cremona transform reduces the degree to eight.

In the present article, we will discuss the asymptotic of the ℚ-rational points of bounded height on the resulting double covering and construct an accumulating subvariety. A final section is devoted to the problem to what extent this subvariety is unique.

1991 Mathematics Subject Classification. Primary 11G35, Secondary 14J20, 14J45, 11G50.

Key words and phrases. Cubic surface, Pentahedral normal form, Discriminant, Rational point, Manin’s conjecture, Accumulating subvariety.

The computer part of this work was executed on the Sun Fire V20z Servers of the Gauß Laboratory for Scientific Computing at the Göttingen Mathematical Institute. Both authors are grateful to Prof. Y. Tschinkel for the permission to use these machines as well as to the system administrators for their support.

2. The discriminant and the index two subgroup

The goal of this section is to fix notation and to recall some facts on cubic surfaces and their discriminant.

2.1. One way to write down a cubic surface in explicit form is the so-called pentahedral normal form. Denote by S^{(a_{0},...,a_{4})} the cubic surface given in P^{4} by the system of equations

a_{0}X_{0}^{3} + a_{1}X_{1}^{3} + a_{2}X_{2}^{3} + a_{3}X_{3}^{3} + a_{4}X_{4}^{3} = 0 ,
X_{0} + X_{1} + X_{2} + X_{3} + X_{4} = 0 .

Remark 2.2. A general cubic surface over an algebraically closed field may be brought into pentahedral normal form over that field. Further, the coefficients are unique up to permutation and scaling. This is a classical result, which was first observed by J. J. Sylvester [8]. A proof is given in [2]. Cubic surfaces in pentahedral normal form with rational coefficients are, however, special to a certain extent.

Definition 2.3. The expression

Δ(S^{(a_{0},...,a_{4})}) :=

a_{0}^{8} · . . . · a_{4}^{8} ·

∏_{i_{1},i_{2},i_{3},i_{4}∈[0,1]}(\frac{1}{\sqrt{a_{0}}} + (−1)^{i_{1}} \frac{1}{\sqrt{a_{1}}} + (−1)^{i_{2}} \frac{1}{\sqrt{a_{2}}} + (−1)^{i_{3}} \frac{1}{\sqrt{a_{3}}} + (−1)^{i_{4}} \frac{1}{\sqrt{a_{4}}} )

is called the discriminant of the cubic surface S^{(a_{0},...,a_{4})}. Instead of Δ(S^{(a_{0},...,a_{4})}), we will usually write Δ(a_{0}, …, a_{4}).

Facts 2.4. a) Δ ∈ ℚ[a_{0}, …, a_{4}] is a symmetric polynomial, homogeneous of degree 32, and absolutely irreducible.

b) The cubic surface S^{(a_{0},...,a_{4})} is non-singular if and only if Δ(a_{0}, …, a_{4}) ≠ 0. Proof. [4, Lemma 2.5 and Corollary 2.10]. □

Theorem 2.5. Let a_{0}, …, a_{4} ∈ ℚ such that Δ(a_{0}, …, a_{4}) ≠ 0. Then, the Galois group operating on the 27 lines on S^{(a_{0},...,a_{4})} is contained in the index two subgroup D^{1}W(E_{6}) ⊂ W(E_{6}) if and only if (−3)Δ(a_{0}, …, a_{4}) ∈ ℚ is a perfect square.

□

Remark 2.6. This result was essentially known to H. Burkhardt [1, p. 341] in 1893. Burkhardt gives credit to C. Jordan [6], who was the first to study the automorphism group of the configuration of the 27 lines on a cubic surface. In [4, Theorem 2.12], we give a modern proof.

Proposition 2.7 (The two constraints). Suppose a_{0}, …, a_{4} ∈ ℤ are such that gcd(a_{0}, …, a_{4}) = 1 and (−3)Δ(a_{0}, …, a_{4}) ≠ 0 is a perfect square in ℚ.

a) Then, a_{0}, …, a_{4} all have the same sign.

b) Further, for every prime number p ≡ 2 (mod 3), all the p-adic valuations ν_{p}(a_{0}), …, ν_{p}(a_{4}) are even.

Proof. This is shown in [4, Proposition 3.3]. □

Fact 2.8. There is a form Δ′ homogeneous of degree 8 such that

Δ(a_{0}, …, a_{4}) = (a_{0} · . . . · a_{4})^{8} · Δ′(1/a_{0}, …, 1/a_{4}) .

Proof. The octic Δ′ is given by the formula

Δ′(x_{0}, …, x_{4}) :=

∏_{i_{1},i_{2},i_{3},i_{4}∈[0,1]} (\sqrt{x_{0}} + (−1)^{i_{1}} \sqrt{x_{1}} + (−1)^{i_{2}} \sqrt{x_{2}} + (−1)^{i_{3}} \sqrt{x_{3}} + (−1)^{i_{4}} \sqrt{x_{4}}) .

□

Definition 2.9. We will call the birational map ι from P^{4} to itself, given by

(a_{0} : …: a_{4}) ↦ (1/a_{0} : …: 1/a_{4}) ,

a generalized Cremona transform. Note that the standard Cremona transform of P^{2} is (a_{0} : a_{1} : a_{2}) ↦ (1/a_{0} : 1/a_{1} : 1/a_{2}).

Lemma 2.10. a) Δ′ ∈ ℚ[x_{0}, …, x_{4}] is a symmetric polynomial, homogeneous of degree eight and absolutely irreducible.

b) One has Δ′(0, x_{1}, …, x_{4}) = D^{2} for a symmetric, homogeneous quartic form D ∈ ℚ[x_{1}, …, x_{4}].

Proof. See [4, Lemma 4.5]. □

3. Rational points on the discriminantal covering

3.1. A point search.

3.1.1. We are interested in smooth cubic surfaces S^{(a_{0},...,a_{4})} such that the Galois group operating on the 27 lines is exactly equal to D^{1}W(E_{6}).

By Theorem 2.5, this implies that (a_{0} : …: a_{4}) ∈ P^{4}(ℚ) gives rise to a ℚ-rational point on the discriminantal covering. Further, according to Fact 2.4.b), (a_{0} : …: a_{4}) is supposed not to lie on the ramification locus.

Finally, if two of the coefficients were the same, say a_{0} = a_{1}, then S^{(a_{0},...,a_{4})} allowed the tritangent plane x_{0}+x_{1} = 0, which was defined over ℚ. Consequently, the order of the group acting on the lines could be at most 1152.

Thus, on the double covering π: O → P_{ℚ}^{4} , given by

w^{2} = (−3)Δ′(x_{0}, …, x_{4}) ,

we searched for rational points such that

i) w ≠ 0, ii) the five coordinates x_{0}, …, x_{4} are pairwise different from each other.

3.1.2. Surprisingly many solutions have been found. It turned out that there are 4 900 907 essentially different solutions up to a height limit of 3000. Under symmetry, they give rise to 120 solutions each. The smallest ones are (1 : 3 : 7 : 9 : 12), (1 : 3 : 4 : 7 : 13), (1 : 3 : 7 : 12 : 13), and (3 : 7 : 9 : 12 : 13). For a few height limits, we indicate the number of solutions up to that limit in the table below.

Table 1. Numbers of solutions up to various height limits

limit  # limit  # limit  # limit  #
25  20 200 10 039 500 93 680 1500 1 111 303
50 209 300 25 778 750 236 403 2000 2 088 752
100 1 481 400 54 331 1000 460 330 3000 4 900 907

Remark 3.1.3. We used the constraints shown above to optimize the searching algorithm. On one hand, it is sufficient to search for solutions such that 0 < x_{0} < x_{1} < x_{2} < x_{3} < x_{4}. On the other hand, only 751 of the positive integers up to 3000 fulfill the condition that all prime divisors p ≡ 2 (mod 3) have an even exponent.

3.2. The conjecture of Manin.

3.2.1. Let X be a non-singular (weak) Fano variety over ℚ. Assume that X(ℚ) ≠ ∅. Then, the conjecture of Manin [5] makes the following prediction for the number of ℚ-rational points on X of bounded anticanonical height.

There exists some τ > 0 such that, for every Zariski open set X^{∘} ⊆ X that is sufficiently small but non-empty,

#{x ∈ X^{∘}(ℚ) | h_{−K}(x) < B} ∼ τB log^{r}B

for r := rk Pic(X) − 1 and B ≫ 0.

Unfortunately, O is singular. In this situation, one has to consider a resolution Õ of singularities and compare heights.

Theorem 3.2.2. i) The singular locus of O is reducible into ten components. The component S_{(x_{0},x_{1})} is given by

x_{0} − x_{1} = 0 , x_{2}^{2} + x_{3}^{2} + x_{4}^{2} − 2x_{2}x_{3} − 2x_{2}x_{4} − 2x_{3}x_{4} = 0 .

The others are obtained by permuting coordinates.

ii) Let pr: Õ → O be the proper and birational morphism obtained by blowing up the ten singular components.

a) Then, Õ is non-singular. I.e., pr is a resolution of singularities.

b) Further, rk Pic(Õ) = 11.

c) The canonical divisor of Õ is K = pr^{∗}K_{O} for K_{O} = −π^{∗}H and H a hyperplane section of P^{4}.

Proof. This is shown in [4, Proposition 5.2 and Theorem 5.3]. □

Remark 3.2.3 (The prediction—Manin’s conjecture for the double covering O). Theorem 3.2.2.ii.c) implies

h_{−K}(y) = h_{−pr^{∗}K_{O}}(y) = h_{−K_{O}}(pr(y)) = h_{naive,P^{4}}(π(pr(y)))

for every y ∈ Õ(ℚ). Manin’s conjecture therefore predicts that, for every sufficiently small, non-empty, Zariski open subset O^{∘} ⊆ O,

#{x ∈ O^{∘}(ℚ) | h_{naive,P^{4}}(π(x)) < B} ∼ τB log^{10}B .

The reader might want to compare Table 2 below, where the actual numbers are given for a reasonably chosen Zariski open subset.

Remark 3.2.4. The proof given in [4] actually shows that Pic(Õ) ≅ ℤ^{11} is a trivial Gal(\overline{ℚ}/ℚ)-module. This implies that there is no Brauer-Manin obstruction present on Õ.

3.3. Infinitely many solutions.

Proposition 3.3.1. There are infinitely many ℚ-rational points on O. In fact, over the quadric surface Q in P^{4}, given by l = q = 0 for

l := x_{0} + x_{1} + x_{2} − 3x_{3} − 3x_{4} ,
q := x_{0}^{2} + x_{1}^{2} + x_{2}^{2} + 9x_{3}^{2} − x_{0}x_{1} − x_{0}x_{2} − 3x_{0}x_{3} − x_{1}x_{2} − 3x_{1}x_{3} − 3x_{2}x_{3} ,

the double covering π: O → P_{ℚ}^{4} splits. In particular, there are one or two ℚ-rational points above each ℚ-rational point of Q. Proof. Modulo ℐ_{Q}, one has actually

(3.1)  (−3)Δ(x_{0}, …, x_{4}) = [\frac{64}{3}(x_{0} − x_{1})(x_{0} − x_{2})(x_{1} − x_{2})(x_{3} − x_{4})]^{2} . □

Remarks 3.3.2. i) The difference of the two octic forms in equation (3.1) consists of 495 monomials. To verify the assertion, one may first use the linear equation to eliminate x_{4} and then check that the remaining octic form in x_{0}, …, x_{3} is divisible by the quadratic form q. Actually, a simple Gröbner base calculation reveals the fact that equation (3.1) is true even modulo ℐ_{Q}^{2}. ii) There is another proof for Lemma 3.3.1, which is somehow easier from the computational point of view but less canonical. In fact, Q is parametrized by the birational map ι: P^{2} [figure] Q,

(t_{0} : t_{1} : t_{2}) ↦ ((t_{0}^{2} + t_{1}^{2} + t_{2}^{2} − t_{0}t_{1} − t_{0}t_{2} − t_{1}t_{2}) : (t_{0}^{2} + t_{1}^{2} + t_{2}^{2} − t_{0}t_{1} + 2t_{0}t_{2} − t_{1}t_{2}) :

: (t_{0}^{2} + t_{1}^{2} + t_{2}^{2} − t_{0}t_{1} − t_{0}t_{2} + 2t_{1}t_{2}) : t_{2}^{2} : (t_{0}^{2} + t_{1}^{2} − t_{0}t_{1})),

being defined over ℚ. The locus where ι is undefined does not contain any ℚ-rational point since the quadratic form t_{0}^{2} + t_{1}^{2} − t_{0}t_{1} does not represent zero over ℚ. A direct calculation shows

(−3)Δ′(ι(t_{0}, t_{1}, t_{2})) = [576t_{0}t_{1}(t_{0} − t_{1})t_{2}^{3}(t_{0}^{2} + t_{1}^{2} − t_{0}t_{1} − t_{2}^{2})]^{2} .

Here, the factor t_{0} corresponds to (x_{0} − x_{1}), t_{1} to (x_{0} − x_{2}), (t_{0} − t_{1}) to (x_{1} − x_{2}), and (t_{0}^{2} + t_{1}^{2} − t_{0}t_{1} − t_{2}^{2}) to (x_{3} − x_{4}). The factor t_{2}^{3} is somehow artificial. For t_{2} = 0, the parametrization is constant to (1 : 1 : 1 : 0 : 1). The parametrization ι is actually constructed in a very naive manner. Start with the point (1 : 1 : 1 : 0 : 1) and determine for which value of τ ≠ 0 the point

(1 : (1 + τt_{0}) : (1 + τt_{1}) : (τt_{2}/3) : (1 + τ(t_{0} + t_{1} − t_{2})/3))

is contained in the quadric surface Q. Many other parametrizations would serve the same purpose.

Remarks 3.3.3. i) The surface Q is obviously symmetric under permutations of {x_{0}, x_{1}, x_{2}}. It is symmetric under switch of x_{3} and x_{4}, too. All in all, there are ten mutually different copies of Q. ii) Q is a smooth quadric surface. The two pencils of lines on Q are defined over ℚ(\sqrt{−3}) and conjugate to each other. iii) This implies that Pic(Q) = ℤ. The Picard group has two generators as soon as the ground field contains ℚ(\sqrt{−3}).

For quadrics such as Q, Manin’s conjecture is proven. The number of points of height ≤ B is asymptotically τ_{Q}B^{2} for some constant τ_{Q}. This means that π^{−1}(Q) ⊂ O is an example of a so-called accumulating subvariety. The growth of the number of rational points on π^{−1}(Q) is faster than predicted for a sufficiently small Zariski open subset of O.

Remark 3.3.4. As the height limit of 3000 is too low, most of the rational points found are actually not contained in π^{−1}(Q) or one of its copies. Cf. Table 2 below for the numbers of points on O with those over the copies of Q excluded.

Table 2. Numbers of solutions, accumulating subvarieties excluded

limit  # limit  # limit  # limit  #
25  12 200 8 989 500 86 897 1500 1 049 502
50 156 300 23 496 750 221 187 2000 1 977 863
100 1 248 400 50 070 1000 432 737 3000 4 651 857

Remarks 3.3.5. i) The smallest ℚ-rational points on Q with no two coordinates equal are (3 : 9 : 12 : 1 : 7) and (1 : 7 : 13 : 3 : 4). [4, Algorithm 3.7] shows that, indeed, these two points yield cubic surfaces such that the 27 lines are acted upon by the simple group D^{1}W(E_{6}). According to B. L. van der Waerden, this is the generic behaviour on Q. ii) When testing the cubic surface corresponding to (3 : 9 : 12 : 1 : 7), [4, Algorithm 3.7] works with the primes 19 and 73. Therefore, we have an explicit infinite set of ℚ-rational points that lead to the group D^{1}W(E_{6}). It is given by those points on Q reducing to (3 : 9 : 12 : 1 : 7) modulo both 19 and 73.

3.3.6. Some of the surprising properties of Q are described by the following two facts.

Fact 3.3.7. Q meets the octic R only within its singular locus. Actually,

Q ∩ R ⊂ S_{(x_{0},x_{1})} ∪ S_{(x_{0},x_{2})} ∪ S_{(x_{1},x_{2})} ∪ S_{(x_{3},x_{4})} .

Proof. Suppose (x_{0} : …: x_{4}) ∈ Q ∩ R. Then, formula (3.1) implies that x_{0} = x_{1}, x_{0} = x_{2}, x_{1} = x_{1}, or x_{3} = x_{4}. The equation x_{0} = x_{1} yields x_{0} = (−x_{2}+3x_{3}+3x_{4})/2. Substituting this into the quadratic relation q(x_{0}, …, x_{4}) = 0 from the definition of Q shows

x_{2}^{2} + x_{3}^{2} + x_{4}^{2} − 2x_{2}x_{3} − 2x_{2}x_{4} − 2x_{3}x_{4} = 0 .

For the relations x_{0} = x_{2}, x_{1} = x_{2}, and x_{3} = x_{4}, the situation is analogous. □

Fact 3.3.8. Q is tangent to all five coordinate hyperplanes. The points of tangency are (0 : 3 : 3 : 1 : 1), (3 : 0 : 3 : 1 : 1), (3 : 3 : 0 : 1 : 1), (1 : 1 : 1 : 0 : 1), and (1 : 1 : 1 : 1 : 0). □

3.3.9. The quadric surface Q determines the linear form l uniquely. On the other hand, the quadratic form q is unique only up to a multiple of l. One might have the idea to fix a canonical representative \underline{q} by the requirement that the quadric threefold, given by \underline{q} = 0, contain some of the singular components entirely. This is possible to a certain extent.

Fact. a) There is no quadric threefold in P^{4} containing the singular components S_{(x_{0},x_{1})} and S_{(x_{3},x_{4})}. b) There is, however, a one-dimensional family of quadric threefolds in P^{4} containing S_{(x_{0},x_{1})} and S_{(x_{0},x_{2})}. It is given by f_{t} = 0 for a parameter t and

f_{t} := −tx_{0}^{2} + x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2} −
− (1 − t)x_{0}x_{1} − (1 − t)x_{0}x_{2} + 2x_{0}x_{3} + 2x_{0}x_{4} +
+ (1 − t)x_{1}x_{2} − 2x_{1}x_{3} − 2x_{1}x_{4} − 2x_{2}x_{3} − 2x_{2}x_{4} − 2x_{3}x_{4} = 0 .

Proof. The statement that a quadric threefold contains S_{(x_{0},x_{1})} is equivalent to saying it is given by an equation of the form \underline{q} = 0 for

\underline{q} := a(x_{2}^{2} + x_{3}^{2} + x_{4}^{2} − 2x_{2}x_{3} − 2x_{2}x_{4} − 2x_{3}x_{4}) + (a_{0}x_{0} + … + a_{4}x_{4}) · (x_{0} − x_{1}) .

The assumptions of a) yield a linear system of equations that is only trivially solvable. On the other hand, the system of equations for b) leads to a two-dimensional vector space. □

Remark 3.3.10. This family is attached to the rational map f: P^{4} [figure] P^{1},

(x_{0} : …: x_{4}) ↦ (x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2} − x_{0}x_{1} − x_{0}x_{2} + 2x_{0}x_{3} + 2x_{0}x_{4} +

+ x_{1}x_{2} − 2x_{1}x_{3} − 2x_{1}x_{4} − 2x_{2}x_{3} − 2x_{2}x_{4} − 2x_{3}x_{4}) : (x_{0}^{2} − x_{0}x_{1} − x_{0}x_{2} + x_{1}x_{2}) .

The map f enjoys the following remarkable properties.

i) Its locus of indeterminacy is equal to S_{(x_{0},x_{1})} ∪ S_{(x_{0},x_{2})}. ii) The fiber at t = −1 is a singular quadric of rank three. The fiber at infinity is reducible into the two hyperplanes x_{0} = x_{1} and x_{0} = x_{2}. All other special fibers are smooth. iii) The special fiber at t = \frac{1}{3} may also be written as

4q + (−7x_{0} + 5x_{1} + 5x_{2} + 9x_{3} − 3x_{4})l = 0 .

In particular, the accumulating subvariety Q is contained within this fiber. iv) The fiber at t = \frac{1}{3} contains more of the rational points known than any other, even after deleting the accumulating subvarieties. The singular fiber at t = −1 follows next.

3.4. A statistical method.

3.4.1. We detected the quadric surface Q by a statistical investigation of the rational points found on O.

The theoretical background for this is a complement to Manin’s conjecture. In fact, it is expected that the rational points on Õ that lie outside the accumulating subvarieties are equidistributed with respect to a certain measure, the so-called Tamagawa measure. This includes that, for p a prime of good reduction, every 𝔽_{p}-rational point on Õ_{p} is equally likely to occur as the reduction of a ℚ-rational point.

Thus, in order to detect an accumulating subvarietiy, one might look at the rational points found and search for irregularities in the distribution of their reductions modulo p.

3.4.2. More precisely, our method was as follows. For a fixed good prime p, we counted how many rational points reduced to each of the points modulo p. This simply meant to group the known rational points into residue classes. Thereby, we ignored the points reducing to the ramification locus.

Example 3.4.3. For p = 53, most residue classes contained less than 500 points. However, there were exceptional residue classes containing between 1150 and 1250 points. Finally, some residue classes were even more exceptional as they contained more than 2000 points. We illustrate the distribution in the histogram below.

Figure 1. Numbers of rational points in the residue classes

When working with a different prime that is approximately of the same size, one ends up with a similar picture.

3.4.4. The ℚ-rational points on Õ are clearly not equidistributed modulo primes such as p = 53. Assuming that this phenomenon is explained by an accumulating subvariety, one would expect this subvariety to reduce only to exceptional residue classes. We therefore filtered our data by extracting the rational points that reduce to exceptional residue classes modulo several primes.

This leads to a list of points that are supposed to lie on one or more accumulating subvarieties. To determine a single one of these, we selected an exceptional residue class modulo 53. Among the coordinates of the corresponding points, we searched for linear and quadratic dependencies. The equations for Q, given above, arised in this way.

Remark 3.4.5. Having removed the points lying above Q and its copies, there is no further statistical abnormality to be seen from our data.

4. Accumulating subvarieties

4.1. The goal of this section is to prove that there are no other accumulating subvarieties that are, in a certain sense, similar to Q. Similarity shall include to be a non-degenerate quadric surface, over which the double covering π: O → P_{ℚ}^{4} splits.

In view of the first constraint established above, this implies that the real points on such a quadric surface S are contained in the 16-ant

{ (x_{0} : …: x_{4}) ∈ P^{4}(ℝ) | x_{0}, …, x_{4} ≥ 0 or x_{0}, …, x_{4} ≤ 0 } .

Further, there are strong restrictions for the behaviour at the boundary. By Lemma 2.10.b), we know that Δ′ is a perfect square on the coordinate hyperplane H_{0}, given by x_{0} = 0. On the other hand, we require (−3)Δ′ to be a perfect square on S.

A way to realize both of these, seemingly contradictory, requirements is to make S ∩ H_{0} a curve of degree two, on which (−3) is the square of a rational function. The only such examples are two lines over ℚ(\sqrt{−3}) that are conjugate to each other. This implies that S must necessarily be tangent to H_{0} and the point of tangency must be a ℚ-rational point on the ramification locus R.

Theorem 4.2. Suppose S ⊂ P_{ℚ}^{4} is a smooth quadric surface such that the double covering π: O → P_{ℚ}^{4} splits over S. Assume further that S is tangent to the five coordinate hyperplanes H_{0}, …, H_{4} and that, for each i, the point of tangency is actually contained in one of the three lines on H_{i} ∩ R. Then, S is equal to Q or one of its copies under permutation of coordinates.

Remark 4.3. On the Steiner surface H_{0} ∩ R, there are two types of ℚ-ratio-nal points. There are the three lines given by (0 : r : r : s : s) and permutations of the four coordinates to the right. The other ℚ-rational points are of the form (0 : t_{1}^{2} : …: t_{4}^{2}) for t_{1}, …, t_{4} ∈ ℚ such that t_{1} + … + t_{4} = 0.

Lemma 4.4. Assume S is as in Theorem 4.2. Further, write

P^{(0)} := (0 : x_{1}^{(0)} : x_{2}^{(0)} : x_{3}^{(0)} : x_{4}^{(0)})

for the point of tangency of S with the coordinate hyperplane H_{0}. Then, x_{1}^{(0)}, x_{2}^{(0)}, x_{3}^{(0)}, x_{4}^{(0)} ≠ 0.

Proof. Assume, to the contrary, that x_{1}^{(0)} = 0. The assumption on the type of the points of tangency made in Theorem 4.2 implies that one more coordinate must vanish. Without restriction, we may assume P^{(0)} = (0 : 0 : 0 : 1 : 1). The tangent plane at P^{(0)} is given by x_{0} = 0 and another linear relation C_{1}x_{1} + … + C_{4}x_{4} = 0. Whatever the coefficients are, there is a tangent vector (v_{0}, …, v_{4}) such that v_{1} < 0 or v_{2} < 0. The implicit function theorem yields a real point (x_{0} : …: x_{3} : 1) ∈ S(ℝ) satisfying x_{1} < 0 or x_{2} < 0. This is a contradiction. □

Lemma 4.5. Assume that the quadric surface S is tangent to the coordinate hyperplanes H_{0}, H_{1}, and H_{2} in (0 : x_{1}^{(0)} : x_{2}^{(0)} : x_{3}^{(0)} : x_{4}^{(0)}), (x_{0}^{(1)} : 0 : x_{2}^{(1)} : x_{3}^{(1)} : x_{4}^{(1)}), and (x_{0}^{(2)} : x_{1}^{(2)} : 0 : x_{3}^{(2)} : x_{4}^{(2)}), respectively. Then,

x_{1}^{(0)}x_{2}^{(1)}x_{0}^{(2)} − x_{2}^{(0)}x_{0}^{(1)}x_{1}^{(2)} = 0

or

x_{1}^{(0)}x_{2}^{(1)}x_{0}^{(2)} + x_{2}^{(0)}x_{0}^{(1)}x_{1}^{(2)} = 0 ,
x_{1}^{(0)}x_{0}^{(1)}x_{3}^{(2)} − x_{1}^{(0)}x_{0}^{(2)}x_{3}^{(1)} − x_{0}^{(1)}x_{1}^{(2)}x_{3}^{(0)} = 0 ,
x_{1}^{(0)}x_{2}^{(1)}x_{3}^{(2)} + x_{2}^{(0)}x_{1}^{(2)}x_{3}^{(1)} − x_{2}^{(1)}x_{1}^{(2)}x_{3}^{(0)} = 0 ,
x_{2}^{(0)}x_{0}^{(1)}x_{3}^{(2)} − x_{2}^{(0)}x_{0}^{(2)}x_{3}^{(1)} + x_{2}^{(1)}x_{0}^{(2)}x_{3}^{(0)} = 0 .

Proof. The linear equation by which S is defined may be written

(4.1)  L_{0}x_{0} + L_{1}x_{1} + L_{2}x_{2} + L_{3}x_{3} + L_{4}x_{4} = 0 .

We distinguish three cases.

First case. L_{4} ≠ 0. Then, we may use the linear equation (4.1) to eliminate x_{4} from the quadratic equation. Write

Q_{0}x_{0}^{2} + Q_{1}x_{1}^{2} + Q_{2}x_{2}^{2} + Q_{3}x_{3}^{2} +

+ Q_{4}x_{0}x_{1} + Q_{5}x_{0}x_{2} + Q_{6}x_{0}x_{3} + Q_{7}x_{1}x_{2} + Q_{8}x_{1}x_{3} + Q_{9}x_{2}x_{3} = 0 .

Tangency of H_{0} at (0 : x_{1}^{(0)} : x_{2}^{(0)} : x_{3}^{(0)} : x_{4}^{(0)}) means that the two linear forms

(Q_{4}x_{1}^{(0)} + Q_{5}x_{2}^{(0)} + Q_{6}x_{3}^{(0)})x_{0} + (2Q_{1}x_{1}^{(0)} + Q_{7}x_{2}^{(0)} + Q_{8}x_{3}^{(0)})x_{1} +
+ (2Q_{2}x_{2}^{(0)} + Q_{7}x_{1}^{(0)} + Q_{9}x_{3}^{(0)})x_{2} + (2Q_{3}x_{3}^{(0)} + Q_{8}x_{1}^{(0)} + Q_{9}x_{2}^{(0)})x_{3} ,
L_{0}x_{0} + L_{1}x_{1} + L_{2}x_{2} + L_{3}x_{3} + x_{4} ,

together generate x_{0}. This enforces the linear relations

2x_{1}^{(0)}Q_{1} + x_{2}^{(0)}Q_{7} + x_{3}^{(0)}Q_{8} = 0 ,
(4.2)  2x_{2}^{(0)}Q_{2} + x_{1}^{(0)}Q_{7} + x_{3}^{(0)}Q_{9} = 0 ,
2x_{3}^{(0)}Q_{3} + x_{1}^{(0)}Q_{8} + x_{1}^{(0)}Q_{9} = 0 .

The two other points of tangency yield relations that are completely analogous. Altogether, we find the homogeneous linear system of equations associated with the 9 × 10-matrix

( \begin{matrix}0 & 2x_{1}^{(0)} & 0 & 0 & 0 & 0 & 0 & x_{2}^{(0)} & x_{3}^{(0)} & 0 \\ 0 & 0 & 2x_{2}^{(0)} & 0 & 0 & 0 & 0 & x_{1}^{(0)} & 0 & x_{3}^{(0)} \\ 0 & 0 & 0 & 2x_{3}^{(0)} & 0 & 0 & 0 & 0 & x_{1}^{(0)} & x_{2}^{(0)} \\ 2x_{0}^{(1)} & 0 & 0 & 0 & 0 & x_{2}^{(1)} & x_{3}^{(1)} & 0 & 0 & 0 \\ 0 & 0 & 2x_{2}^{(1)} & 0 & 0 & x_{0}^{(1)} & 0 & 0 & 0 & x_{3}^{(1)} \\ 0 & 0 & 0 & 2x_{3}^{(1)} & 0 & 0 & x_{0}^{(1)} & 0 & 0 & x_{2}^{(1)} \\ 2x_{0}^{(2)} & 0 & 0 & 0 & x_{1}^{(2)} & 0 & x_{3}^{(2)} & 0 & 0 & 0 \\ 0 & 2x_{1}^{(2)} & 0 & 0 & x_{0}^{(2)} & 0 & 0 & 0 & x_{3}^{(2)} & 0 \\ 0 & 0 & 0 & 2x_{3}^{(2)} & 0 & 0 & x_{0}^{(2)} & 0 & x_{1}^{(2)} & 0 \end{matrix} ) .

If this matrix is of rank 9 then the quadratic equation defining S is, up to scaling, determined uniquely. In fact, this case is degenerate. There is a linear form in x_{0}, …, x_{3} only, vanishing on the three points given. The unique solution of the system corresponds to the square of this linear form.

Consequently, the rank is at most 8. The ten 9 × 9-minors must all vanish. These minors are polynomials in x_{0}^{(0)}, …, x_{3}^{(2)} having

(x_{1}^{(0)}x_{2}^{(1)}x_{0}^{(2)} − x_{2}^{(0)}x_{0}^{(1)}x_{1}^{(2)})

as their greatest common divisor. After division by this, we are left with ten sextics. It turns out that they are precisely the squares and pairwise products of the four cubics x_{1}^{(0)}x_{2}^{(1)}x_{0}^{(2)} + x_{2}^{(0)}x_{0}^{(1)}x_{1}^{(2)}, x_{1}^{(0)}x_{0}^{(1)}x_{3}^{(2)} − x_{1}^{(0)}x_{0}^{(2)}x_{3}^{(1)} − x_{0}^{(1)}x_{1}^{(2)}x_{3}^{(0)}, x_{1}^{(0)}x_{2}^{(1)}x_{3}^{(2)}+x_{2}^{(0)}x_{1}^{(2)}x_{3}^{(1)}−x_{2}^{(1)}x_{1}^{(2)}x_{3}^{(0)}, and x_{2}^{(0)}x_{0}^{(1)}x_{3}^{(2)}− x_{2}^{(0)}x_{0}^{(2)}x_{3}^{(1)}+ x_{2}^{(1)}x_{0}^{(2)}x_{3}^{(0)}.

Second case. L_{4} = 0 and L_{3} ≠ 0. As the roles of the third and fourth coordinates may be interchanged, we have, as in the first case, x_{1}^{(0)}x_{2}^{(1)}x_{0}^{(2)} − x_{2}^{(0)}x_{0}^{(1)}x_{1}^{(2)} = 0 or

x_{1}^{(0)}x_{2}^{(1)}x_{0}^{(2)} + x_{2}^{(0)}x_{0}^{(1)}x_{1}^{(2)} = 0 .

Suppose that the second variant is present. Then, the linear equation (4.1) implies that the vector (x_{3}^{(0)}, x_{3}^{(1)}, x_{3}^{(2)})^{t} is linearly dependent of (0, x_{0}^{(1)}, x_{0}^{(2)})^{t}, (x_{1}^{(0)}, 0, x_{1}^{(2)})^{t}, and (x_{2}^{(0)}, x_{2}^{(1)}, 0)^{t}. For these vectors instead of (x_{3}^{(0)}, x_{3}^{(1)}, x_{3}^{(2)})^{t}, the three more relations asserted are clearly true.

Third case. L_{3} = L_{4} = 0. In this situation, we may write the three points of tangency in the form (0 : L_{2} :(−L_{1}):x_{3}^{(0)} :x_{4}^{(0)}), (L_{2} : 0 :(−L_{0}):x_{3}^{(1)} :x_{4}^{(1)}), and (L_{1} :(−L_{0}) : 0:x_{3}^{(2)} :x_{4}^{(2)}). It turns out that the relation

x_{1}^{(0)}x_{2}^{(1)}x_{0}^{(2)} + x_{2}^{(0)}x_{0}^{(1)}x_{1}^{(2)} = 0

is automatically fulfilled. Further, L_{0}, L_{1}, L_{2} ≠ 0. Each of the three equations still to be proven reduces to L_{0}x_{3}^{(0)} − L_{1}x_{3}^{(1)} + L_{2}x_{3}^{(2)} = 0.

We may use the linear equation (4.1) to eliminate x_{0} from the quadratic equation. Write

Q_{0}x_{1}^{2} + Q_{1}x_{2}^{2} + Q_{2}x_{3}^{2} + Q_{3}x_{4}^{2} +

+ Q_{4}x_{1}x_{2} + Q_{5}x_{1}x_{3} + Q_{6}x_{1}x_{4} + Q_{7}x_{2}x_{3} + Q_{8}x_{2}x_{4} + Q_{9}x_{3}x_{4} = 0 .

Tangency of H_{0} at (0 : L_{2} : (−L_{1}) : x_{3}^{(0)} : x_{4}^{(0)}) yields the linear relations

L_{2}(2L_{2}Q_{0} − L_{1}Q_{4} + x_{3}^{(0)}Q_{5} + x_{4}^{(0)}Q_{6}) −

− L_{1}(−2L_{1}Q_{1} + L_{2}Q_{4} + x_{3}^{(0)}Q_{7} + x_{4}^{(0)}Q_{8}) = 0 ,
2x_{3}^{(0)}Q_{2} + x_{1}^{(0)}Q_{5} + x_{2}^{(0)}Q_{7} + x_{4}^{(0)}Q_{9} = 0 ,
2x_{4}^{(0)}Q_{3} + x_{1}^{(0)}Q_{6} + x_{2}^{(0)}Q_{8} + x_{3}^{(0)}Q_{9} = 0 .

Tangency of H_{1} and H_{2} leads to linear relations completely analogous to those given in (4.2). Altogether, we find the homogeneous linear system of equations associated with the 9 × 10-matrix

( \begin{matrix}2L_{2}^{2} & 2L_{1}^{2} & 0 & 0 & −2L_{1}L_{2} & x_{3}^{(0)}L_{2} & x_{4}^{(0)}L_{2} & −x_{3}^{(0)}L_{1} & −x_{4}^{(0)}L_{1} & 0 \\ 0 & 0 & 2x_{3}^{(0)} & 0 & 0 & x_{1}^{(0)} & 0 & x_{2}^{(0)} & 0 & x_{4}^{(0)} \\ 0 & 0 & 0 & 2x_{4}^{(0)} & 0 & 0 & x_{1}^{(0)} & 0 & x_{2}^{(0)} & x_{3}^{(0)} \\ 0 & −2L_{0} & 0 & 0 & 0 & 0 & 0 & x_{3}^{(1)} & x_{4}^{(1)} & 0 \\ 0 & 0 & 2x_{3}^{(1)} & 0 & 0 & 0 & 0 & −L_{0} & 0 & x_{4}^{(1)} \\ 0 & 0 & 0 & 2x_{4}^{(1)} & 0 & 0 & 0 & 0 & −L_{0} & x_{3}^{(1)} \\ −2L_{0} & 0 & 0 & 0 & 0 & x_{3}^{(2)} & x_{4}^{(2)} & 0 & 0 & 0 \\ 0 & 0 & 2x_{3}^{(2)} & 0 & 0 & −L_{0} & 0 & 0 & 0 & x_{4}^{(2)} \\ 0 & 0 & 0 & 2x_{4}^{(2)} & 0 & 0 & −L_{0} & 0 & 0 & x_{3}^{(2)} \end{matrix} ) .

If this matrix is of rank 9 then, again, we have a degenerate case. There is a linear form in x_{1}, …, x_{4} only, vanishing on the three points given. The unique solution of the system corresponds to the square of this linear form.

Consequently, all the ten 9 × 9-minors must vanish. Actually, when deleting the fourth column, the corresponding minor is

−16L_{0}^{4}L_{1}L_{2}(L_{0}x_{3}^{(0)} − L_{1}x_{3}^{(1)} + L_{2}x_{3}^{(2)})^{2} .  □

Remark 4.6 (Interpretation). The relations established in Lemma 4.5 may be interpreted as follows. The coordinates of three points of tangency form a 3 × 5matrix

( \begin{matrix}0 & x_{1}^{(0)} & x_{2}^{(0)} & x_{3}^{(0)} & x_{4}^{(0)} \\ x_{0}^{(1)} & 0 & x_{2}^{(1)} & x_{3}^{(1)} & x_{4}^{(1)} \\ x_{0}^{(2)} & x_{1}^{(2)} & 0 & x_{3}^{(2)} & x_{4}^{(2)} \end{matrix} ) .

We may scale such that x_{0}^{(1)} = x_{1}^{(0)} and x_{0}^{(2)} = x_{2}^{(0)}.

i) Then, the leftmost 3 × 3-block is either symmetric, i.e., x_{1}^{(2)} = x_{2}^{(1)}, or symmetric up to sign. Then, x_{1}^{(2)} = −x_{2}^{(1)}.

ii) In the latter case, the column vector (x_{3}^{(0)}, x_{3}^{(1)}, x_{3}^{(2)})^{t} is a linear combination of the column vectors (0, x_{0}^{(1)}, x_{0}^{(2)})^{t}, (x_{1}^{(0)}, 0, x_{1}^{(2)})^{t}, and (x_{2}^{(0)}, x_{2}^{(1)}, 0)^{t}.

Remarks 4.7. i) In the non-symmetric variant, (x_{4}^{(0)}, x_{4}^{(1)}, x_{4}^{(2)})^{t} is a linear combination of the column vectors (0, x_{0}^{(1)}, x_{0}^{(2)})^{t}, (x_{1}^{(0)}, 0, x_{1}^{(2)})^{t}, and (x_{2}^{(0)}, x_{2}^{(1)}, 0)^{t}, too. The roles of the third and fourth coordinates may be interchanged.

ii) Actually, in this variant, linear dependence of the three vectors (0, x_{0}^{(1)}, x_{0}^{(2)})^{t}, (x_{1}^{(0)}, 0, x_{1}^{(2)})^{t}, and (x_{2}^{(0)}, x_{2}^{(1)}, 0)^{t} is a non-trivial condition. Observe, they do not form a base of ℝ^{3}. In the symmetric variant, an analogous condition would be empty.

Remark 4.8. For each triple consisting of points of tangency of S with a coordinate hyperplane, relations of the same kind must be fulfilled.

Proof of Theorem 4.2. For each of the five points of tangency, we have at least two pairs {i, j} ⊂ {0, …, 4} such that x_{i} = x_{j}. There are two cases.

First case. Each of the ten pairs of {0, …, 4} appears exactly once.

Without restriction, the point of tangency to H_{0} is (0 : 1 : 1 : t : t). Again without loss of generality, (1 : 0 : s : 1 : s) is the point of tangency to H_{1}. The structure of the remaining three points of tangency is then fixed. The five points form a matrix as follows,

( \begin{matrix}0 & 1 & 1 & t & t \\ 1 & 0 & s & 1 & s \\ 1 & r & 0 & r & 1 \\ t & q & t & 0 & q \\ t & t & p & p & 0 \end{matrix} ) .

Lemma 4.5 implies that r = s. Indeed, r = −s would enforce that both (t, 1, −s)^{t} and (t, s, 1)^{t} are linearly dependent of (0, 1, 1)^{t}, (1, 0, −s)^{t}, and (1, s, 0)^{t}. This is a contradiction, since (0, s − 1, s + 1)^{t} is not in the span of these three.

For the same reason, p = q. Further, we have q = ±1 and s = ±t such that we end up with four one-parameter families,

( \begin{matrix}0 & 1 & 1 & t & t \\ 1 & 0 & t & 1 & t \\ 1 & t & 0 & t & 1 \\ t & 1 & t & 0 & 1 \\ t & t & 1 & 1 & 0 \end{matrix} ), (\begin{matrix}0 & 1 & 1 & t & t \\ 1 & 0 & −t & 1 & −t \\ 1 & −t & 0 & −t & 1 \\ t & 1 & t & 0 & 1 \\ t & t & 1 & 1 & 0 \end{matrix}), (\begin{matrix}0 & 1 & 1 & t & t \\ 1 & 0 & t & 1 & t \\ 1 & t & 0 & t & 1 \\ t & −1 & t & 0 & −1 \\ t & t & −1 & −1 & 0 \end{matrix}), (\begin{matrix}0 & 1 & 1 & t & t \\ 1 & 0 & −t & 1 & −t \\ 1 & −t & 0 & −t & 1 \\ t & −1 & t & 0 & −1 \\ t & t & −1 & −1 & 0 \end{matrix}).

The linear equation of S requires that the matrices considered are of rank at most 4. However, in the second and third families, the determinants (t^{2} − t − 1)(t^{3} + 2t − 1) and (t^{2} + t − 1)(t^{3} − 2t^{2} − 1) have no rational zeroes. For the fourth family, we find (t + 1)(t^{2} − t + 1)(t^{2} + 3t + 1) for the determinant. But, for t = −1, we had four equal coordinates in several of the points of tangency. Finally, for the first family, the determinant is (t + 1)(t^{2} − 3t + 1)^{2} and the value t = −1 could be possible.

The corresponding data lead to systems of equations that are uniquely solvable up to scaling. The resulting quadric surface is given by l = q = 0 for

l := x_{0} + x_{1} + x_{2} + x_{3} + x_{4} ,
q := x_{0}^{2} + x_{1}^{2} − x_{2}^{2} − x_{3}^{2} + 3x_{0}x_{1} + x_{0}x_{2} − x_{0}x_{3} − x_{1}x_{2} + x_{1}x_{3} − 3x_{2}x_{3} .

This surface is indeed smooth and tangent to all five coordinate hyperplanes but the double covering π: O → P_{ℚ}^{4} does not split over it.

Second case. One of the ten pairs of {0, …, 4} appears at least twice.

Without loss of generality, the points of tangency to H_{0} and H_{1}, respectively, are (0 : 1 : 1 : t : t) and (1 : 0 : 1 : s : s). If the point of tangency to H_{2} were (1 : (−1) : 0 : 1 : (−1)) then, by Lemma 4.5, both (t, s, 1)^{t} and (t, s, −1)^{t} had to be linear combinations of (0, 1, 1)^{t}, (1, 0, −1)^{t}, and (1, 1, 0)^{t}. This is a contradiction, since (0, 0, 2)^{t} is not in the span of these three.

Consequently, the five points of tangency form a matrix as follows,

( \begin{matrix}0 & 1 & 1 & t & t \\ 1 & 0 & 1 & s & s \\ 1 & 1 & 0 & r & r \\ t & ±s & ±r & 0 & q \\ t & ±s & ±r & ±q & 0 \end{matrix} ) .

Assume that one of the “r” or “s” actually carries a minus sign. Without restriction, there is “−r” in the fourth line. Then, Lemma 4.5 yields the contradiction that (t, r, q)^{t} must be a linear combination of (0, 1, t)^{t}, (1, 0, −r)^{t}, and (t, r, 0)^{t}. Further, if there were a “−q” in the fifth line then (1, r, r)^{t} had to be a linear combination of (0, t, t)^{t}, (t, 0, −q)^{t}, and (t, q, 0)^{t}, which is not the case, either.

Finally, in the fourth line, we must have two pairs of equal entries. Without restriction, suppose that q = t and r = s. All in all, we find a matrix of the form

( \begin{matrix}0 & 1 & 1 & t & t \\ 1 & 0 & 1 & s & s \\ 1 & 1 & 0 & s & s \\ t & s & s & 0 & t \\ t & s & s & t & 0 \end{matrix} ) .

For the determinant, one calculates 2t^{2}(4s − t − 1). We may conclude that s = \frac{t+1}{4}.

For every t ≠ 0, these data lead to systems of equations that are uniquely solvable up to scaling. The result is the one-parameter family S_{t} of quadric surfaces given by l_{t} = q_{t} = 0 for

l_{t} := (t − 1)x_{0} − 2tx_{1} − 2tx_{2} + 2x_{3} + 2x_{4} ,
q_{t} := (t + 1)^{2}x_{0}^{2} + 4(t + 1)tx_{1}^{2} + 4(t + 1)tx_{2}^{2} + 16x_{3}^{2} −
− 4(t + 1)tx_{0}x_{1} − 4(t + 1)tx_{0}x_{2} + 8(t − 1)x_{0}x_{3} +
+ 8(t − 1)tx_{1}x_{2} − 16tx_{1}x_{3} − 16tx_{2}x_{3} .

For each t ≠ 0, the quadric surface S_{t} is indeed smooth and tangent to all five coordinate hyperplanes.

In order to check for which values of t the double covering π: O → P_{ℚ}^{4} splits over S_{t}, we first restrict to the intersection C_{t} := S_{t} ∩ “x_{1} = x_{0} +x_{2}”. This is a smooth conic for each t ≠ 0. A parametrization ι_{t}: P^{1} → C_{t} is given by

(u : v) ↦ (16tu^{2} : ((t^{2} + 18t + 1)u^{2} + 8(t + 1)tuv + 16t^{2}v^{2}) :

: ((t^{2}+2t+1)u^{2} + 8(t+1)tuv + 16t^{2}v^{2}) : ((t^{2}+2t+1)tu^{2} + 8(t−1)t^{2}uv + 16t^{3}v^{2}) :

: ((t^{2} + 10t + 9)tu^{2} + 8(t + 3)t^{2}uv + 16t^{3}v^{2})) .

The binary form (−3)Δ′(ι_{t}(u, v)) of degree 16 factors into u^{6}((t + 1)u + 4tv)^{4} and a form of degree six that is irreducible for general t. We ask for the values of t, for which this sextic is a perfect square. According to magma, its discriminant is equal to

C(t − 3)(t − 1)^{6}(3t − 1)^{6}t^{83}(t^{2} + 8t − 1)^{4}(19t^{3} − 82t^{2} + 59t − 16)^{2}

for C a 103-digit integer. Over S_{1}, the double covering π: O → P_{ℚ}^{4} does not split. The cases t = 3 and t = \frac{1}{3} both yield the accumulating subvariety Q studied in subsection 3.3. They are equivalent to each other under the permutation (0)(13)(24) of coordinates. □

References

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[3] Elsenhans, A.-S. and Jahnel, J.: Experiments with general cubic surfaces, in: Tschinkel, Y. and Zarhin, Y. (Eds.): Algebra, Arithmetic, and Geometry, In Honor of Yu. I. Manin, Volume I, Progress in Mathematics 269, Birkhäuser, Boston 2007, 637–654

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Universität Bayreuth, Mathematisches Institut, Universitätsstraße 30, D-95447 Bayreuth, Germany

E-mail address: stephan.elsenhans@uni-bayreuth.de

Universität Siegen, Département Mathematik, Walter-Flex-Straße 3, D-57068 Siegen, Germany

E-mail address: ~jjahnel@mathematik.uni-siegen.de