COMPOSITION OF POINTS

AND MORDELL–WEIL PROBLEM FOR CUBIC SURFACES

D. Kanevsky^{1,2}, Yu. Manin^{2}

1T.J. Watson Research Center, P.O. Box 218, 23-116A, Yorktown Heights, New York 10598, US 2Max–Planck–Institut für Mathematik, Bonn, Germany

Abstract. Let V be a plane smooth cubic curve over a finitely generated field k. The Mordell–Weil theorem for V states that there is a finite subset P ⊂ V (k) such that the whole V (k) can be obtained from P by drawing secants and tangents through pairs of previously constructed points and consecutively adding their new intersection points with V. Equivalently, the group of birational transformations of V generated by reflections with respect to k–points is finitely generated. In this paper, elaborating an idea from [M3], we establish a Mordell–Weil type finite generation result for some birationally trivial cubic surfaces W. To the contrary, we prove that the birational automorphism group generated by reflections cannot be finitely generated if W (k) is infinite.

§1. Introduction

1.1. Composition of points. Let V be a cubic hypersurface without multiple components over a field k in P^{d}, d ≥ 2. Three points x, y, z ∈ V (k) (possibly coinciding) are called collinear if either x + y + z is the intersection cycle of V with a line in P^{d} (with correct multiplicities), or x, y, z lie on a k–line belonging to V. If x, y, z are collinear, we write x = y ∘ z. Thus ∘ is a (partial and multivalued) composition law on V (k). We will also consider its restriction on subsets of V (k), e.g. that of smooth points.

If x ∈ V (k) is smooth, and does not lie on a hyperplane component of V, the birational map t_{x} : V → V, y ↦ x ∘ y, is well defined. It is called reflection with respect to x. Denote by Bir V the full group of birational automorphisms of V.

The following two results summarize the properties of {t_{x}} for curves and surfaces respectively. The first one is classical, and the second is proved in [M1], Chapter V.

1.2. Theorem. Let V be a smooth cubic curve. Then:

(a) Bir V is a semidirect product of a finite group and the subgroup consisting of products of an even number of reflections {t_{x} | x ∈ V (k)}.

(b) We have identically

t_{x}^{2} = (t_{x}t_{y}t_{z})^{2} = 1  (1.1)

for all x, y, z ∈ V (k).

If in addition k is finitely generated over a prime field, then:

(c) Bir V is finitely generated.

(d) All points of V (k) can be obtained from a finite subset of them by drawing secants and tangents and adding the intersection points.

1.3. Theorem. Let V be a minimal smooth cubic surface over a perfect non– closed field k. Then:

(a) Bir V is a semi–direct product of the group of projective automorphisms and the subgroup generated by

{t_{x} | x ∈ V (k)} and {s_{u,v} | u, v ∈ V (K); [K : k] = 2; u, v are conjugate over k}

where

s_{u,v} := t_{u}t_{u∘v}t_{v},

and u, v do not lie on lines of V.

(b) We have identically

t_{x}^{2} = (t_{x}t_{x∘y}t_{y})^{2} = (s_{u,v})^{2} = 1, st_{x}s^{−1} = t_{s(x)},  (1.2)

for all pairs u, v not lying on lines in V, and projective automorphisms s. (c) The relations (1.2) form a presentation of Bir V.

We remind that V is called minimal if one cannot blow down some lines of V by a birational morphism defined over k. The opposite class consists of split surfaces upon which all lines are k–rational.

1.4. Main results of the paper. Although Theorems 1.2 and 1.3 look very similar, there is an important difference between finiteness properties in one– and two–dimensional cases.

Basically, (1.1) means only that x + y := e ∘ (x ∘ y) is an Abelian group law with identity e: see [M1], Theorem I.2.1. The statements c) and d) of the Theorem 1.2 additionally assert that this group is finitely generated. Therefore, (1.1) generally is not a complete system of relations between {t_{x}}.

On the contrary, (1.2) is complete, and in §2 we will see that this prevents Bir V from being finitely generated if V (k) is infinite. This answers one of the questions raised in [M3].

Therefore, any reasonable analog of the Mordell–Weil problem must address the problem of finite generation for (V (k), ∘) or of quotients of V (k) with respect to various equivalence relations compatible with ∘. This is the subject of §§3–5.

As in [M1], Chapter II, we can start with the universal equivalence relation U. By definition, this is the finest equivalence relation compatible with collinearity and such that ∘ induces a well defined operation on V (k)/U also denoted ∘. Then one of the Mordell–Weil type questions asks about finite generation (= finiteness) of the CH–quasigroup (V (k)/U, ∘) (see [M1], Chapter I.).

In §3 and §4 we give a description of U refining earlier results of [M1]. Consider the set of intersections of V with tangent planes at points of V (k) and add to it all images of these curves with respect to the group generated by all t_{x}, x ∈ V (k). Then one class of U consists of points that can be pairwise joined by a chain of curves belonging to this set of curves. This is the content of Theorem 3.3 below. We then discuss various versions of finite generation of (V (k), ∘). One essential choice is whether to allow to apply ∘ only to the different previously constructed points (for minimal surfaces, the result will then be uniquely defined). Another option giving more flexibility is to allow expressions x ∘ x and treat them as multivalued, thus adding at one step all the intersection points of V with a tangent plane at x. Finally, in §4 we extend the group–theoretic description of U given in [M1], II.13.10.

The results of §3 and §4 are essentially algebraic and do not add any new cases of finite generation of (V (k), ∘) to the short list of locally compact local fields already treated in [M1]. (In fact, [M1] proves the finiteness of V (k)/U over such fields by establishing that V (k) is covered by a finite number of sets of the form (x ∘ x) ∘ (y ∘ y)).

In §5 we study modified composition laws of points introduced in [Ma3]. The idea behind this development is to reinterpret the classical theorem on the structure of abstract projective planes as a finiteness result.

Namely, let k be a finitely generated field. Start with a finite subset S ⊂ P^{2}(k) and add to it pairwise intersections of all lines passing through two points of S thus getting a new finite set S′. Apply the same procedure to S′, and so on. If S is large enough, in the limit we will get the whole P^{2}(k). This easily follows from the fact that if we start with S consisting of ≥ 4 points in general position, in the limit we will get an abstract projective plane satisfying the Desargues axiom and therefore coinciding with P^{2}(k′) for k′ ⊂ k up to a projective coordinate change.

A trick, first introduced in [M3], allowed us to translate this remark into a finiteness theorem for V (k) assuming the existence of a birational morphism p : V → P^{2} defined over k. However, this required dealing with modified composition laws: roughly speaking, instead of looking at the collinearity relation induced by that in P^{3}, we now have to use the collinearity relations determined by the morphism p.

In this paper we make some steps towards eliminating this complication. Although the final result falls short of what we would like to prove, we feel that the connection and analogies with the theory of abstract projective planes deserve further study.

Acknowledgement. The first named author would like to thank V. Berkovich and J-L. Colliot-Thélène for useful discussions. The work was partially supported by the Humboldt Foundation during the author’s stay at the Max–Planck–Institut für Mathematik.

§2. Cardinality of generators of subgroups in a reflection group

2.1. Notation. We shall call an abstract cubic a set S with a ternary relation L ⊂ S × S × S, satisfying the following axioms:

(a) L is invariant with respect to permutations of factors.

(b) If (x, y, z), (x, y, z′) ∈ L and x ≠ y, then z = z′.

The reflection group G_{S} of an abstract cubic S is generated by symbols t_{x}, x ∈ S, subject to the following relations:

t_{x}^{2} = 1 for all x ∈ S;

(t_{x}t_{y}t_{z})^{2} = 1 for all (x, y, z) ∈ L.

The following result is proved in [K1].

2.2. Theorem. (a). Any element of finite order in G_{S} is conjugate to either t_{x} or to t_{x}t_{y}t_{z} for appropriate x, y, z ∈ S.

Let S be given effectively and L ⊂ S × S × S be decidable. Then:

(b) The word problem in G_{S} is decidable.

(c) The conjugacy problem in G_{S} is decidable.

The proof is based on a direct description of G_{S} as a limit of amalgamated sums. In [K2] it is shown that S can be sometimes reconstructed from G_{S}. Moreover, under some additional assumptions it is proved that Aut G_{S} is generated by G_{S} and permutations of S preserving L.

A different interesting description of G_{S} and another proof of the Theorem 2.2 is given in [P].

For the purposes of our paper we need the following description of G_{S} that is a special case of the general structure theorem 1.4 in [K1].

2.3. Structural Theorem. Let x ∈ S be an arbitrary point and S′ := {S \ x}. Then G_{S} is canonically isomorphic to G_{S′} ∗_{Π} K (the free product of G_{S′} and K with the amalgamated subgroup Π). The groups in this product can be described as follows.

(a) G_{S′} is the reflection group of the cubic S′ with the ternary relation induced by L on S′.

(b) The amalgamated subgroup Π is a free group generated by free generators a_{u,v} = t_{u}t_{v} for all distinct pairs u, v ∈ S′ such that (u, v, x) ∈ L and u < v (for some fixed ordering of S).

(c) K →^{∼} Z_{2} ∗ Z_{2} ∗ ⋯ ∗ Z_{2}. Generators of the subgroups Z_{2} in this free product are t_{x} and t_{x}a_{u,v}.

(d) Π is of index 2 in K. The quotient group K/Π is generated by the class t_{x}.

This structural result leads to the following auxiliary statement, which we need to prove our main results in this section.

2.4. Definition–Lemma. (a) In the situation of Theorem 2.3 a family W = ⟨R_{1}, t_{x}, R_{2}, t_{x}, …, t_{x}, R_{n}⟩, where R_{i} ∈ G_{S′}, is called a reduced t_{x}–partition of g = R_{1}t_{x}R_{2}t_{x} … R_{n} if R_{i} ∉ Π for 1 < i < n.

(b) Let W be a reduced t_{x}–partition of g ∈ G_{S}. Let us define ord_{x}(g) as the number of t_{x} in W. This number depends on g and x and is the same for different reduced t_{x}–partitions of g.

(c) Let g ∈ G_{S} be such that ord_{x}(g) = 0. Then g ∈ G_{S′}.

(d) ord_{x}(g_{1}g_{2}) ≡ (ord_{x}(g_{1}) + ord_{x}(g_{2})) mod 2.

(e) ord_{x}(aga^{−1}) ≡ ord_{x}(g) mod 2 for any a, g ∈ G_{S}.

(f) Let g ∈ G_{S}. We put δ(g) := {x ∈ S | ord_{x}(g) ≠ 0}. The set δ(g) is finite.

(g) δ(g_{1}g_{2}) ⊂ δ(g_{1}) ∪ δ(g_{2}).

(i) Let ⟨h_{1}, h_{2}, … ⟩ be a family generating a subgroup H. Then ∪_{h∈H}δ(h) = ∪_{i}δ(h_{i}) .

Now we can formulate the main theorem of this section.

2.5. Theorem. Consider a subgroup H = ⟨g_{1}, g_{2}, … g_{n}, … ⟩ ⊂ G_{S} generated by an infinite family of elements such that δ = ∪_{i}δ(g_{i}) is infinite. Then H is not finitely generated.

Proof. Assume that H is finitely generated by h_{1}, …, h_{k}. Then δ′ = ∪_{i=1,..k}δ(h_{i}) is finite. Therefore there exists some g_{r} and x ∈ S such that the following holds: ord_{x}(g_{r}) ≠ 0 and x ∉ δ′. Hence ord_{x}(h_{i}) = 0 for all i = 1, …, k. By 2.4(i), H ⊂ G_{S′} if H is generated by h_{1}, …, h_{k}. Since ord_{x}(g_{r}) ≠ 0, g_{r} ∉ H. This contradiction proves the theorem.

The following extension of Theorem 2.5 can be applied to various subgroups of Bir (V).

2.6. Corollary. Let G_{S} be the reflection group of an abstract cubic S and let W be the group of permutations of S, preserving its ternary relation L. Let G ≅ W ∗G_{S} be the semi-direct product of W and G_{S}, such that wt_{x}w^{−1} = t_{w(x)} for any w ∈ W and x ∈ S. Let a subgroup H ⊂ G be generated by a finite subgroup W′ ⊂ W and an infinite family of elements g_{i} ∈ G_{S} such that δ = ∪_{i=1,2,..}δ(g_{i}) is infinite. Then H is not finitely generated.

Proof. Let us assume that H is generated by a finite number of elements h_{1}, …, h_{k} ∈ G_{S} and a finite number of elements from W′. Let δ = ∪_{i}δ(g_{i}) and let δ′ = ∪_{w∈W′}wδ be obtained by applications of all w ∈ W′ to δ. Since δ and W′ are finite, δ′ is finite. Therefore there exists a generator g_{i} ∈ H such that δ(g_{i}) ⊈ δ′. Therefore, g_{i} ∉ G_{δ′}, i.e. it cannot be obtained as a product of elements from h_{1}, …, h_{k} and w ∈ W′. This contradiction proves the corollary.

2.7. Examples. In the situation of Theorem 1.3 assume that S = V (k) is infinite. Then the following subgroups of Bir (V) cannot be finitely generated:

(a) Bir (V), B(V) := ⟨t_{x} | x ∈ V (k)⟩ and

G := ⟨t_{x}, s_{u,v} | x ∈ V (k), u, v ∈ V (K); [K : k] = 2; u, v are conjugate over k⟩.

(b) The commutant of any of subgroups described in (a).

(c) Let B_{0}(V) denote the normal subgroup of B(V) generated by elements of the form t_{x}t_{y}t_{z}t_{x′}t_{y}t_{z′}, where (x, y, z) and (x′, y, z′) run through triples of collinear points of V (k). This subgroup was introduced in [M1] (II.13.9; beware of a misprint there: the second y carries a superfluous prime). It is closely related to the universal equivalence on V (k) (see the section 4 below).

(d) Let B_{1}(V) denote the normal subgroup of B(V) generated by elements of the form t_{x}t_{y}t_{z}, where (x, y, z) run through all possible triples of collinear points of V (k). This subgroup was introduced in [M2] because it is closely related to some admissible equivalence relations on V (k)

We will now show that Theorem 2.3 implies the statement for the case (b). Other cases will be discussed later and stronger statements will be proved.

Proof of (b). The commutant of B(V) contains elements t_{x}t_{y}t_{x}^{−1}t_{y}^{−1} = t_{x}t_{y}t_{x}t_{y} = a_{x,y}^{2}. Let us consider an infinite family of elements a_{x_{i}y_{i}}^{2}, x_{i}, y_{i} ∈ S. The statement for (b) will follow if we show that δ(a_{x,y}^{2}) contains x, since it will follow that ∪_{i}δ(a_{x_{i}y_{i}}) is infinite. For this it is enough to show that t_{x}t_{y}t_{x}t_{y} has the following reduced t_{x}–partition: (t_{x}, t_{y}, t_{x}, t_{y}). Indeed, in the notation of 2.4 one has to check that t_{y} ∉ Π. But this fact follows immediately from 2.3 if one notes that Π is a free group (hence it contains no nontrivial elements of finite order) and t_{y}^{2} = 1. This implies that ord_{x}(a_{x,y}^{2}) > 0, i.e. x ∈ δ(a_{x,y}^{2}). Q.E.D.

Our next theorem provides a lower bound for the number of generators in the normal closure.

2.8. Theorem. For any g ∈ G_{S} let δ̃(g) = {x ∈ S | ord_{x}(g) ≢ 0 mod 2}. Let H be the normal closure in G_{S} generated by a family of elements that contains a subfamily of elements h = (h_{1}, …, h_{i}, …) such that the following condition holds.

(J): For any i there exist x_{i} ∈ δ̃(h_{i}) such that x_{i} ∉ δ̃(h_{j}) if i ≠ j.

Then H cannot be the normal closure in G_{S} of less than card h generators.

This theorem immediately implies

2.9. Corollary. In the situation of Theorem 2.8, H cannot be the normal closure of a finite number of elements if there is an infinite subsystem h satisfying (J).

Proof of Theorem 2.8. Define a map of G_{S} into the vector space F_{2}^{S} as follows:

ψ : G_{S} → V, ψ(g) = (…, ord_{x}(g) mod 2, …).

It follows from 2.4(d) that ψ is a group homomorphism so that it maps conjugacy classes in G_{S} into one element. The theorem will follow if one shows that the image ψ(H) cannot be generated by less than card h vectors. But this follows immediately from the condition (J) in the theorem that guarantees that each image ψ(h_{i}) has a non–zero x_{i}–component while all other vectors ψ(h_{j}) have a zero x_{i}–component.

2.10. Corollary. None of the subgroups that are defined in (a),(c) and (d) in 2.7 can be obtained as the normal closure of a finite number of generators.

Proof. (a) follows from the fact that δ̃(t_{x}) = x. Since G_{S} contains the infinite number of t_{x}, G_{S} cannot be obtained as the normal closure of a finite number of elements.

(c) will follow similarly to (a) if we show that δ̃(t_{x}t_{y}t_{z}t_{x′}t_{y}t_{z′}) contains x for x ≠ y, z, x′, z′. This follows from the fact that the t_{x}–partition of t_{x}t_{y}t_{z}t_{x′}t_{y}t_{z′} is (t_{x}, t_{y}t_{z}t_{x′}t_{y}t_{z′}) (where t_{y}t_{z}t_{x′}t_{y}t_{z′} ∈ G_{S′}). Since V (k) has infinitely many collinear triples (x, y, z), such that x ≠ y ≠ z ≠ x, one can find infinitely many generators in B_{0}(V) satisfying the condition (J).

The case (d) can be treated similarly.

§3. Structure of universal equivalence

3.1. Setup. Let P be an abstract cubic with the collinearity relation L ⊂ P × P × P, such that for any x, y ∈ P, there exists z ∈ P with (x, y, z) ∈ P.

An equivalence relation R on P is called admissible if the relation L/R induced on P/R has the following property: for any X, Y ∈ P/R, there exists a unique Z with (X, Y, Z) ∈ L/R. An admissible equivalence relation is called universal if it is finer that any other admissible relation.

In [M1] it was proved that the universal relation exists (and of course, is unique) by a simple argument: just take the intersection of all admissible relations. Here we will clarify its structure by representing it as a limit of a sequence of explicitly constructed equivalence relations of which every next one is less fine than the previous one.

3.2. Approximations. For every i ≥ 0, we will describe inductively a symmetric and reflexive binary relation ∼_{i} on P and its transitive closure ≈_{i} . By definition, ∼_{0} and ≈_{0} are simply identical relations x = x′.

3.2.1. Definition. If ∼_{i} and ≈_{i} are already defined, we put x ∼_{i+1} x′ iff x = x′ or there exist u, v, u′, v′ ∈ P such that u ≈_{i} u′, v ≈_{i} v′, and (u, v, x) ∈ L, (u′, v′, x′) ∈ L.

Furthermore, we put x ≈_{i+1} x′ iff there is a sequence of points x = y_{0}, y_{1}, …, y_{r} = x′ such that y_{a} ∼_{i+1} y_{a+1} for all a < r.

Let us consider the case i = 1. By definition, x ∼_{1} x′ iff there exist u, v ∈ P such that (u, v, x), (u, v, x′) ∈ L. Let P be the set of k–points of a cubic surface V and L the usual collinearity relation. Assume for simplicity that V does not contain lines defined over k. Then x ∼_{1} x′ means that x = x′ or x and x′ lie on the intersection of V with the tangent plane at some k–point u (with u deleted if the double tangent lines to u in this plane are not defined over k). So one equivalence class for ≈_{1} consists of one point or of a maximal connected union of such quasiprojective curves, two of them being connected if they have an intersection point defined over k. The case of general cubic surface allows a similar description, but points of k–lines in V must be added as subsets of equivalence classes.

3.3. Theorem.(a) If x ≈_{i} x′ then x ≈_{i+1} x′.

(b) Denote by ≈ the equivalence relation

x ≈ x′ ⟺ ∃i, x ≈_{i} x′.

Then it is admissible and universal.

Proof. (a) It suffices to prove that if x ≠ x′, x ∼_{i} x′ then x ≈_{i+1} x′. For i = 0 this is clear. Assume that we have proved that u ∼_{i−1} u′ implies u ≈_{i} u′.

If x ∼_{i} x′ then by definition (u, v, x) ∈ L and (u′, v′, x′) ∈ L for some u ≈_{i−1} u′ and v ≈_{i−1} v′. ¿From the inductive assumption it follows that u ≈_{i} u′ and v ≈_{i} v′. By definition, then x ≈_{i+1} x′.

(b) Let us first prove that ≈ is admissible, in other words, if (u, v, x) ∈ L, (u′, v′, x′) ∈ L, and u ≈ u′, v ≈ v′, then x ≈ x′. In fact, for some i we have u ≈_{i} u′, v ≈_{i} v′, so that x ≈_{i+1} x′ and x ≈ x′.

Now denote temporarily the universal equivalence relation by ≈_{U} . The previous argument shows that x ≈_{U} x′ ⇒ x ≈ x′. It remains to prove that x ≈_{i} x′ ⇒ x ≈_{U} x′. We argue by induction. Again, it suffices to check that x ∼_{i} x′ ⇒ x ≈_{U} x′ assuming x ≠ x′. We can then find (u, v, x) ∈ L, (u′, v′, x′) ∈ L such that u ≈_{i−1} u′, v ≈_{i−1} v′. Therefore u ≈_{U} u′, v ≈_{U} v′, and finally x ≈_{U} x′.

3.4. Types of finite generation. Let us say, as in [M3], that P is ∘–generated by (x_{α} | α ∈ A) if for any y ∈ P there is a non–associative commutative word in x_{α}’s such that, informally, y is one of the values of this word. This means that when we calculate this word in the order determined by the brackets, every time that we have to calculate some u ∘ v, we may replace it by any x such that (u, v, x) ∈ L.

3.4.1. Claim. If P is ∘–generated by (x_{α}| α ∈ A), then the CH–quasigroup P/U is generated by the classes X_{α} of x_{α}.

We consider the following different types of ∘-generation.

3.4.2. Values of nonassociative words. Let W be a non-associative commutative word in finite number of variables X_{i}, P as in 3.1, and x_{i} a family of elements of P with the same set of indices. We define different rules of computing values of W on (x_{i}) in the order determined by the brackets inductively as follows for i = 0, 1, … ∞. We set x ≈_{∞} y if x ≈ y (i.e. x ≈_{j} y for some j).

Rule A_{i}. If the word W = X has length 1, then a value of W at any point x ∈ P is any y ∈ P such that x ≈_{i} y. In particular, A_{0} means that the value of W at x coincides with x. The rule A_{1} means that the set of values of W consists of those y for which which there are points u_{j}, y_{j}, j = 0, …, r, y_{0} = x, y_{r} = y such that the following holds: (u_{j}, u_{j}, y_{j−1}) ∈ L, (u_{j}, u_{j}, y_{j}) ∈ L for j = 1, …, r.

If the word W = X ∘ Y has length 2, its set P (x, y) of values of W at x, y ∈ P^{2} is defined as follows.

P (x, y) = {z ∈ P | z ≈_{i} z′, (x, y, z′) ∈ L}.

If the word W has length more than two, it is a product of two non empty words W = W_{1} ∘W_{2}. Let P (W_{i}) be a set of values of W_{i} that is defined inductively. Then the set of values P (W) is defined as ∪P (x, y) for all (x, y) ∈ P (W_{1}) × P (W_{2}).

We say that P is ∘_{A_{i}} generated by P′ = (x_{α} | α ∈ A) if it is generated by application of the rule A_{i} to points in P′.

The inverse statement of 3.4.1 is valid for ∘_{A_{∞}} by trivial reasons.

3.4.2. Claim. If CH-quasigroup P/U ≈ is ∘-generated by classes of (x_{α} | α ∈ A), then P is ∘_{A_{∞}} generated by x_{α}.

3.4.3. Questions. Let us define the generation index i(P) of P as the smallest i such that P is ∘_{A_{i}}–generated by a finite number of points in P. Let P = V (k) for some cubic surface.

(1) For which fields k and for which classes of cubic surfaces i(P) is finite? In particular, is i(P) = 0 for V defined over a number field (the original Mordell-Weil problem)?

(2) If the the CH–quasigroup P/U is finite, is the index i(P) finite?

It would be worthwhile to study (2) for an abstract cubic P that has an additional property: every three points of it generate an Abelian group like points on a plane cubic curve.

§4. A group–theoretic description of universal equivalence

In [M1], II.13.10 a group–theoretic description of universal equivalence was given for a cubic surface that is defined over an infinite field and has a point of general type. In this section we extend this description of universal equivalence. We relate the sequence of explicitly constructed equivalence relations from §3 to a filtration by subgroups in the reflection group associated with a minimal cubic surface.

Let B(V) and B_{0}(V) be the groups described in the examples 2.7. Here the field k over which the cubic surface V is defined can be finite and therefore we do not assume that V (k) is infinite.

Define x ∼ y mod U if t_{x}t_{y} ∈ B_{0}(V). It is clear that U is an equivalence relation on V (k). The proof of the following theorem differs from the proof of the corresponding theorem 13.10 in [M1] in the following respects. It uses the explicit description of the universal admissible equivalence from the section 3 and the structural description of the reflection group of S = V (k).

4.1. Theorem. U is the universal admissible equivalence relation.

Proof. We will check in turn that each of the equivalence relations is finer than the other one.

Assume first that z′ and z are universally equivalent. We want to show that z′ ∼ z mod U.

According to Theorem 3.3, z′ ≈_{i} z for some i. Since U is an equivalence relation, it is sufficient to treat the case z′ ∼_{i} z. The following Lemmma does the job.

4.2. Lemma. Denote by B^{i}(V), i = 0, 1, …, the normal closure of the family {t_{x}t_{x′} | x ∼_{i} x′} in B(V). Let x ∼_{i} x′, y ∼_{i} y′, (x, y, z) ∈ L and (x′, y′, z′) ∈ L . Then the following holds:

t_{z}t_{z′} ∈ t_{z}t_{z′}B^{i}(V) = t_{x}t_{y}t_{z}t_{x′}t_{y′}t_{z′}B^{i}(V) ⊂ B^{i+1} ⊂ B_{0}(V)

Proof of Lemma 4.2. Using relations t_{x}^{2} = 1 and t_{x}t_{y}t_{z} = t_{z}t_{y}t_{x} we get b = t_{z}t_{y}t_{x}t_{x′}t_{y′}t_{z′} = t_{z}t_{z′}b′ where b′ = t_{z′}t_{y}t_{x}t_{x′}t_{y′}t_{z′}. Next, b′ is conjugate to b′′ = t_{y}t_{x}t_{x′}t_{y′}. And, finally, b′′ is a product of t_{y}t_{y′} ∈ B^{i}(V) and t_{y}t_{x}t_{x′}t_{y} which is conjugate to t_{x}t_{x′} ∈ B^{i}(V). This proves the equality t_{z}t_{z′}B^{i}(V) = t_{x}t_{y}t_{z}t_{x′}t_{y′}t_{z′}B^{i}(V). It remains to show the inclusion B^{i+1}(V) ⊂ B_{0}(V). We will prove this inductively.

B^{1}(V) is generated by t_{z}t_{z′} such that z′ and z lie on the intersection of V with a tangent plane at some k–point u. In this case t_{z′}t_{z} = t_{z′}t_{u}t_{u}t_{z}t_{u}t_{u} ∈ B_{0}(V).

Assume that we already proved that B^{i}(V) ⊂ B_{0}(V) and let us prove that t_{z}t_{z′} ∈ B_{0}(V). Let z′′ ∈ V (k) be such that (x′, y, z′′) ∈ L. Then t_{z}t_{z′} = t_{z}t_{z′′}t_{z′′}t_{z′} and the following inclusions hold:

t_{z}t_{z′′} ∈ t_{z}t_{y}t_{x}t_{x′}t_{y}t_{z′′}B^{i}(V) ⊂ B_{0}(V)B^{i}(V) ⊂ B_{0}(V),
t_{z′′}t_{z′} ∈ t_{z′′}t_{x′}t_{y}t_{z′}t_{x′}t_{y′}B^{i}(V) ⊂ B_{0}(V)B^{i}(V) ⊂ B_{0}(V).

Since t_{z}t_{z′} ∈ B^{i+1}(V), this proves the inductive statement, establishes the Lemma and the first part of the Theorem.

We turn now to the second part. Let A be any admissible equivalence relation. We shall show that x ∼ y mod U implies x ∼ y mod A. Let X, Y, Z be the A– classes of x, y, z. Then Z = X ∘ Y in the sense of the composition law induced by collinearity relation on S = V (k). Denote by E = V (k)/A the set of classes with the induced structure of the symmetric quasigroup. Let t_{X} : E → E be the map t_{X}(Y) = X ∘ Y. The map t_{x} ↦ t_{X} extends to an epimorphism of groups φ : B(V) → T (E). We will show that its kernel contains B_{0}(V). Therefore if t_{x}t_{y} ∈ B_{0}(V) then φ(t_{x}t_{y}) = t_{X}t_{Y} = 1. This implies that t_{X} = t_{Y} and that X = Y. To prove this property of φ we need to extend the Theorem 13.1 (ii),(iii) in [M1] to our case. Recall that the Theorem 13.1 uses assumptions for cubic hypersurfaces that implies the fact that every equivalence class is dense in the Zariski topology. This is not true any more in general in our case.

4.3. Lemma. (a) φ : B(V) → T (E) is well defined and is an epimorphism of groups.

(b) In T (E) the following equality holds: t_{X}t_{Y} t_{Z} = t_{Y ∘Y}.

Proof. (a) Our proof is based on the representation of elements in B_{0}(V) as “minimal” words in the group K^{S}, the free product of groups Z_{2} generated by symbols T_{x}, one for each point x with the relations T_{x}^{2} = 1 (cf. [K1], 2.6 and §6). In order to construct the homomorphism B(V) → T (E), we first define the action of B(V) on E. Denote by T_{x_{1}}T_{x_{2}} … T_{x_{n}} a minimal representation in K^{S} of some s ∈ B(V). Choose Y ∈ E and put s(Y) = X_{1} ∘ (X_{2} ∘ … (X_{n} ∘ Y) …) where X_{i} are classes of x_{i} in E.

One can show that this definition does not depend on the choice of a minimal representation of s in K^{S}. This can be done inductively on the length of minimal words in K^{S}. All minimal words of length one representing the same element in B(V) coincide. Let us assume that the statement is proved for minimal words of the length i − 1. Consider now two different minimal words w = T_{1} … T_{i}, w′ = T′_{1} … T′_{i} of the length i representing s ∈ B(V). (Minimal words representing the same element have the same length). If T_{i} = T′_{i} then the action of w (resp. w′) on E can be factored through the actions of T_{i} and w_{1} = T_{1} … T_{i−1} (resp. w′_{1} = T′_{1} … T′_{i−1}) . Since w_{1} and w′_{1} represent the same element in B(V) and have the length i − 1, the statement follows by the inductive assumption.

Otherwise, if T_{i} ≠ T′_{i}, consider a T_{i}–partition of w′ (it is defined in the same way as t_{x}–partition above): (R_{1}, T_{i}, … R_{k−1}, T_{i}, R_{k}). From [K1] it follows that R_{k} = T_{u_{1}}T_{v_{1}}T_{u_{2}}T_{v_{2}} … T_{u_{r}}T_{v_{r}} and (u_{j}, v_{j}, u) ∈ L for all j = 1, … r and T_{u} = T_{i}. Moreover, if we replace T_{i}R_{k} in w′ with R′_{k}T_{i} where R′_{k} = T_{v_{1}}T_{u_{1}}T_{v_{2}}T_{u_{2}} … T_{v_{r}}T_{u_{r}}, then we get a new word w′′ that is already a minimal representation of s. Since w′′ and w both end with the same element T_{i} = T_{u}, they act in the same way on T (E). In order to prove that w′ and w′′ also act identically on T (E) it is enough to check that T_{u}R_{k} and R′_{k}T_{u} act in the same way on T (E). This can be shown using the fact that t_{u_{j}}t_{v_{j}}t_{u} = t_{u}t_{v_{j}}t_{u_{j}}.

To complete (a) we need to show that for any two elements s_{1}, s_{2} ∈ B(V) and Z ∈ E we have s_{1}(s_{2}(Z)) = (s_{1}s_{2})(Z). We will prove this statement by induction on the sum of lengths of minimal representation of s_{1} and s_{2}. The statement is obvious if s_{1} has length 0. Assume now that s_{1} has a minimal representation w_{1} = T_{x_{1}} … T_{x_{i}}, i ≥ 1, and s_{2} has a minimal representation w_{2} = T_{y_{1}} … T_{y_{k}}. If w = w_{1}w_{2} is the minimal representation of s = s_{1}s_{2} than the action of s on E is defined via the action of w by the rule X_{1} ∘ (… X_{i} ∘ (Y_{1} ∘ … (Y_{k} ∘ Z) …) where X_{i} (resp. Y_{j}) are the classes of x_{i} (resp. y_{j}) and Z ∈ E. Therefore s_{1}(s_{2}(Z)) = (s_{1}s_{2})(Z). Assume now that w_{1}w_{2} is not minimal.

Consider first the case when there exists such minimal representation of w_{1}, w_{2} that T_{x_{i}} = T_{y_{1}} (i.e. the last element in w_{1} coincides with the first element in w_{2}). Let s′_{1} ∈ B(V) be represented by w_{1} = T_{x_{1}} … T_{x_{i−1}} and s′_{2} ∈ B(V) be represented by w′_{2} = T_{y_{1}} … T_{y_{k−1}}. Then s′_{1}(s′_{2}(Z)) = s_{1}(s_{2}(Z)) and one can apply the inductive statement to s′_{1} and s′_{2}.

Otherwise, let us assume that the word w_{1}w_{2} has the following T_{x}–partition;

R_{1}T_{x}R_{2} … T_{x}R_{l}R_{l+1}T_{x}R_{l+2}T_{x} … T_{x}R_{m}

where R_{1}T_{x}R_{2} … T_{x}R_{l} (resp. R_{l+1}T_{x}R_{l+2}T_{x} … T_{x}R_{m}) is a minimal partition of w_{1} (resp. w_{2}). Since w_{1}w_{2} is not minimal, T_{x} can be chosen in such a way that R_{l}R_{l+1} = T_{u_{1}}T_{v_{1}}T_{u_{2}}T_{v_{2}} … T_{u_{r}}T_{v_{r}}, where (u_{s}, v_{s}, x) ∈ L for s = 1, …, r. As in the case of minimal words above one can replace T_{x}R_{l}R_{l+1} in w_{1}w_{2} with

T_{v_{1}}T_{u_{1}}T_{v_{2}}T_{u_{2}} … T_{v_{r}}T_{u_{r}}T_{x}

and obtain a new word w′ that has the same action on E that w_{1}w_{2}. Since w′ has two subsequent elements T_{x}, we can split it into a product of w′_{1} that ends with T_{x} and w′_{2} that starts with T_{x}. This case was already considered in this proof.

(b) follows from properties of the group law on plane cubic curves. This proves the Lemma 4.2.

To finish the proof of Theorem 4.1, we use the following identity:

φ(t_{x}t_{y}t_{z}t_{x′}t_{y}t_{z′}) = t_{X}t_{Y} t_{X∘Y} t_{X′}t_{Y} t_{X′∘Y′} = t_{Y ∘Y}^{2} = 1.

Here X, Y, … are the classes of x, y, … mod A. As a consequence, B_{0}(V) ⊂ Ker φ, proving the theorem.

4.4. Corollary. Let V be a minimal cubic surface over a finite field with q elements. Then B(V)/B_{0}(V) = Z_{2}, except when all points of V (k) are Eckardt points. In the later case we have either q = 2, card V (k) = 3, or q = 4, card V (k) = 9.

Proof. This follows from the description of the universal equivalence for V over finite fileds in [Sw–D].

4.5. Remarks. (a) As it follows from the proof of Theorem 4.1, it can be extended to an abstract cubic for which every three points generate an abelian group, in the same sense as for a plane cubic curve. We believe that this theorem can be proved also for an abstract cubic using only a structural description of G_{S} without this additional assumption. We plan to address this problem elsewhere.

(b) Groups G_{S} were studied in [P] using different methods. [P] asked whether the dependency problem DP (n) is decidable for reflection groups of an abstract cubic for n ≥ 3 or n = ∞. DP (n) can be formulated as follows.

We will say that g_{0} is dependent on (g_{1}, …, g_{k}) if there is a family (g_{i_{1}}, …, g_{i_{p}} and elements u_{1}, …, u_{p} of G such that

g_{0}(u_{1}g_{i_{1}}u_{1}^{−1}) … (u_{p}g_{i_{p}}u_{p}^{−1}) = 1.

If n is a positive number or infinity then the dependence problem DP (n) asks for an algorithm to decide for any sequence (g_{0}, …, g_{k}), 0 ≤ k < n, of elements of G whether or not g_{0} is dependent on (g_{1}, …, g_{k}). The problems D(1), D(2) are usually called the word problem and the conjugacy problem.

A special case of the dependence problem for t_{x}t_{y} ∈ B_{0}(V) can be related to the decidability of universal equivalence. Namely, if DP (∞) is decidable for g_{i} = t_{x_{i}}t_{y_{i}}t_{z_{i}}t_{x′_{i}}t_{y_{i}}t_{z′_{i}} and g_{0} = t_{x}t_{y} than one can efficiently define whether x, y are universally equivalent.

Since the decidability of the universal equivalence seems to be a very difficult problem in general, one can infer about the difficulty of the DP (∞) for B_{0}(V).

Question. Let an abstract cubic S be decidable. Is DP (n) decidable for arbitrary t_{x}t_{y} and generators of the subgroup B_{0}(V) described in 2.7(c)?

(c) Another construction of a filtration of the group of birational automorphism of V reflecting the structure of admissible equivalences is given in [M2]. One can apply the method from [M2] to the classes of universal equivalence. One can show that there exist classes of universal equivalence that are abstract cubics. One can consider universal equivalence on the set of points of such a class (considered as the abstract cubic). Applying this construction iteratively one can get a set of abstract cubics that corresponds to a filtration of subgroups in reflection groups. As in [M2] one can ask whether this sequence of subgroups stabilizes and what is its intersection.

§5. Birationally trivial cubic surfaces: a finiteness theorem

5.1. Modified composition. Let V be a smooth cubic surface, and x, y ∈ V (k). Let C ⊂ V be a curve on V passing through x, y, and p : C → P^{2} an embedding of C into a projective plane such that p(C) is again a cubic, and p(x) ∘ p(y) is defined in p(C). We assume that C and p are defined over k. In this situation, following [M3], we will put

x ∘_{(C,p)} y := p^{−1}(p(x) ∘ p(y)).

Example 1. Choose C = a plane section of V containing x, y. If p is the embedding of C into the secant plane, then x ∘_{(C,p)} y = x ∘ y in the standard notation. Notice that the result does not depend on C if x ≠ y. If x = y, then the choice of C determines a choice of one or two tangent lines to V at x so that the multivaluedness of ∘ is taken care of by the introduction of this new parameter.

Example 2. Assume now that V admits a birational morphism p : V → P^{2} defined over k (e.g., V is split). We will choose and fix p once for all. Then any plane section C of V not containing one of the blown down lines as a component is embedded by p into P^{2} as a cubic curve. Therefore we can apply to (C, p) the previous construction. Notice that this time x ∘_{(C,p)} y depends on C even if x ≠ y.

5.2. Theorem. Assume that k is a finitely generated field. In the situation of Example 2, the complement to the blown down lines in V (k) is finitely generated with respect to operations ∘_{(C,p)} with the additional restriction:

(C) the operation x ∘_{(C,p)} y is applied only to the different previously constructed points.

Proof. This theorem was stated and proved in [M3] without the additional condition (C). It uses the following auxiliary construction. Choose a k–rational line l ⊂ P^{2}. Then Γ := p^{−1}(l) is a twisted rational cubic in V. The family of all such cubics reflects properties of that of lines: a) any two different points a, b of V (k) belong to a unique Γ(a, b); b) any two different Γ’s either have one common k–point, or intersect a common blown down line. The proof of this theorem is based on generation of points by adding intersections of lines l passing through pairs of previously constructed points in a projective plane. This induces generation of points on V that are intersections of p^{−1}(l). Analysis of this proof in [M3] shows that it considers only different points in pairs of previously constructed points hereby providing the statement of the theorem with the condition (C).

If one drops the condition (C) one can prove the stronger statement.

5.3. Theorem. Let V be a smooth cubic surface over an arbitrary field k. Assume that V admits a birational morphism p : V → P^{2}. Then the complement P to all blown down lines in V (k) is generated by any single point from P (in the sense of the composition ∘_{(C,p)}).

Proof. Let us choose a point x ∈ P. The theorem will follow if we prove that the set of points x ∘_{(C,p)} x contains P (here C runs through all k–rational plane sections of V passing through x). Let us show that for any other point y in P there exists such C that y = x ∘_{(C,p)} x. Indeed, following arguments of [M3], for y ∈ P there exists a twisted cubic curve G(x, y) := p^{−1}(l) where l is the line through p(x), p(y) in P^{2}. Let l_{1} be the tangent line to G(x, y) at x. Let a plane through points x, y and l_{1} cut a curve C on V. Then l_{1} is a tangent line to C at x, i.e. G(x, y) is tangent to C at x. Hence l in P^{2} is tangent to p(C) at p(x). Since this line l passes through p(y), on p(C) we have p(y) = p(x) ∘ p(x). This gives y ∈ x ∘_{(C,p)} x proving the statement.

One can apply this theorem to the proof of the triviality of the 3–component of the universal equivalence on P = V (k). 3–component of the universal equivalence can be defined as the finest admissible equivalence U_{3} for which the following condition holds:

For any class X ∈ P/U_{3}, X ∘ X = X.

Simillarly one can define the 2–component of the universal equivalence as the finest admissible equivalence for which the following condition holds:

For any class X ∈ P/U_{2}, X ∘ X = O for some fixed class O ∈ P.

It follows from [M1] that U = U_{3}∩U_{2}, where U denotes the universal equivalence.

5.4. Corollary. Let V be a smooth cubic surface over an arbitrary field k. Assume that V admits a birational morphism p : V → P^{2}. Then U_{3} is trivial on V (k).

The corollary can be deduced from the following two lemmas.

5.5. Lemma. Let C be a smooth plane cubic curve defined over a field k such that C(k) is non–empty. Let p be another plane embedding of C over k. Then

x ∘_{(C,p)} y := p^{−1}(p(x) ∘ p(y)) = t^{−1}((t(x) ∘ t(y))

where t ∈ Bir C is some birational automorphism of C over k which can be represented as a product of reflections of C defined over k.

Proof. The statement easily follows from the following fact: p can be decomposed into a product of reflections of C over k and a projective isomorphism of C and p(C). Indeed, let us choose a point 0 ∈ C(k). Isomorphism classes of invertible sheaves of degree 3 are parametrized by the jacobian of C of degree 3, say, T, and T is a principal homogeneous space over C. This means that C(k) acts transitively on T (k), i.e. any two sheaves L_{1}, L_{2} differ by a translation by a point a ∈ C(k). Any translation is a product of two reflections, whereas a projective isomorphism preserves collinearity.

5.6. Lemma. In the same notation, for any two points x, y ∈ C(k) the following holds:

t^{−1}(t(x) ∘ t(y)) ∼ x ∘ y mod U_{3}.

Proof. Let t = t_{x_{1}} … t_{x_{n}} where x_{i} ∈ C(k). It is enough to check the statement for n = 1 since the general statement can be obtained by induction. Let t = t_{z}. We have: t^{−1}(t(x)∘t(y)) = t_{z}(t_{z}(x)∘t_{z}(y)) = z ∘((z ∘x)∘(z ∘y)) = z ∘((z ∘z)∘(x∘y)) ∼ z ∘ (z ∘ (x ∘ y)) mod U_{3} ∼ x ∘ y mod U_{3}. Here we used z ∘ z ∼ z mod U_{3}. Q.E.D.

We can now deduce the Corollary 5.4.

Fix some x ∈ P, where P is the complement to all blown down lines in V (k). By the Theorem 5.3, any point z ∈ P can be represented as x ∘_{(C,p)} x. Let z = x ∘_{C,p} x for some C. If C is singular then all points on C(k) are equivalent mod U_{3} (this is a general property of any singular plane cubic curve that does not have a line as a component). Otherwise, by lemmas 5.5 and 5.6

z = x ∘_{(C,p)} x ∼ x ∘ x mod U_{3} ∼ x mod U_{3}.

5.7. Elimination of ∘_{(C,p)}. The use of the modified operation ∘_{(C,p)} is somewhat annoying, and we would like to replace it by the standard composition ∘. For example, in the setup of the Theorem 5.2 for any three points x, y, z on a plane smooth section C ⊂ V the following equality holds:

(x ∘_{(C,p)} y) ∘_{(C,p)} z = (x ∘ y) ∘ z.

This naturally leads to the question whether one can obtain the traditional Mordell– Weil statement for the composition ∘ using our finiteness results for ∘_{(C,p)} and some tricks like the formula above.

The remaining part of the paper is dedicated to the description of our, not altogether successful, attempts to eliminate ∘_{(C,p)}. We reformulate the finiteness theorem above in terms that do not use explicitly compositions ∘_{(C,p)} and a morphism p of a cubic surface into a projective plane. We only use the standard operation ∘ and implicitly use some intersections of planes with lines that belong to this cubic surface.

Before we can state a new statement we need to define a new kind of operation on a cubic surface that involves lines belonging to this cubic surface.

5.7.1. Definition. Let V be a smooth cubic surface over an arbitrary field k. Let Λ = {l_{1}, l_{2}, m} be three (not necessary k-rational) lines belonging to V and such that the following properties hold:

(A) l_{1} and l_{2} are skew lines (i.e. they do not have a common point) and m intersects l_{1} and l_{2}.

Given a triple of lines Λ satisfying (A) and an arbitrary plane T not containing lines in V, let us define a new composition of points u, and w on T ∩ V as follows:

(B) u ∘_{(T,Λ)} w = (x ∘ y) ∘ [z ∘ (u ∘ w)],

where x = l_{1} ∩ T, y = l_{2} ∩ T and z = m ∩ T.

Of course, the point u ∘_{(T,Λ)} w is not necessarily k–rational even u, w, and T are k–rational. But there is a special case when the composition ∘_{(T,Λ)} produces rational points (over k) when u, w, and T are defined over k (whereas lines in Λ are not necessarily defined over k). This case is described in the following statement that reformulates the Theorem 5.2 in terms of the composition ∘_{(T,Λ)}.

5.7.2. Theorem. Let V be a smooth cubic surface. Assume that V admits a birational morphism to a projective plane defined over k. Assume that k is finitely generated field. Then there exists a triplet of lines on V satisfying the property (A) such that the following statement holds: the complement to the blown down lines in V (k) is finitely generated with respect to operations ∘_{(T,Λ)} with the additional restriction:

(D) the operation x ∘_{(T,Λ)} y is applied only to different previously constructed points. (Here Λ is fixed and T runs through some set of k-rational planes).

Similarly, one can reformulate Theorem 5.3 in terms of new operations.

5.7.3. Theorem. Let V be a smooth cubic surface over an arbitrary field k. Assume that V admits a birational morphism to a projective plane defined over k. Then the complement P to all blown down lines in V (k) is generated by any single point from P in the sense of compositions ∘_{(T,Λ)} for some fixed triple of lines Λ in V.

Below we will show how to replace operations ∘_{(C,p)} by operations ∘_{(T,Λ)}.

5.7.4. Lemma. Let V be a smooth cubic surface defined over a field k and k̄ be an algebraic closure of k. Let p : V → P^{2} be a birational morphism over k̄. Then there exists a triplet of lines Λ satisfying the property (A ) such that for any plane section C of V not containing one of the blown down lines as a component and for any two points u, w ∈ V (k̄) lying on C the following holds:

u ∘_{(C,p)} w = u ∘_{(T,Λ)} w where T is a plane that cuts the curve C on V.

5.7.5. Corollary. Assume that the birational morphism p in Lemma 5.7.4 is defined over k. Then a triplet Λ can be chosen in such a way that the point u∘_{(T,Λ)}w is k–rational if u, w and the plane T are k–rational.

The proof of Lemma 5.7.4 is a consequence of the following claims which might be of independent interest.

5.7.6. Claim. In the conditions of Lemma 5.7.4, let x, y, u, w be some points on C. Then the following equality holds:

u ∘_{(C,p)} w = (x ∘ y) ∘ [(x ∘_{(C,p)} y) ∘ (u ∘ w)].

In other words, if we know how to compute z = x ∘_{(C,p)} y at least for some two points x, y in C then operation ∘_{(C,p)} for all other points in C can be computed in terms of ∘ only.

5.7.7. Claim. In the conditions of Lemma 5.7.4, let Λ = {l_{1}, l_{2}, m} be a triplet of lines satisfying (A) and such that p(m) is a line on the plane P^{2}, and l_{1}, l_{2} are blown down lines. Let x = l_{1} ∩ T, y = l_{2} ∩ T and z = m ∩ T, where the plane T cuts a curve C on V. Then z = x ∘_{(C,p)} y.

In other words, one can easily compute an operation ∘_{(C,p)} for intersection of lines l_{1} and l_{2} with a plane T. The result of this composition is an intersection of a third line m with T !

To show that the Lemma 5.7.4 follows from these claims, it is sufficient to note the following. By Claim 5.7.6, the operation u ∘_{(C,p)} w can be replaced by (x ∘ y) ∘ [(x ∘_{(C,p)} y) ∘ (u ∘ w)] where x, y are any points on C. There exists a triplet of lines Λ on V satisfying (A), such that p(m) is a line on the plane P^{2}, and l_{1}, l_{2} are the blown down lines. By the Claim 5.7.7, x, y can be chosen as intersections of lines l_{1}, l_{2} with a plane T that cuts C on V and in this case x ∘_{(C,p)} y = m ∩ T.

Now we prove our Claims.

Proof of the Claim 5.7.6. Step 1: Since C and p are fixed, one can simplify our notation by putting x ∗ y =: x ∘_{(C,p)} y. In this step we show that for any points x, y, u, w on C the following equality holds:

u ∗ w = (x ∗ y)(x ∘ y)^{−1}(u ∘ w),  (5.1)

where the expressions in brackets are multiplied by using an Abelian structure on C: xy = a ∘ (x ∘ y) for some point a in C(k).

First, we consider the case when C is smooth. In this case by the Lemma 5.5 p in the formula p(p^{−1}(u) ∘ p^{−1}(w)) can be replaced by a product of reflections of C. Let us check (5.1) for the case when p can be replaced by one reflection t_{b}:

u ∗ w = p(p^{−1}(u) ∘ p^{−1}(w)) = b ∘ ((b ∘ u) ∘ (b ∘ w)) = b ∘ ((b ∘ b) ∘ (u ∘ w)).

The general case can be obtained by iterating this argument.

Using the identity u ∘ w = (a ∘ a)u^{−1}w^{−1} we get:

u ∗ w = b ∘ ((b ∘ b) ∘ (u ∘ w)) = b^{−1}(b ∘ b)(u ∘ w)

Similarly we have for other two points: x∗y = b^{−1}(b∘b)(x∘y). Replacing b^{−1}(b∘b) with (x ∗ y)(x ∘ y)^{−1} in b ∘ ((b ∘ b) ∘ (u ∘ w)) gives (5.1).

Step 2: Replacing the Abelian multiplication operation in (5.1) by a ∘ (…) we can rewrite (5.1) as as:

u ∗ w = a ∘ (r ∘ (u ∘ w)),

where r = a ∘ {(x ∗ y) ∘ [(a ∘ a) ∘ (x ∘ y)]}. Since the point a is arbitrary, one can choose a = x ∘ y. This gives r = x ∗ y and immediately implies the formula in the Claim 5.7.6.

In order to complete the proof of the Claim we need to consider the case when C is a singular plane cubic curve that does not contain a line. This can be done by appealing to an obvious limiting construction in the case of topological field k, or to a similar argument using the Zariski topology in general.

Proof of the Claim 5.7.7. Since l_{1}, l_{2} are blown down lines and p(m) is a line in P^{2}, the points p(x), p(y), p(z) are intersections of the line p(m) with the curve p(C) in P^{2}. This means that on p(C) we have p(x)∘p(y) = p(z). This is equivalent to the equality z = x ∘_{(C,p)} y in the Claim.

References

[K1] D. S. Kanevski. Structure of groups, related to cubic surfaces, Mat. Sb. 103:2, (1977), 292–308 (in Russian); English. transl. in Mat. USSR Sbornik, Vol. 32:2 (1977), 252–264.

[K2] D. S. Kanevsky, On cubic planes and groups connected with cubic surfaces. J. Algebra 80:2 (1983), 559–565.

[M1] Yu. I. Manin. Cubic Forms: Algebra, Geometry, Arithmetic. North Holland, 1974 and 1986.

[M2] Yu. I. Manin. On some groups related to cubic surfaces. In: Algebraic Geometry. Tata Press, Bombay, 1968, 255–263.

[M3] Yu. I. Manin. Mordell–Weil problem for cubic surfaces. In: Advances in the Mathematical Sciences—CRM’s 25 Years (L. Vinet, ed.) CRM Proc. and Lecture Notes, vol. 11, Amer. Math. Soc., Providence, RI, 1997, pp. 313–318.

[P] S. J. Pride. Involutary presentations, with applications to Coxeter groups, NEC-Groups, and groups of Kanevsky. J. of Algebra 120 (1989), 200–223.

[Sw–D] H. P. F. Swinnerton–Dyer. Universal equivalence for cubic surfaces over finite and local fields. Symp. Math., Bologna 24 (1981), 111–143.

E-mail addresses: kanevsky@us.ibm.com manin@mpim–bonn.mpg.de