Rational Real Algebraic Models

of Topological Surfaces

Indranil Biswas and Johannes Huisman

Received: September 20, 2007

Communicated by Thomas Peternell

Abstract. Comessatti proved that the set of all real points of a rational real algebraic surface is either a nonorientable surface, or diffeomorphic to the sphere or the torus. Conversely, it is well known that each of these surfaces admits at least one rational real algebraic model. We prove that they admit exactly one rational real algebraic model. This was known earlier only for the sphere, the torus, the real projective plane and the Klein bottle.

2000 Mathematics Subject Classification: 14P25, 14E07

Keywords and Phrases: Real algebraic surface, topological surface, rational surface, rational model, birational map, algebraic diffeomorphism, transitivity, geometrically rational surface, geometrically rational model

1 Introduction

Let X be a rational nonsingular projective real algebraic surface. Then the set X(ℝ) of real points of X is a compact connected topological surface. Comessatti showed that X(ℝ) cannot be an orientable surface of genus bigger than 1. To put it otherwise, X(ℝ) is either nonorientable, or it is orientable and diffeomorphic to the sphere S^{2} or the torus S^{1} × S^{1} [Co2, p. 257]. Conversely, each of these topological surfaces admits a rational real algebraic model, or rational model for short. In other words, if S is a compact connected topological surface which is either nonorientable, or orientable and diffeomorphic to the sphere or the torus, then there is a nonsingular rational projective real algebraic surface X such that X(ℝ) is diffeomorphic to S. Indeed, this is clear for the sphere, the torus and the real projective plane: the real projective surface defined by the affine equation x^{2} + y^{2} + z^{2} = 1 is a rational model of the sphere S^{2}, the real algebraic surface ℙ^{1} × ℙ^{1} is a rational model of the torus S^{1} × S^{1}, and the real projective plane ℙ^{2} is a rational model of the topological real projective plane ℙ^{2}(ℝ). If S is any of the remaining topological surfaces, then S is diffeomorphic to the n-fold connected sum of the real projective plane, where n ≥ 2. A rational model of such a topological surface is the real surface obtained from ℙ^{2} by blowing up n−1 real points. Therefore, any compact connected topological surface which is either nonorientable, or orientable and diffeomorphic to the sphere or the torus, admits at least one rational model. Now, if S is a compact connected topological surface admitting a rational model X, then one can construct many other rational models of S. To see this, let P and \overline{P} be a pair of complex conjugate complex points on X. The blow-up X̃ of X at P and \overline{P} is again a rational model of S. Indeed, since P and \overline{P} are nonreal points of X, there are open subsets U of X and V of X̃ such that

• X(ℝ) ⊆ U(ℝ), X̃(ℝ) ⊆ V (ℝ), and

• U and V are isomorphic.

In particular, X(ℝ) and X̃(ℝ) are diffeomorphic. This means that X̃ is a rational model of S if X is so. Iterating the process, one can construct many nonisomorphic rational models of S. We would like to consider all such models of S to be equivalent. Therefore, we introduce the following equivalence relation on the collection of all rational models of a topological surface S.

Definition 1.1. Let X and Y be two rational models of a topological surface S. We say that X and Y are isomorphic as rational models of S if there is a sequence

\xymatrix{ & X_{1} \ar[dl] \ar[dr] &  & X_{3} \ar[dl] \ar[dr] &  & X_{2n−1} \ar[dl] \ar[dr] \\ X = X_{0} &  & X_{2} &  & ⋯ &  & X_{2n} = Y}

where each morphism is a blowing-up at a pair of nonreal complex conjugate points.

We note that the equivalence relation, in Definition 1.1, on the collection of all rational models of a given surface S is the smallest one for which the rational models X and X̃ mentioned above are equivalent. Let X and Y be rational models of a topological surface S. If X and Y are isomorphic models of S, then the above sequence of blowing-ups defines a rational map

f: X −−→ Y

having the following property. There are open subsets U of X and V of Y such that • the restriction of f to U is an isomorphism of real algebraic varieties from

U onto V, and

• X(ℝ) ⊆ U(ℝ) and Y (ℝ) ⊆ V (ℝ).

It follows, in particular, that the restriction of f to X(ℝ) is an algebraic diffeomorphism from X(ℝ) onto Y (ℝ), or in other words, it is a biregular map from X(ℝ) onto Y (ℝ) in the sense of [BCR]. Let us recall the notion of an algebraic diffeomorphism. Let X and Y be smooth projective real algebraic varieties. Then X(ℝ) and Y (ℝ) are compact manifolds, not necessarily connected or nonempty. Let

f : X(ℝ) ⟶ Y (ℝ)  (1)

be a map. Choose affine open subsets U of X and V of Y such that X(ℝ) ⊆ U(ℝ) and Y (ℝ) ⊆ V (ℝ). Since U and V are affine, we may assume that they are closed subvarieties of 𝔸^{m} and 𝔸^{n}, respectively. Then X(ℝ) is a closed submanifold of ℝ^{m}, and Y (ℝ) is a closed submanifold of ℝ^{n}. The map f in (1) is algebraic or regular if there are real polynomials p_{1}, …, p_{n}, q_{1}, …, q_{n} in the variables x_{1}, …, x_{m} such that none of the polynomials q_{1}, …, q_{n} vanishes on X(ℝ), and

f(x) = (\frac{p_{1}(x)}{q_{1}(x)}, …, \frac{p_{n}(x)}{q_{n}(x)})

for all x ∈ X(ℝ). One can check that the algebraicity of f depends neither on the choice of the affine open subsets U and V nor on the choice of the embeddings of U and V in affine space. Note that the algebraicity of f immediately implies that f is a C^{∞}-map. The map f in (1) is an algebraic diffeomorphism if f is algebraic, bijective, and f^{−1} is algebraic. Again let X and Y be rational models of a topological surface S. As observed above, if X and Y are isomorphic models of S, then there is an algebraic diffeomorphism

f: X(ℝ) ⟶ Y (ℝ) .

Conversely, if there is an algebraic diffeomorphism f: X(ℝ) ⟶ Y (ℝ), then X and Y are isomorphic models of S, as it follows from the well known Weak Factorization Theorem for birational maps between real algebraic surfaces(see [BPV, Theorem III.6.3] for the WFT over ℂ, from which the WFT over ℝ follows). Here we address the following question. Given a compact connected topological surface S, what is the number of nonisomorphic rational models of S? By Comessatti’s Theorem, an orientable surface of genus bigger than 1 does not have any rational model. It is known that the topological surfaces S^{2}, S^{1} × S^{1} and ℙ^{2}(ℝ) have exactly one rational model, up to isomorphism (see also Remark 3.2). Mangolte has shown that the same holds for the Klein bottle [Ma, Theorem 1.3] (see again Remark 3.2).

Mangolte asked how large n should be so that the n-fold connected sum of the real projective plane admits more than one rational model, up to isomorphism; see the comments following Theorem 1.3 in [Ma]. The following theorem shows that there is no such integer n.

Theorem 1.2. Let S be a compact connected real two-manifold.

1. If S is orientable of genus greater than 1, then S does not admit any rational model.

2. If S is either nonorientable, or it is diffeomorphic to one of S^{2} and S^{1} × S^{1}, then there is exactly one rational model of S, up to isomorphism. In other words, any two rational models of S are isomorphic.

Of course, statement 1 is nothing but Comessatti’s Theorem referred to above. Our proof of statement 2 is based on the Minimal Model Program for real algebraic surfaces developed by János Kollár in [Ko1]. Using this Program, we show that a rational model X of a nonorientable topological surface S is obtained from ℙ^{2} by blowing it up successively in a finite number of real points (Theorem 3.1). The next step of the proof of Theorem 1.2 involves showing that the model X is isomorphic to a model X′ obtained from ℙ^{2} by blowing up ℙ^{2} at real points P_{1}, …, P_{n} of ℙ^{2}. At that point, the proof of Theorem 1.2 would have been finished if we were able to prove that the group Diff_{alg}(ℙ^{2}(ℝ)) of algebraic diffeomorphisms of ℙ^{2}(ℝ) acts n-transitively on ℙ^{2}(ℝ). However, we were unable to prove such a statement. Nevertheless, a statement we were able to prove is the following.

Theorem 1.3. Let n be a natural integer. The group Diff_{alg}(S^{1} × S^{1}) acts n-transitively on S^{1} × S^{1}.

We conjecture, however, the following.

Conjecture 1.4. Let X be a smooth projective rational surface. Let n be a natural integer. Then the group Diff_{alg}(X(ℝ)) acts n-transitively on X(ℝ).^{1}

The only true evidence we have for the above conjecture is that it holds for X = ℙ^{1} × ℙ^{1} according to Theorem 1.3. Now, coming back to the idea of the proof of Theorem 1.2, we know that any rational model of S is isomorphic to one obtained from ℙ^{2} by blowing up ℙ^{2} at real points P_{1}, …, P_{n}. Since we have established n-transitivity of the group of algebraic diffeomorphisms of S^{1} × S^{1}, we need to realize X′ as a blowing-up of ℙ^{1} × ℙ^{1} at a finite number of real points. Let L be the real projective line in ℙ^{2} containing P_{1} and P_{2}. Applying a nontrivial algebraic diffeomorphism of ℙ^{2} into itself, if necessary, we may assume that P_{i} ∉ L for i ≥ 3. Then we can do the usual transformation of ℙ^{2} into ℙ^{1} × ℙ^{1} by first blowing-up P_{1} and P_{2}, and then contracting the strict transform of L. This realizes X′ as a surface obtained from ℙ^{1} ×ℙ^{1} by blowingup ℙ^{1} × ℙ^{1} at n − 1 distinct real points. Theorem 1.2 then follows from the (n − 1)-transitivity of Diff_{alg}(S^{1} × S^{1}). We will also address the question of uniqueness of geometrically rational models of a topological surface. By yet another result of Comessatti, a geometrically rational real surface X is rational if X(ℝ) is nonempty and connected. Therefore, Theorem 1.2 also holds when one replaces “rational models” by “geometrically rational models”. Since the set of real points of a geometrically rational surface is not neccesarily connected, it is natural to study geometrically rational models of not necessarily connected topological surfaces. We will show that such a surface has an infinite number of geometrically rational models, in general. The paper is organized as follows. In Section 2 we show that a real Hirzebruch surface is either isomorphic to the standard model ℙ^{1} × ℙ^{1} of the real torus S^{1} × S^{1}, or isomorphic to the standard model of the Klein bottle. The standard model of the Klein bottle is the real algebraic surface B_{P} (ℙ^{2}) obtained from the projective plane ℙ^{2} by blowing up one real point P. In Section 3, we use the Minimal Model Program for real algebraic surfaces in order to prove that any rational model of any topological surface is obtained by blowing up one of the following three real algebraic surfaces: ℙ^{2}, 𝕊^{2} and ℙ^{1} ×ℙ^{1} (Theorem 3.1). Here 𝕊^{2} is the real algebraic surface defined by the equation x^{2} + y^{2} + z^{2} = 1. As a consequence, we get new proofs of the known facts that the sphere, the torus, the real projective plane and the Klein bottle admit exactly one rational model, up to isomorphism of course. In Section 4 we prove a lemma that will have two applications. Firstly, it allows us to conclude the uniqueness of a rational model for the “next” topological surface, the 3-fold connected sum of the real projective plane. Secondly, it also allows us to conclude that a rational model of a nonorientable topological surface is isomorphic to a model obtained from ℙ^{2} by blowing up a finite number of distinct real points P_{1}, …, P_{n} of ℙ^{2}. In Section 5 we prove n-transitivity of the group of algebraic diffeomorphisms of the torus S^{1} × S^{1}. In Section 6 we construct a nontrivial algebraic diffeomorphism f of ℙ^{2}(ℝ) such that the real points f(P_{i}), for i = 3, …, n, are not on the real projective line through f(P_{1}) and f(P_{2}). In Section 7 we put all the pieces together and complete the proof of Theorem 1.2. In Section 8 we show by an example that the uniqueness does not hold for geometrically rational models of nonconnected topological surfaces.

1This conjecture is now proved [HM].

Documenta Mathematica 12 (2007) 549–567

Acknowledgement. The second author thanks the Tata Institute of Fundamental Research for its hospitality.

2 Real Hirzebruch surfaces

The set of real points of the rational real algebraic surface ℙ^{1} × ℙ^{1} is the torus S^{1}×S^{1}. We call this model the standard model of the real torus. Fix a real point O of the projective plane ℙ^{2}. The rational real algebraic surface B_{O}(ℙ^{2}) obtained from ℙ^{2} by blowing up the real point O is a model of the Klein bottle K. We call this model the standard model of the Klein bottle. Let d be a natural integer. Let 𝔽_{d} be the real Hirzebruch surface of degree d. Therefore, 𝔽_{d} is the compactification ℙ(O_{ℙ^{1}}(d)⊕O_{ℙ^{1}}) of the line bundle O_{ℙ^{1}}(d) over ℙ^{1}. Recall that the real algebraic surface 𝔽_{d} is isomorphic to 𝔽_{e} if and only if d = e. The restriction of the line bundle O_{ℙ^{1}}(d) to the set of real points ℙ^{1}(ℝ) of ℙ^{1} is topologically trivial if and only if d is even. Consequently, 𝔽_{d} is a rational model of the torus S^{1} × S^{1} if d is even, and it is a rational model of the Klein bottle K if d is odd (see [Si, Proposition VI.1.3] for a different proof). The following statement is probably well known, and is an easy consequence of known techniques (compare the proof of Theorem 6.1 in [Ma]). We have chosen to include the statement and a proof for two reasons: the statement is used in the proof of Theorem 3.1, and the idea of the proof turns out also to be useful in Lemma 4.1.

Proposition 2.1. Let d be a natural integer.

1. If d is even, then 𝔽_{d} is isomorphic to the standard model ℙ^{1} × ℙ^{1} of S^{1} × S^{1}.

2. If d is odd, then 𝔽_{d} is isomorphic to the standard model B_{O}(ℙ^{2}) of the Klein bottle K.

(All isomorphisms are in the sense of Definition 1.1.)

Proof. Observe that

• the real algebraic surface ℙ^{1} × ℙ^{1} is isomorphic to 𝔽_{0}, and

• that the real algebraic surface B_{O}(ℙ^{2}) is isomorphic to 𝔽_{1}.

Therefore, the proposition follows from the following lemma.

Lemma 2.2. Let d and e be natural integers. Then the two models 𝔽_{d} and 𝔽_{e} are isomorphic if and only if d ≡ e (mod 2).

Proof. Since the torus is not diffeomorphic to the Klein bottle, the rational models 𝔽_{d} and 𝔽_{e} are not isomorphic if d ≢ e (mod 2). Conversely, if d ≡ e (mod 2), then 𝔽_{d} and 𝔽_{e} are isomorphic models, as follows from the following lemma using induction.

Lemma 2.3. Let d be a natural integer. The two rational models 𝔽_{d} and 𝔽_{d+2} are isomorphic.

Proof. Let E be the section at infinity of 𝔽_{d}. The self-intersection of E is equal to −d. Choose nonreal complex conjugate points P and \overline{P} on E. Let F and \overline{F} be the fibers of the fibration of 𝔽_{d} over ℙ^{1} that contain P and \overline{P}, respectively. Let X be the real algebraic surface obtained from 𝔽_{d} by blowing up P and \overline{P}.

Denote again by E the strict transform of E in X. The self-intersection of E is equal to −d − 2. The strict transforms of F and \overline{F}, again denoted by F and \overline{F} respectively; they are disjoint smooth rational curves of self-intersection −1, and they do not intersect E. The real algebraic surface Y obtained from X by contracting F and \overline{F} is a smooth ℙ^{1}-bundle over ℙ^{1}. The image of E in Y has self-intersection −d − 2. It follows that Y is isomorphic to 𝔽_{d+2} as a real algebraic surface. Therefore, we conclude that 𝔽_{d} and 𝔽_{d+2} are isomorphic models.

3 Rational models

Let Y be a real algebraic surface. A real algebraic surface X is said to be obtained from Y by blowing up if there is a nonnegative integer n, and a sequence of morphisms

\xymatrix{X = X_{n} \ar[r]^{f_{n}} & X_{n−1} \ar[r]^{f_{n−1}} & ⋯ \ar[r]^{f_{1}} & X_{0} = Y ,}

such that for each i = 1, …, n, the morphism f_{i} is either the blow up of X_{i−1} at a real point, or it is the blow up of X_{i−1} at a pair of distinct complex conjugate points. The surface X is said to be obtained from Y by blowing up at real points only if for each i = 1, …, n, the morphism f_{i} is a blow up of X_{i−1} at a real point of X_{i−1}. One defines, similarly, the notion of a real algebraic surface obtained from Y by blowing up at nonreal points only. The real algebraic surface defined by the affine equation

x^{2} + y^{2} + z^{2} = 1

will be denoted by 𝕊^{2}. Its set of real points is the two-sphere S^{2}. The real Hirzebruch surface 𝔽_{1} will be simply denoted by 𝔽. Its set of real points is the Klein bottle K. Thanks to the Minimal Model Program for real algebraic surfaces due to János Kollár [Ko1, p. 206, Theorem 30], one has the following statement:

Theorem 3.1. Let S be a compact connected topological surface. Let X be a rational model of S.

1. If S is not orientable then X is isomorphic to a rational model of S obtained from ℙ^{2} by blowing up at real points only.

2. If S is orientable then X is isomorphic to 𝕊^{2} or ℙ^{1} × ℙ^{1}, as a model.

Proof. Apply the Minimal Model Program to X in order to obtain a sequence of blowing-ups as above, where Y is one of the following:

1. a minimal surface, 2. a conic bundle over a smooth real algebraic curve,

3. a Del Pezzo surface of degree 1 or 2, and

4. ℙ^{2} or 𝕊^{2}.

(See [Ko1, p. 206, Theorem 30].) The surface X being rational, we know that X is not birational to a minimal surface. This rules out the case of Y being a minimal surface. Since X(ℝ) is connected, it can be shown that X is not birational to a Del Pezzo surface of degree 1 or 2. Indeed, such Del Pezzo surfaces have disconnected sets of real points [Ko1, p. 207, Theorem 33(D)(c– d)]. This rules out the case of Y being a Del Pezzo surface of degree 1 or 2. It follows that

• either Y is a conic bundle, or

• Y is isomorphic to ℙ^{2}, or

• Y is isomorphic to 𝕊^{2}.

We will show that the statement of the theorem holds in all these three cases. If Y is isomorphic to ℙ^{2}, then Y (ℝ) is not orientable. Since X is obtained from Y by blowing up, it follows that X(ℝ) is not orientable either. Therefore, the surface S is not orientable, and also X is isomorphic to a rational model of S obtained from ℙ^{2} by blowing up. Moreover, it is easy to see that X is then isomorphic to a rational model of S obtained from ℙ^{2} by blowing up at real points only. This settles the case when Y is isomorphic to ℙ^{2}. If Y is isomorphic to 𝕊^{2}, then there are two cases to consider: (1) the case of S being orientable, (2) and the case of S being nonorientable. If S is orientable, then X(ℝ) is orientable too, and X is obtained from Y by blowing up at nonreal points only. It follows that X is isomorphic to 𝕊^{2} as a model. If S is nonorientable, then X(ℝ) is nonorientable too, and X is obtained from 𝕊^{2} by blowing up a nonempty set of real points. Therefore, the map X ⟶ Y factors through a blow up 𝕊̃^{2} of 𝕊^{2} at a real point. Now, 𝕊̃^{2} contains two smooth disjoint complex conjugated rational curves of self-intersection −1. When we contract them, we obtain a real algebraic surface isomorphic to ℙ^{2}. Therefore, X is obtained from ℙ^{2} by blowing up. It follows again that X is isomorphic to a rational model of S obtained from ℙ^{2} by blowing up at real points only. This settles the case when Y is isomorphic to 𝕊^{2}. The final case to consider is the one where Y is a conic bundle over a smooth real algebraic curve B. Since X is rational, B is rational. Moreover, B has real points because X has real points. Hence, the curve B is isomorphic to ℙ^{1}. The singular fibers of the the conic bundle Y over B are real, and moreover, the number of singular fibers is even. Since X(ℝ) is connected, we conclude that Y (ℝ) is connected too. it follows that the conic bundle Y over B has either no singular fibers or exactly 2 singular fibers. If it has exactly 2 singular fibers, then Y is isomorphic to 𝕊^{2} [Ko2, Lemma 3.2.4], a case we have already dealt with.

Therefore, we may assume that Y is a smooth ℙ^{1}-bundle over ℙ^{1}. Therefore, Y is a real Hirzebruch surface. By Proposition 2.1, we may suppose that Y = ℙ^{1} × ℙ^{1}, or that Y = 𝔽. Since 𝔽 is obtained from ℙ^{2} by blowing up one real point, the case Y = 𝔽 follows from the case of Y = ℙ^{2} which we have already dealt with above. Therefore, we may assume that Y = ℙ^{1} × ℙ^{1}. Again, two cases are to be considered: (1) the case of S being orientable, and (2) the case of S being nonorientable. If S is orientable, X(ℝ) is orientable, and X is obtained from Y by blowing up at non real points only. It follows that X is isomorphic as a model to ℙ^{1} × ℙ^{1}. If S is not orientable, X is obtained from Y by blowing up, at least, one real point. Since Y = ℙ^{1} × ℙ^{1}, a blow-up of Y at one real point is isomorphic to a blow-up of ℙ^{2} at two real points. We conclude again by the case of Y = ℙ^{2} dealt with above.

Note that Theorem 3.1 implies Comessatti’s Theorem referred to in the introduction, i.e., the statement to the effect that any orientable compact connected topological surface of genus greater than 1 does not admit a rational model (Theorem 1.2.1).

Remark 3.2. For sake of completeness let us show how Theorem 3.1 implies that the surfaces S^{2}, S^{1} × S^{1}, ℙ^{2}(ℝ) and the Klein bottle K admit exactly one rational model. First, this is clear for the orientable surfaces S^{2} and S^{1} × S^{1}. Let X be a rational model of ℙ^{2}(ℝ). From Theorem 3.1, we know that X is isomorphic to a rational model of ℙ^{2}(ℝ) obtained from ℙ^{2} by blowing up at real points only. Therefore, we may assume that X itself is obtained from ℙ^{2} by blowing up at real points only. Since X(ℝ) is diffeomorphic to ℙ^{2}(ℝ), it follows that X is isomorphic to ℙ^{2}. Thus any rational model of ℙ^{2}(ℝ) is isomorphic to ℙ^{2} as a model. Let X be a rational model of the Klein bottle K. Using Theorem 3.1 one may assume that X is a blowing up of ℙ^{2} at real points only. Since X(ℝ) is diffeomorphic to the 2-fold connected sum of ℙ^{2}(ℝ), the surface X is a blowing up of ℙ^{2} at exactly one real point. It follows that X is isomorphic to 𝔽. Therefore, any rational model of the Klein bottle K is isomorphic to 𝔽, as a model; compare with [Ma, Theorem 1.3].

One can wonder whether the case where S is a 3-fold connected sum of real projective planes can be treated similarly. The first difficulty is as follows. It is, a priori, not clear why the following two rational models of #^{3}ℙ^{2}(ℝ) are isomorphic. The first one is obtained from ℙ^{2} by blowing up two real points of ℙ^{2}. The second one is obtained by a successive blow-up of ℙ^{2}: first blow up ℙ^{2} at a real point, and then blow up a real point of the exceptional divisor. In the next section we prove that these two models are isomorphic.

4 The 3-fold connected sum of the real projective plane

We start with a lemma.

Lemma 4.1. Let P be a real point of ℙ^{2}, and let B_{P} (ℙ^{2}) be the surface obtained from ℙ^{2} by blowing up P. Let E be the exceptional divisor of B_{P} (ℙ^{2}) over P. Let L be any real projective line of ℙ^{2} not containing P. Consider L as a curve in B_{P} (ℙ^{2}). Then there is a birational map

f: B_{P} (ℙ^{2}) −−→ B_{P} (ℙ^{2})

whose restriction to the set of real points is an algebraic diffeomorphism such that f(L(ℝ)) = E(ℝ).

Proof. The real algebraic surface B_{P} (ℙ^{2}) is isomorphic to the real Hirzebruch surface 𝔽 = 𝔽_{1}, and any isomorphism between them takes the exceptional divisor of B_{P} (ℙ^{2}) to the section at infinity of the conic bundle 𝔽/ℙ^{1} = ℙ(O_{ℙ^{1}}(1) ⊕ O_{ℙ^{1}}). The line L in B_{P} (ℙ^{2}) is given by a unique section of O_{ℙ^{1}}(1) over ℙ^{1}; this section of O_{ℙ^{1}}(1) will also be denoted by L. We denote again by E the section at infinity of 𝔽. We have to show that there is a birational self-map f of 𝔽 such that the equality f(L(ℝ)) = E(ℝ) holds. Let R be a nonreal point of L. Let F be the fiber of the conic bundle 𝔽 passing through R. The blowing-up of 𝔽 at the pair of points R and \overline{R} is a real algebraic surface in which we can contract the strict transforms of F and \overline{F}. The real algebraic surface one obtains after these two contractions is again isomorphic to 𝔽. Therefore, we have a birational self-map f of 𝔽 whose restriction to the set of real points is an algebraic diffeomorphism. The image, by f, of the strict transform of L in 𝔽 has self-intersection −1. Therefore, the image, by f, of the strict transform of L coincides with E. In particular, we have f(L(ℝ)) = E(ℝ).

Proposition 4.2. Let S be the 3-fold connected sum of ℙ^{2}(ℝ). Then S admits exactly 1 rational model.

Proof. Fix two real points O_{1}, O_{2} of ℙ^{2}, and let B_{O_{1},O_{2}}(ℙ^{2}) be the real algebraic surface obtained from ℙ^{2} by blowing up O_{1} and O_{2}. The surface B_{O_{1},O_{2}}(ℙ^{2}) is a rational model of the 3-fold connected sum S of ℙ^{2}(ℝ). Let X be a rational model of S. We prove that X is isomorphic to B_{O_{1},O_{2}}(ℙ^{2}), as a model. By Theorem 3.1, we may assume that X is obtained from ℙ^{2} by blowing up real points only. Since X(ℝ) is diffeomorphic to a 3-fold connected sum of the real projective plane, the surface X is obtained from ℙ^{2} by blowing up twice real points. More precisely, there is a real point P of ℙ^{2} and a real point Q of the blow-up B_{P} (ℙ^{2}) of ℙ^{2} at P, such that X is isomorphic to the blow-up B_{Q}(B_{P} (ℙ^{2})) of B_{P} (ℙ^{2}) at Q. Choose any real projective line L in ℙ^{2} not containing P. Then, L is also a real curve in B_{P} (ℙ^{2}). We may assume that Q ∉ L. By Lemma 4.1, there is a birational map f from B_{P} (ℙ^{2}) into itself whose restriction to the set of real points is an algebraic diffeomorphism, and such that

f(L(ℝ)) = E(ℝ) ,

where E is the exceptional divisor on B_{P} (ℙ^{2}). Let R = f(Q). Then R ∉ E, and f induces a birational isomorphism

f̃: B_{Q}(B_{P} (ℙ^{2})) ⟶ B_{R}(B_{P} (ℙ^{2}))

whose restriction to the set of real points is an algebraic diffeomorphism. Since R ∉ E, the point R is a real point of ℙ^{2} distinct from P, and the blow-up B_{R}(B_{P} (ℙ^{2})) is equal to the blow up B_{P,R}(ℙ^{2}) of ℙ^{2} at the real points P, R of ℙ^{2}. It is clear that B_{P,R}(ℙ^{2}) is isomorphic to B_{O_{1},O_{2}}(ℙ^{2}). It follows that X is isomorphic to B_{O_{1},O_{2}}(ℙ^{2}) as rational models of the 3-fold connected sum of ℙ^{2}(ℝ).

Lemma 4.3. Let S be a nonorientable surface and let X be a rational model of S. Then, there are distinct real points P_{1}, …, P_{n} of ℙ^{2} such that X is isomorphic to the blowing-up of ℙ^{2} at P_{1}, …, P_{n}, as a model.

Proof. By Theorem 3.1, we may assume that X is obtained from ℙ^{2} by blowing up at real points only. Let

\xymatrix{X = X_{n} \ar[r]^{f_{n}} & X_{n−1} \ar[r]^{f_{n−1}} & ⋯ \ar[r]^{f_{1}} & X_{0} = ℙ^{2}.}  (2)

be a sequence of blowing ups, where for each i = 1, …, n, the map f_{i} is a blowing up of X_{i−1} at a real point P_{i} of X_{i−1}. To a sequence of blowing-ups as in (2) is associated a forest F of trees. The vertices of F are the centers P_{i} of the blow-ups f_{i}. For i > j, there is an edge between the points P_{i} and P_{j} in F if

• the composition f_{j+1} ∘ ⋯ ∘ f_{i−1} is an isomorphism at a neighborhood of P_{i}, and

• maps P_{i} to a point belonging to the exceptional divisor f_{j}^{−1}(P_{j}) of P_{j} in X_{j}.

Let ℓ be the sum of the lengths of the trees belonging to F. We will show by induction on ℓ that X is isomorphic, as a model, to the blowing-up of ℙ^{2} at a finite number of distinct real points of ℙ^{2}. This is obvious if ℓ = 0. If ℓ ≠ 0, let P_{j} be the root of a tree of nonzero length, and let P_{i} be the vertex of that tree lying immediately above P_{j}. By changing the order of the blowing-ups f_{i}, we may assume that j = 1 and i = 2. Choose a real projective line L in ℙ^{2} which does not contain any of the roots of the trees of F. By Lemma 4.1, there is a birational map g_{1} from X_{1} = B_{P_{1}}(ℙ^{2}) into itself whose restriction to the set of real points is an algebraic diffeomorphism and satisfies the condition g_{1}(L(ℝ)) = E(ℝ), where E is the exceptional divisor of X_{1}. Put X′_{0} = ℙ^{2}, X′_{1} = X_{1}, and f′_{1} = f_{1}. We consider g_{1} as a birational map from X_{1} into X′_{1}. Put P′_{2} = g_{1}(P_{2}). Let X′_{2} be the blowing-up of X′_{1} at P′_{2}, and let

f′_{2} : X′_{2} ⟶ X′_{1}

be the blowing-up morphism. Then, g_{1} induces a birational map g_{2} from X_{2} into X′_{2} which is an algebraic diffeomorphism on the set of real points. By iterating this construction, one gets a sequence of blowing ups

f′_{i}: X′_{i} ⟶ X′_{i−1} ,

where i = 1, …, n, and birational morphisms g_{i} from X_{i} into X′_{i} whose restrictions to the sets of real points are algebraic diffeomorphisms. In particular, the rational models X = X_{n} and X′ = X′_{n} of S are isomorphic. Let F′ be the forest of the trees of centers of X′. Then the sum of the lengths ℓ′ of the trees of F′ is equal to ℓ − 1. Indeed, one obtains F′ from F by replacing the tree T of F rooted at P_{1} by the disjoint union of the tree T \ P_{1} and the tree {P_{1}}. This follows from the fact that P′_{2} does not belong to the exceptional divisor of f′_{1}, and that, no root of the other trees of F belongs to the exceptional divisor of f′_{1} either.

As observed in the Introduction, if we are able to prove the n-transitivity of the action of the group Diff_{alg}(ℙ^{2}(ℝ)) on ℙ^{2}(ℝ), then the statement of Theorem 1.2 would follow from Lemma 4.3. However, we did not succeed in proving so. Nevertheless, we will prove the n-transitivity of Diff(S^{1} × S^{1}), which is the subject of the next section. Now that we know that the topological surfaces S^{1}, S^{1} × S^{1} and #^{n}ℙ^{2}(ℝ), for n = 1, 2, 3, admit exactly one rational model, one may also wonder whether Lemma 4.3 allows us to tackle the “next” surface, which is the 4-fold connected sum of ℙ^{2}(ℝ). We note that Theorem 1.2 and Lemma 4.3 imply that a rational model of such a surface is isomorphic to a surface obtained from ℙ^{2} by blowing up 3 distinct real points. However, it it is not clear why the two surfaces of the following type are isomorphic as models. Take three distinct non-collinear real points P_{1}, P_{2}, P_{3}, and three distinct collinear real points Q_{1}, Q_{2}, Q_{3} of ℙ^{2}. Then the surfaces X = B_{P_{1},P_{2},P_{3}}(ℙ^{2}) and Y = B_{Q_{1},Q_{2},Q_{3}}(ℙ^{2}) are rational models of #^{4}ℙ^{2}(ℝ) (the 4-fold connected sum of ℙ^{2}(ℝ)), but it is not clear why they should be isomorphic. One really seems to need some nontrivial algebraic diffeomorphism of ℙ^{2}(ℝ), that maps P_{i} to Q_{i} for i = 1, 2, 3, in order to show that X and Y are isomorphic models. We will come back to this in Section 6 (Lemma 6.1).

5 Algebraic diffeomorphisms of S^{1} × S^{1} and n-transitivity

The following statement is a variation on classical polynomial interpolation.

Lemma 5.1. Let m be a positive integer. Let x_{1}, …, x_{m} be distinct real numbers, and let y_{1} …, y_{m} be positive real numbers. Then there is a real polynomial p of degree 2m that does not have real zeros, and satisfies the condition p(x_{i}) = y_{i} for all i.

Proof. Set

p(ζ) := ∑_{j=1}^{m} ∏_{k≠j} \frac{(ζ − x_{k})^{2}}{(x_{j} − x_{k})^{2}} · y_{j}.

Then p is of degree 2m, and p does not have real zeros. Furthermore, we have p(x_{i}) = y_{i} for all i.

Corollary 5.2. Let m be a positive integer. Let x_{1}, …, x_{m} be distinct real numbers, and let y_{1} …, y_{m}, z_{1}, …, z_{m} be positive real numbers. Then there are real polynomials p and q without any real zeros such that degree(p) = degree(q), and

\frac{p(x_{i})}{q(x_{i})} = \frac{y_{i}}{z_{i}}

for all 1 ≤ i ≤ m.

The interest in the rational functions p/q of the above type lies in the following fact.

Lemma 5.3. Let p and q be two real polynomials of same degree that do not have any real zeros. Define the rational map f: ℙ^{1} × ℙ^{1} −−→ ℙ^{1} × ℙ^{1} by

f(x, y) = (x, \frac{p(x)}{q(x)} · y) .

Then f is a birational map of ℙ^{1} × ℙ^{1} into itself whose restriction to the set of real points is an algebraic diffeomorphism.

Theorem 5.4. Let n be a natural integer. The group Diff_{alg}(ℙ^{1} × ℙ^{1}) acts n-transitively on ℙ^{1}(ℝ) × ℙ^{1}(ℝ).

Proof. Choose n distinct real points P_{1}, …, P_{n} and n distinct real points Q_{1}, …, Q_{n} of ℙ^{1} × ℙ^{1}. We need to show that there is a birational map f from ℙ^{1} × ℙ^{1} into itself, whose restriction to (ℙ^{1} × ℙ^{1})(ℝ) is an algebraic diffeomorphism, such that f(P_{i}) = Q_{i}, for i = 1, …, n. First of all, we may assume that P_{1}, …, P_{n}, Q_{1}, …, Q_{n} are contained in the first open quadrant of ℙ^{1}(ℝ) × ℙ^{1}(ℝ). In other words, the coordinates of P_{i} and Q_{i} are strictly positive real numbers. Moreover, it suffices to prove the statement for the case where Q_{i} = (i, i) for all i. By the hypothesis above, there are positive real numbers x_{i}, y_{i} such that P_{i} = (x_{i}, y_{i}) for all i. By Corollary 5.2, there are real polynomials p and q without any real zeros such that degree(p) = degree(q), and such that the real numbers

\frac{p(x_{i})}{q(x_{i})} · y_{i}

are positive and distinct for all i. Define f: ℙ^{1} × ℙ^{1} −−→ ℙ^{1} × ℙ^{1} by

f(x, y) := (x, \frac{p(x)}{q(x)} · y) .

By Lemma 5.3, f is birational, and its restriction to (ℙ^{1} × ℙ^{1})(ℝ) is an algebraic diffeomorphism. By construction, the points f(P_{i}) have distinct second coordinates. Therefore, replacing P_{i} by f(P_{i}) if necessary, we may assume that the points P_{i} have distinct second coordinates, which implies that y_{1}, …, y_{m} are distinct positive real numbers. By Corollary 5.2, there are real polynomials p, q without any real zeros such that degree(p) = degree(q), and

\frac{p(y_{i})}{q(y_{i})} · x_{i} = i.

Define f: ℙ^{1} × ℙ^{1} −−→ ℙ^{1} × ℙ^{1} by

f(x, y) = (\frac{p(y)}{q(y)} · x, y) .

By Lemma 5.3, f is birational and its restriction to the set of real points is an algebraic diffeomorphism. By construction, one has f(P_{i}) = (i, y_{i}) for all i. Therefore, we may assume that P_{i} = (i, y_{i}) for all i. Now, again by Corollary 5.2, there are two real polynomials p and q without any real zeros such that degree(p) = degree(q), and

\frac{p(i)}{q(i)} · y_{i} = i

for all i. Define f: ℙ^{1} × ℙ^{1} −−→ ℙ^{1} × ℙ^{1} by

f(x, y) = (x, \frac{p(x)}{q(x)} · y) .

By Lemma 5.3, f is birational, and its restriction to the set of real points is an algebraic diffeomorphism. By construction f(P_{i}) = Q_{i} for all i.

Remark 5.5. One may wonder whether Theorem 5.4 implies that the group Diff_{alg}(ℙ^{2}(ℝ)) acts n-transitively on ℙ^{2}(ℝ). We will explain the implication of Theorem 5.4 in that direction. Let P_{1}, …, P_{n} be distinct real points of ℙ^{2}, and let Q_{1}, …, Q_{n} be distinct real points of ℙ^{2}. Choose a real projective line L in ℙ^{2} not containing any of the points P_{1}, …, P_{n}, Q_{1}, …, Q_{n}. Let O_{1} and O_{2} be distinct real points of L. Identify ℙ^{1} × ℙ^{1} with the surface obtained from ℙ^{2} by, first, blowing up O_{1}, O_{2} and, then, contracting the strict transform of L. Denote by E_{1} and E_{2} the images of the exceptional divisors over O_{1} and O_{2} in ℙ^{1} × ℙ^{1}, respectively. We denote again by P_{1}, …, P_{n}, Q_{1}, …, Q_{n} the real points of ℙ^{1} × ℙ^{1} that correspond to the real points P_{1}, …, P_{n}, Q_{1}, …, Q_{n} of ℙ^{2}. Now, the construction in the proof of Theorem 5.4 gives rise to a birational map f from ℙ^{1} × ℙ^{1} into itself which is an algebraic diffeomorphism on (ℙ^{1} × ℙ^{1})(ℝ) and which maps P_{i} onto Q_{i}, for i = 1, …, n. Moreover, if one carries out carefully the construction of f, one has that f(E_{1}(ℝ)) = E_{1}(ℝ) and f(E_{2}(ℝ)) = E_{2}(ℝ) and that the real intersection point O of E_{1} and E_{2} in ℙ^{1} × ℙ^{1} is a fixed point of f. Note that one obtains back ℙ^{2} from ℙ^{1} × ℙ^{1} by blowing up O and contracting the strict transforms of E_{1} and E_{2}. Therefore, the birational map f of ℙ^{1} × ℙ^{1} into itself induces a birational map g of ℙ^{2} into itself. Moreover, g(P_{i}) = Q_{i}. One may think that g is an algebraic diffeomorphism on ℙ^{2}(ℝ). However, the restriction of g to the set of real points is not necessarily an algebraic diffeomorphism! In fact, g is an algebraic diffeomorphism on ℙ^{2}(ℝ) \ {O_{1}, O_{2}}. The restriction of g to ℙ^{2}(ℝ) \ {O_{1}, O_{2}} does admit a continuous extension g̃ to ℙ^{2}(ℝ), and g̃ is obviously a homeomorphism. One may call g̃ an algebraic homeomorphism, but g̃ is not necessarily an algebraic diffeomorphism. It is not difficult to find explicit examples of such algebraic homeomorphisms that are not diffeomorphisms. That is the reason why we do not claim to have proven n-transitivity of Diff_{alg}(ℙ^{2}(ℝ). The only statement about ℙ^{2}(ℝ) the above arguments prove is the n-transitivity of the group Homeo_{alg}(ℙ^{2}(ℝ)) of algebraic homeomorphisms.

6 A nontrivial algebraic diffeomorphism of ℙ^{2}(ℝ)

The nontrivial diffeomorphisms we have in mind are the following. They have been studied in another recent paper as well [RV]. Let Q_{1}, …, Q_{6} be six pairwise distinct complex points of ℙ^{2} satisfying the following conditions:

1. the subset {Q_{1}, ⋯, Q_{6}} is closed under complex conjugation,

2. the subset {Q_{1}, ⋯, Q_{6}} does not lie on a complex conic,

3. the complex conic passing through any 5 of these six points is nonsingular.

Denote by C_{1}, …, C_{6} the nonsingular complex conics one thus obtains. These conics are pairwise complex conjugate. Consider the real Cremona transformation f = f_{Q} of ℙ^{2} defined by first blowing-up ℙ^{2} at Q_{1}, …, Q_{6} and then contracting the strict transforms of C_{1}, …, C_{6}. Let R_{1}, …, R_{6} denote the points of ℙ^{2} that correspond to the contractions of the conics C_{1}, …, C_{6}. The restriction to ℙ^{2}(ℝ) of the birational map f from ℙ^{2} into itself is obviously an algebraic diffeomorphism. The Cremona transformation f maps a real projective line, not containing any of the points Q_{1}, …, Q_{6}, to a real rational quintic curve having 6 distinct nonreal double points at the points R_{1}, …, R_{6}. Moreover, it maps a real rational quintic curve in ℙ^{2} having double points at Q_{1}, …, Q_{6} to a real projective line in ℙ^{2} that does not contain any of the points R_{1}, …, R_{6}. Observe that the inverse of the Cremona transformation f_{Q} is the Cremona transformation f_{R}. It follows that f = f_{Q} induces a bijection from the set of real rational quintics having double points at Q_{1}, …, Q_{6} onto the set of real projective lines in ℙ^{2} that do not contain any of R_{1}, …, R_{6}.

This section is devoted to the proof of following lemma.

Lemma 6.1. Let n be a natural integer bigger than 1. Let P_{1}, …, P_{n} be distinct real points of ℙ^{2}. Then there is a birational map of ℙ^{2} into itself, whose restriction to the set of real points is an algebraic diffeomorphism, such that the image points f(P_{3}), …, f(P_{n}) are not contained in the real projective line through f(P_{1}) and f(P_{2}).

Proof. Choose complex points Q_{1}, …, Q_{6} of ℙ^{2} as above. As observed before, the Cremona transformation f = f_{Q} induces a bijection from the set of real rational quintic curves having double points at Q_{1}, …, Q_{6} onto the set of real projective lines of ℙ^{2} not containing any of the above points R_{1}, …, R_{n}. In particular, there is a real rational quintic curve C in ℙ^{2} having 6 nonreal double points at Q_{1}, …, Q_{6}. We show that there is a real projectively linear transformation α of ℙ^{2} such that α(C) contains P_{1} and P_{2}, and does not contain any of the points P_{3}, …, P_{n}. The Cremona transformation f_{α(Q)} will then be a birational map of ℙ^{2} into itself that has the required properties. First of all, let us prove that there is α ∈ PGL_{3}(ℝ) such that P_{1}, P_{2} ∈ α(C). This is easy. Since C is a quintic curve, C(ℝ) is infinite. In particular, C contains two distinct real points. It follows that there is α ∈ PGL_{3}(ℝ) such that P_{1}, P_{2} ∈ α(C). Replacing C by α(C) if necessary, we may suppose that P_{1}, P_{2} ∈ C. We need to show that there is α ∈ PGL_{3}(ℝ) such that α(P_{1}) = P_{1}, α(P_{2}) = P_{2} and α(C) does not contain any of the points P_{3}, …, P_{n}. To prove the existence of α by contradiction, assume that there is no such automorphism of ℙ^{2}. Therefore, for all α ∈ PGL_{3}(ℝ) having P_{1} and P_{2} as fixed points, the image α(C) contains at least one of the points of P_{3}, …, P_{n}. Let G be the stabilizer of the pair (P_{1}, P_{2}) for the diagonal action of PGL_{3} on ℙ^{2} × ℙ^{2}. It is easy to see that G is a geometrically irreducible real algebraic group. Let

ρ: C × G ⟶ ℙ^{2}

be the morphism defined by ρ(P, α) = α(P). Let

X_{i} := ρ^{−1}(P_{i})

be the inverse image, where i = 3, …, n. Therefore, X_{i} is a real algebraic subvariety of C × G. By hypothesis, for every α ∈ G(ℝ), there is an integer i such that α(C) contains P_{i}. Denoting by p the projection on the second factor from C × G onto G, this means that

⋃_{i=3}^{n} p(X_{i}(ℝ)) = G(ℝ).

Since G(ℝ) is irreducible, there is an integer i_{0} ∈ [3, n] such that the semialgebraic subset p(X_{i_{0}}(ℝ)) is Zariski dense in G(ℝ). Since G is irreducible and p is proper, one has p(X_{i_{0}}) = G. In particular, P_{i_{0}} ∈ α(C) for all α ∈ G(ℂ). To put it otherwise, α^{−1}(P_{i_{0}}) ∈ C for all α ∈ G(ℂ), which means that the orbit of P_{i_{0}} under the action of G is contained in C. In particular, the dimension of the orbit of P_{i_{0}} is at most one. It follows that P_{1}, P_{2} and P_{i_{0}} are collinear. Let L be the projective line through P_{1}, P_{2}. Then the orbit of P_{i_{0}} coincides with L \ {P_{1}, P_{2}}. It now follows that L ⊆ C. This is in contradiction with the fact that C is irreducible.

7 Proof of Theorem 1.2.2

Let S be a topological surface, either nonorientable or of genus less than 2. We need to show that any two rational models of S are isomorphic. By Remark 3.2, we may assume that S is the n-fold connected sum of ℙ^{2}(ℝ), where n ≥ 3. Let O_{1}, …, O_{n−2} be fixed pairwise distinct real points of ℙ^{1} × ℙ^{1}, and let B_{n−2}(ℙ^{1} × ℙ^{1}) be the surface obtained from ℙ^{1} × ℙ^{1} by blowing up the points O_{1}, …, O_{n−2}. It is clear that B_{n−2}(ℙ^{1} × ℙ^{1}) is a rational model of S. Now, it suffices to show that any rational model of S is isomorphic to B_{n−2}(ℙ^{1}× ℙ^{1}), as a model. Let X be any rational model of S. By Lemma 4.3, we may assume that there are distinct real points P_{1}, …, P_{m} of ℙ^{2} such that X is the surface obtained from ℙ^{2} by blowing up P_{1}, …, P_{m}. Since X is a rational model of an n-fold connected sum of ℙ^{2}(ℝ), one has m = n − 1. In particular, m ≥ 2. By Lemma 6.1, we may assume that the points P_{3}, …, P_{m} are not contained in the real projective line L through P_{1} and P_{2}. The blow-up morphism X ⟶ ℙ^{2} factors through the blow up ℙ̃^{2} = B_{P_{1},P_{2}}(ℙ^{2}). The strict transform L̃ of L has self-intersection −1 in ℙ̃^{2}. If we contract L̃, then we obtain a surface isomorphic to ℙ^{1} × ℙ^{1}. Therefore, X is isomorphic to a model obtained from ℙ^{1} × ℙ^{1} by blowing up m − 1 = n − 2 distinct real points of ℙ^{1} × ℙ^{1}. It follows from Theorem 5.4 that X is isomorphic to B_{n−2}(ℙ^{1} × ℙ^{1}).

8 Geometrically rational models

Recall that a nonsingular projective real algebraic surface X is geometrically rational if the complex surface X_{ℂ} = X ×_{ℝ} ℂ is rational. Comessatti showed that, if X is a geometrically rational real algebraic surface with X(ℝ) connected, then X is rational; see Theorem IV of [Co1] and the remarks thereafter (see also [Si, Corollary VI.6.5]). Therefore, the main result, namely Theorem 1.2, also applies to geometrically rational models. More precisely, we have the following consequence.

Corollary 8.1. Let S be a compact connected real two-manifold.

1. If S is orientable and the genus of S is greater than 1, then S does not admit a geometrically rational real algebraic model.

2. If S is either nonorientable, or it is diffeomorphic to one of S^{2} and S^{1} × S^{1}, then there is exactly one geometrically rational model of S, up to isomorphism. In other words, any two geometrically rational models of S are isomorphic.

Now, the interesting aspect about geometrically rational real surfaces is that their set of real points can have an arbitrary finite number of connected components. More precisely, Comessati proved the following statement [Co2, p. 263 and further] (see also [Si, Proposition VI.6.1]).

Theorem 8.2. Let X be a geometrically rational real algebraic surface such that X(ℝ) is not connected. Then each connected component of X(ℝ) is either nonorientable or diffeomorphic to S^{2}. Conversely, if S is a nonconnected compact topological surface each of whose connected components is either nonorientable or diffeomorphic to S^{2}, then there is a geometrically rational real algebraic surface X such that X(ℝ) is diffeomorphic to S.

Let S be a nonconnected topological surface. One may wonder whether the geometrically rational model of S whose existence is claimed above, is unique up to isomorphism of models. The answer is negative, as shown by the following example.

Example 8.3. Let S be the disjoint union of a real projective plane and 4 copies of S^{2}. Then, any minimal real Del Pezzo surface of degree 1 is a geometrically rational model of S [Ko2, Theorem 2.2(D)]. Minimal real Del Pezzo surfaces of degree 1 are rigid; this means that any birational map between two minimal real Del Pezzo surfaces of degree 1 is an isomorphism of real algebraic surfaces [Is, Theorem 1.6]. Now, the set of isomorphism classes of minimal real Del Pezzo surfaces of degree 1 is in one-to-one correspondence with an open dense subset of the quotient ℙ^{2}(ℝ)^{8}/PGL_{3}(ℝ) for the diagonal action of the group PGL_{3}(ℝ). It follows that the topological surface S admits a 8-dimensional continuous family of nonisomorphic geometrically rational models. In particular, the number of nonisomorphic geometrically rational models of S is infinite.

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Indranil Biswas School of Mathematics Tata Institute of Fundamental Research, Homi Bhabha Road Mumbai 400005 India indranil@math.tifr.res.in Johannes Huisman Département de Mathématiques Laboratoire CNRS UMR 6205 Université de Bretagne Occidentale 6 avenue Victor Le Gorgeu CS 93837 29238 Brest cedex 3 France johannes.huisman@univ-brest.fr

http://pageperso.univ-brest.fr/∼huisman

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Documenta Mathematica 12 (2007)