F R Éd

Éric Mangolte
email: frederic.mangolte@univ-savoie.fr

TOPOLOGICAL TYPES OF REAL REGULAR JACOBIAN ELLIPTIC SURFACES FR ÉD ÉRIC BIHAN AND

Keywords: elliptic surface, real algebraic variety, trigonal curve

We present the topological classification

Introduction

By a real algebraic variety X, we mean a pair (X, σ) where X is a complex algebraic variety and σ an antiholomorphic involution acting on X. The involution σ is often called a real structure and the fixed point set X σ is called the real part of X and is denoted by X(R). From now on, a surface will be assumed to be a nonsingular surface and a topological type of real surfaces will be a class of surfaces with homeomorphic real parts.

Our main purpose is to make a step towards the Enriques-Kodaira classification of real algebraic surfaces. More precisely, we are interested in the classification of the topological types of real algebraic surfaces in a given complex deformation family. The case of real rational surfaces goes back to Comessatti, [START_REF] Comessatti | Sulla connessione delle superfizie razionali reali[END_REF] as well as the case of real abelian varieties [START_REF] Comessatti | Reele Fragen in der algebraischen Geometrie[END_REF], (see also [23, Chap. III, IV]). For real ruled surfaces, see [START_REF] Silhol | Real Algebraic Surfaces[END_REF]Chap. V]. The case of K3 surfaces is due to Nikulin and Kharlamov [START_REF] Kharlamov | The topological type of non singular surfaces in RP 3 of degree 4, English transl[END_REF][START_REF] Nikulin | Integral symmetric bilinear forms and some of their applications[END_REF]. More recently, the classification of topological types of real Enriques surfaces has been obtained by Degtyarev and Kharlamov [START_REF] Degtyarev | Topological classification of real Enriques surfaces[END_REF], while the classification for real hyperelliptic surfaces has been achieved by Catanese and Frediani [START_REF] Catanese | Real hyperelliptic surfaces and the orbifold fundamental group[END_REF].

The remaining class of real algebraic surfaces of special type is made by the so-called properly elliptic surfaces. In this direction there is the fundamental classification of the real singular fibres of an elliptic fibration by R. Silhol [START_REF] Silhol | Real algebraic surfaces with rational or elliptic fiberings[END_REF] and some partial results about the global classification in [START_REF] Mangolte | Surfaces elliptiques réelles et inégalité de Ragsdale-Viro[END_REF] and [START_REF] Akriche | Nombres de Betti des surfaces elliptiques réelles, Beitrage zur Algebra und Geometrie[END_REF].

A real elliptic surface will be a morphism π : X → P 1 defined over R, where X is a real algebraic surface such that for all but finitely many points u ∈ P 1 , the fibre X u = π -1 (u) is a nonsingular curve of genus one. When π admits at least one singular fibre, the surface X is regular, which means that H 1 (X, O X ) = {0}. The elliptic fibration π : X -→ P 1 will be called relatively minimal if no fibre of π contains an exceptional curve of the first kind. When a (real) relatively minimal elliptic surface π : X → P 1 admits a (real) section s : P 1 → X we call X a (real) jacobian elliptic surface.

Over C, two regular relatively minimal elliptic surfaces with no multiple fibres are deformation equivalent if and only if their holomorphic Euler characteristics are equal, see [START_REF] Kas | On the deformation types of regular elliptic surfaces[END_REF].

The main goal of this paper is to present the classification of the topological types of real regular jacobian elliptic surfaces in each complex deformation family of regular jacobian elliptic surfaces.

In order to describe the topological types, we denote by S g the smooth orientable surface of genus g, by S the two dimensional sphere (S = S 0 ) and by V q the non orientable surface of Euler characteristic 2q (V q is diffeomorphic to the connected sum of q copies of P 2 (R)). In what follows, the disjoint union of two surfaces A and B is denoted by A ⊔ B while the disjoint union of a homeomorphic copies of A is denoted by a A.

An abstract Morse simplification on a topological type is a transformation which decreases the total Betti number by 2. There are two kinds of Morse simplifications:

• removing one spherical component, S → ∅,

• contracting one handle S g+1 → S g or V q+2 → V q . A topological type of a real regular jacobian elliptic surface is called extremal if it cannot be obtained by a Morse simplification from the topological type of another real regular jacobian elliptic surface in the same complex deformation family (i.e. with the same holomorphic Euler characteristic).

Theorem 1.1. Let k ≥ 1 be an integer. The extremal topological types of real regular jacobian elliptic surfaces with holomorphic Euler characteristic k are:

(1) M -surfaces,

a = k + 4λ -1, l = 5k -4λ, λ = 0, 1, . . . , k • S l ⊔ a S, k even or • V 2l ⊔ aS, k odd.
(

) (M -2)-surfaces, a = k + 4λ, l = 5k -4λ -3, λ = 0, 1, . . . , k -1 • S l ⊔ a S, k even or • V 2l ⊔ aS, k odd. (3) • S 1 ⊔ S 1 , k even or • V 2 ⊔ V 2 , k odd. 2
Any topological type of a real regular jacobian elliptic surface with holomorphic Euler characteristic k is the result of a sequence of Morse simplifications from one of the previous extremal types.

Conversely, any topological type corresponding to a surface with first Betti number ≥ 2 and which can be obtained by a sequence of Morse simplifications from one of the previous extremal topological types is the topological type of a real regular jacobian elliptic surface X with k = χ(O X ).

Note that in the definition of a jacobian surface, we supposed that the elliptic fibration was relatively minimal. In fact, the analogue of Theorem 1.1 without this hypothesis, that is, the classification of topological types of real regular elliptic surfaces with a real section and contained in a given complex deformation family, can be deduced directly from Theorem 1.1. Indeed, over C, two regular elliptic surfaces with no multiple fibres are deformation equivalent if and only if their holomorphic Euler characteristics are equal and their canonical classes have the same degree. To realize a topological type in one complex family, say k = χ(O X ) and K 2 X = -m < 0, take any real regular jacobian elliptic surface Y with holomorphic Euler characteristic k, then by definition, K 2 Y = 0. Let X be the blow-up of m points which are globally fixed by the real structure. Then K X = -m. Each blowup centered in a real point gives rise to a connected sum with a real projective plane. Conversely, any topological type of a real regular elliptic surface with a real section can be obtained in this way.

The rest of the paper is devoted to the proof of Theorem 1.1 (see Subsection 4.3).

Let π : X -→ P 1 be a real regular jacobian elliptic surface. Denote by k = χ(O X ) the holomorphic Euler characteristic of X. The Betti numbers of X(R) are subject to prohibitions. Classical inequalities and congruences on real regular algebraic surfaces (see [START_REF] Degtyarev | Topological properties of real algebraic varieties: du côté de chez Rokhlin[END_REF] for a survey) yield a finite list for the allowed pairs of topological Euler characteristic χ(X(R)) and total Betti number h * (X(R)) with coefficients in Z/2, see Section 2. The resulting list is shortened thanks to the following inequalities which arise essentially because the surface is elliptic. Namely the first mod 2 Betti number h 1 (X(R)) = rk (H 1 (X(R), Z/2)) and the number of connected components are bounded by

(1.2) h 1 (X(R)) ≤ 10k , #X(R) ≤ 5k .
The upper bound on the first Betti number is due to a more general result of V. Kharlamov (see Theorem 2.7). The upper bound on the number of connected components is due to M. Akriche and the second author, see [START_REF] Akriche | Nombres de Betti des surfaces elliptiques réelles, Beitrage zur Algebra und Geometrie[END_REF].

Using the general prohibitions on real algebraic surfaces and the two specific inequalities (1.2), we draw for each k, the diagram of the allowed pairs (χ, h * ), see Figure 1. The existence of a real section imposes strong conditions on the real part of X. Namely, X(R) = ∅, one connected component of X(R) has Euler characteristic ≤ 0 and other components are diffeomorphic to spheres, unless X(R) is the disjoint union of two tori (if k even) or two Klein bottles (if k odd), see Proposition 2.9. Hence, except for the pair (0, 8), there is a one-to-one correspondence between allowed pairs (χ, h * ) and allowed topological types. The list given in Theorem 1.1 is exactly the list of topological types obtained from the diagram in Figure 1 using this correspondence.

Conversely, for each k ≥ 1 and for each topological type listed in Theorem 1.1, we prove the existence of a real regular jacobian elliptic surface with the desired topological type. These surfaces are obtained in the following way.

Fix an integer k ≥ 1 and let R = F 2k be the Hirzebruch surface of degree 2k endowed with the real ruling π : R → P 1 . Let E be the unique nonsingular algebraic curve of R with nonzero negative selfintersection. By unicity, E is defined over R. Consider a nonsingular real algebraic curve C of bidegree (3, 0) on R (see Section 3 for the definition of bidegree). Such curves are often called trigonal curves. Let ρ : X → R be the double covering ramified over the union C ∪ E which is of even bidegree (4, -2k). Standard calculation on double coverings yields χ(O X ) = k, and it is an easy exercise to check that π = ρ • π is an elliptic fibration. In fact, X is a jacobian elliptic surface.

Since the curve C is real, the surface X becomes real with respect to the two complex conjugations which are lifts of the one on R (they are interchanged by the covering involution). The topological type of the elliptic surface with respect to either complex conjugations is determined by the real scheme of C, i.e. the pair (F (R), C(R)) up to homeomorphism, and the parity of k (see Lemma 3.3). A Morse simplification for either real elliptic surfaces corresponds then to collapsing one oval of the curve C. Hence, the list in Theorem 1.1 results in a list of real schemes of real trigonal curves on R, where extremal ones are those which cannot be obtained from another by collapsing one oval.

Trigonal curves which produce the exceptional topological types of two tori (if k is even) or two Klein bottles (if k is odd) are easy to construct, see Example 3.2. The other extremal cases are constructed by means of the combinatorial patchworking. The combinatorial patchworking is a combinatorial version of the Viro method, which is a powerful construction method of real algebraic varieties with prescribed topology, see [START_REF] Viro | Gluing of algebraic hypersurfaces, smoothing of singularities and construction of curves[END_REF][START_REF] Viro | Gluing of plane algebraic curves and construction of curves of degree 6 and 7[END_REF][START_REF] Viro | Patchworking real algebraic varieties[END_REF]. For the remaining real schemes, we use the fact that ovals of trigonal curves can be collapsed independently 1 . This can be shown using the theory of dessins d'enfants. Though this statement can be easily deduced from [START_REF] Orevkov | Riemann existence theorem and construction of real algebraic curves[END_REF], we give here a detailed proof for the reader's convenience (see Proposition 4.3). The emergence of dessins d'enfants in real algebraic geometry is recent [START_REF] Natanzon | Topological classification of generic real rational functions[END_REF][START_REF] Orevkov | Riemann existence theorem and construction of real algebraic curves[END_REF] (see also, 1 It is worth noting that this is not true for other types of curves, e.g plane projective curves of a given degree for example, [START_REF] Bertrand | A Viro theorem without convexity hypothesis for trigonal curves[END_REF][START_REF] Bihan | Polynomial systems supported on circuits and dessins d'enfants[END_REF]). It is S. Orevkov [START_REF] Orevkov | Riemann existence theorem and construction of real algebraic curves[END_REF] who discovered that dessins d'enfants can be used to construct real trigonal curves with given real scheme. Recently, B. Bertrand and E. Brugalle [START_REF] Bertrand | A Viro theorem without convexity hypothesis for trigonal curves[END_REF] showed how one can recover the dessin d'enfant of a trigonal curve constructed by the combinatorial patchworking.

One advantage of the construction used in Section 4 comes from the obvious observation that a Morse simplification may not be obtained a priori in a continuous family of real regular jacobian elliptic surfaces. This construction shows that such a family does exist in each case except possibly for the exceptional extremal pair of tori or Klein bottles.

Let us finally point out the recent appearance of [START_REF] Degtyarev | On deformation types of real elliptic surfaces[END_REF] in which dessins d'enfants are used in order to study real elliptic surfaces up to equivariant deformation.

We want to thank E. Brugallé, F. Catanese, A. Degtyarev, I. Itenberg and V. Kharlamov for fruitful discussions. The second author thanks also F. Catanese for a stimulating visit in Bayreuth.

Restrictions on real regular elliptic surfaces

In this section, we collect some classical restrictions on the topology of a real algebraic surface (X, σ) in term of the numerical invariants of the complex surface X. Next, we apply them to a real regular elliptic surface. For a complex regular elliptic surface X, most of the numerical invariants can be deduced from the holomorphic Euler characteristic χ(O X ). Recall that in fact, when X is jacobian, its complex deformation class is determined by χ(O X ), see [START_REF] Kas | On the deformation types of regular elliptic surfaces[END_REF].

Let (X, σ) be a real algebraic manifold. Denote by B i (X) = rk H i (X, Z/2 the i-th mod 2 Betti number of X and by

h i (X(R)) = rk (H i (X(R), Z/2)) the i-th mod 2 Betti number of X(R). Let h * (X(R)) = h i (X(R)) and B * (X) = B i (X).
The first important prohibition is given by the Milnor-Smith-Thom inequality.

Proposition 2.1. Let (X, σ) be a real algebraic manifold. Then

h * (X(R)) ≤ B * (X) ; B * (X) -h * (X(R)) is even.
Proof. See e.g. [START_REF] Silhol | Real Algebraic Surfaces[END_REF]Chap. II] or [11, 3.1.1].

Let us denote by τ (X) the signature of the cup-product form on the real vector space H 2 (X, R) and by τ -(X) = 1 2 (B 2 (X)-τ (X)) its negative index. The following inequality, often called Comessatti's inequality, is well-known. Proposition 2.2. Let (X, σ) be a real algebraic surface, then

|χ(X(R)) -1| ≤ τ -(X)
Proof. See e.g. [11, 3.1.2].

An important class of restrictions on extremal real surfaces is given by the Rokhlin congruences and their generalizations by Gudkov and Kharlamov. Recall that

B * (X) -h * (X(R)) is even and let 2d = B * (X) -h * (X(R)). A real algebraic surface is an M -surface if d = 0 and an (M -d)-surface if d = 0. Proposition 2.3. If (X, σ) is an M -surface (that is d = 0), then χ(X(R)) ≡ τ (X) mod 16 . When (X, σ) is an (M -1)-surface (that is d = 1), then χ(X(R)) ≡ τ (X) ± 2 mod 16 .
Proof. See e.g. [START_REF] Silhol | Real Algebraic Surfaces[END_REF]Chap. II] or [11, 2.7.1].

Proposition 2.4. Let π : X → P 1 be a real regular relatively minimal elliptic surface with no multiple fibres When (X, σ) is an M -surface, then

χ(X(R)) ≡ 8 χ(O X ) mod 16 . When (X, σ) is an (M -1)-surface, then χ(X(R)) ≡ 8 χ(O X ) ±2 mod 16 .
Proof. The elliptic fibration has no multiple fibres thus the homology of the complex surface X has no torsion. Whence we will not distinguish between the Betti numbers of X and the mod 2 Betti numbers of X. As π is relatively minimal, the first Chern number c 2 1 (X) is zero and the Noether's formula yields (2.5)

χ top (X) = 12 χ(O X ) .
For X a regular algebraic surface (i.e. with q(X) = dim

H 1 (X, O X ) = 0), the signature of the cup-product form is τ (X) = 4 χ(O X ) -2 -B 2 (X), see e.g. [4, Thm. 2.7]. Moreover, a regular surface has Betti numbers B 1 (X) = B 3 (X) = 0 whence τ (X) = -8 χ(O X ). The conclusion follows from Proposition 2.3. Remark 2.6. Let k = χ(O X ).
From the previous congruences, the Comessatti inequality 2.2, and the Theorems 2.7 and 2.7 below, we get the list of the k + 1 admissible pairs of Betti numbers for M -surfaces:

(#X(R), h 1 (X(R))) = (k + 4λ, 10k -8λ) , λ ∈ {0, 1, . . . , k} .
From the Betti numbers, we will deduce the jacobian topological type by Proposition 2.9.

By Morse simplifications, any (M -1)-jacobian topological type allowed by the Proposition 2.4 comes from an M -jacobian topological type, otherwise said, there are no extremal (M -1)-jacobian topological type. There are k extremal (M -2)jacobian topological type, see Theorem 1.1. To prove that the extremal topological types given in Theorem 1.1 are the only ones, the former classical restrictions on real algebraic surfaces applied to a real regular relatively minimal elliptic surface with no multiple fibres are not sufficient. We need the following additional restrictions. Theorem 2.7. [START_REF] Akriche | Nombres de Betti des surfaces elliptiques réelles, Beitrage zur Algebra und Geometrie[END_REF][START_REF] Mangolte | Surfaces elliptiques réelles et inégalité de Ragsdale-Viro[END_REF] Let X → P 1 be a real regular relatively minimal elliptic surface with no multiple fibres, then

h 1 (X(R)) ≤ 10 χ(O X ) . Moreover, if π is jacobian, then #X(R) ≤ 5 χ(O X ) .
Proposition 2.8. Let π : X → P 1 be a real regular jacobian elliptic surface. We have

4 ≤ h * (X(R)) ≤ 12 χ(O X ) 2 -10 χ(O X ) ≤ χ(X(R)) ≤ 10 χ(O X ) -2
Proof. The inequality (2.1) and the Noether formula for a relatively minimal elliptic fibration (2.5) yield

h * (X(R)) ≤ 12 χ(O X ) .
Furthermore, there is a connected component N 0 of X(R) containing the real part S(R) of the image of the real section s : P 1 → X. Let α ∈ H 1 (N 0 , Z/2) be the fundamental class of S(R) and β ∈ H 1 (N 0 , Z/2) the class of the restriction to N 0 of a smooth fibre, by hypothesis α ∩ β = 1 mod 2 whence h 1 (N 0 ) ≥ 2.

From B * (X) = 12 χ(O X ), a regular relatively minimal elliptic surface with no multiple fibres X has B 2 (X) = 12 χ(O X ) -2 and τ -(X) = 10 χ(O X ) -1. We get from Proposition 2.2,

2 -10 χ(O X ) ≤ χ(X(R)) ≤ 10 χ(O X ) .
We want to prove that, in fact,

χ(X(R)) ≤ 10 χ(O X ) -2. Suppose for a moment that χ(X(R)) = 10 χ(O X ), that is #X(R) = 5 χ(O X ) + 1 2 h 1 (X(R)). Now from the Theorem 2.7, we get the inequality #X(R) ≤ 5 χ(O X ) thus #X(R) = 5 χ(O X ) and h 1 (X(R)) = 0. But we have seen previously that X(R) contains a component N 0 with h 1 (N 0 ) ≥ 2; a contradiction.
For X a real relatively minimal elliptic surface, the Euler characteristic of X(R) is even. Indeed, by Proposition 2.1, χ(X(R)) ≡ χ top (X) mod 2 and by (2.5) χ top (X) is even. Thus χ(X(R)) = 10 χ(O X ) -1 does not occur and finally this yields χ(X(R)) ≤ 10 χ(O X ) -2.

The previous proof shows in particular that if X is a real regular jacobian elliptic surface, then X(R) = ∅ and h 1 (X(R)) ≥ 2. This yields

4 -χ(X(R)) ≤ h * (X(R)) and χ(X(R)) ≤ h * (X(R)) -4.
We also know that χ(X(R)) and h * (X(R)) are even integer numbers. Together with Propositions 2.4 and 2.8 and Theorem 2.7, the preceding analysis allows us to draw, for each k, the diagram of possible values for the pair (χ(X(R)), h * (X(R)) when X is a real regular jacobian elliptic surface. From the diagram, we will infer the list of allowable topological types of real regular jacobian elliptic surfaces. For that we use the following result. Proposition 2.9. Let π : X → P 1 be a real regular jacobian elliptic surface. Let k = χ(O X ). Then X(R) is non empty and it is orientable if k is even or non orientable if k is odd. Moreover, X(R) has one connected component of Euler characteristic ≤ 0 and other components are diffeomorphic to a sphere, unless X(R) is the disjoint union of two tori (if k even) or two Klein bottles (if k odd).

Proof. From [23, Chap. VII], we get that when the elliptic fibration admits a real section, any real singular fibre has connected real part except if the fibre is real isomorphic to the real cubic curve with one isolated node. Recalling that the real part of a smooth real curve of genus one has at most two connected components, the result easily follows except for the assertion about orientability which is deduced from the following Lemma.

Lemma 2.10. Let X → P 1 be a real regular relatively minimal elliptic surface with no multiple fibres.

If χ(O X ) is even, X is a Spin-surface, and if

X(R) = ∅, X(R) is orientable. If χ(O X ) is odd, X is a non Spin-surface, and if X → P 1 admits a real section or if (X, σ) is an M -surface, X(R) = ∅ and X(R) is non orientable.
Proof. By [4, p. 162], we have for a canonical divisor of X :

K X = (χ(O X ) -2) F
where F is any fibre of X → P 1 . From c 1 (T X) = -c 1 (K X ) we obtain for the second Stiefel-Whitney class of the tangent bundle of X w 2 (X) = χ(O X ) mod 2 .

We use [15, 5.7] for the first two assertions. We use [11, 3.4.1 [START_REF] Akriche | Nombres de Betti des surfaces elliptiques réelles, Beitrage zur Algebra und Geometrie[END_REF]] for the last assertion on M -surfaces.

Proposition 2.9 implies that, with the only exception of the point (0, 8), each point in the diagram depicted in Figure 1 corresponds to at most one topological type of a real regular jacobian elliptic surface.

Namely, if k is even the point (0, 8) corresponds to S 2 ⊔ S (non extremal) and to S 1 ⊔S 1 (extremal). If k is odd, the point (0, 8) corresponds to V 4 ⊔S (non extremal) and to V 2 ⊔ V 2 (extremal). Any point (χ, h * ) = (0, 8) of the diagram corresponds to the topological type S l ⊔ aS if k is even, or to V 2l ⊔ aS if k is odd, where a and l are determined by χ = 2(a + 1) -2l and h * = 2(a + 1) + 2l.

Collecting the results of this section, we thus obtain a list of non prohibited topological types, which coincides with the one given in Theorem 1.1.

Hirzebruch surfaces and real elliptic surfaces

Let n ≥ 1, denote by π : F n → P 1 the P 1 -bundle over P 1 associated with the Hirzebruch surface F n . Let (u, x) be affine coordinates on F n so that π can be written locally as (u, x) → u. Denote by B and F the classes in the second homology group H 2 (F n , Z) of the image of a section disjoint from the exceptional divisor E and a fibre of π, respectively. As it is well-known, the homology group H 2 (F n , Z) is generated by B and F with intersection numbers The Hirzebruch surface F n inherits a real structure coming from the usual complex conjugation on some affine chart of F n . This is the standard real structure on F n and we will consider only this one. The real part F n (R) of F n is homeomorphic to a two-dimensional torus if n is even, or to a Klein bottle if n is odd.

B •B = n, F •F = 0 and B •F = 1. A curve C on F n is of bidegree (a, b) if [C] = aB + bF in H 2 (F n , Z),
A trigonal curve is a curve of bidegree (3, 0) on F n . Let r ∈ R[u, x] be a polynomial whose Newton polygon (the convex hull of the exponent vectors of the monomials appearing with non zero coefficients) is the triangle with vertices (0, 0), (3n, 0) and (0, 3). Such a polynomial defines (affinely) a trigonal curve C. A classical (non linear) elimination process allows to eliminate the monomials u l x 2 , l ∈ {0, . . . , n}, from r in order to get a reduced affine equation for C (3.1)

x 3 + p(u)x + q(u) = 0, where p and q are polynomials of degrees 2n and 3n, respectively. The discriminant of this polynomial, viewed as a polynomial in

R[u][x], is ∆ = 4p 3 + 27q 2 ∈ R[u] .
The roots of ∆ are those u 0 for which the vertical line u = u 0 is either tangent to, or contains a singular point of, the curve C. We have ∆(u 0 ) < 0 exactly when the line u = u 0 intersects C(R) in three points. Consider the Hirzebruch surface F 2k of even degree 2k and let C be a real nonsingular trigonal curve on F 2k . A connected component of C(R) is called a pseudo-line if it intersects each vertical line u = u 0 , otherwise it is called an oval. An oval separates F 2k (R) into two connected components, one of them being homeomorphic to a disk called interior of the oval. The image of a pseudo-line by the projection F 2k (R) → P 1 (R) induced by (u, x) → u is the whole P 1 (R), while the image of an oval is an interval whose endpoints are real roots of the discriminant ∆.

With one exception, all but one connected components of C(R) are ovals, while the remaining one is a pseudo-line. The exception arises when C(R) consists of three (non intersecting) pseudo-lines. This happens only if ∆ < 0 on P 1 (R). We give here an explicit example which will be used for the construction of extremal elliptic surfaces with topological type

S 1 ⊔ S 1 (if k even) or V 2 ⊔ V 2 (if k is odd), see Lemma 3.3 below. Example 3.2. Consider the non reduced polynomial r(u, x) = (x -g 1 (u)) • (x -g 2 (u)) • (x -g 3 (u)),
where g 1 , g 2 and g 3 are real polynomials of degree 2k such that g 2g 1 , g 3g 1 and g 3g 2 have each 2k (simple) non real roots and the resulting 6k roots are pairwise distinct. In particular, g 2g 1 , g 3g 1 and g 3g 2 are polynomials of degree 2k with no real roots. This implies that r defines a trigonal curve whose real part consists of three non intersecting pseudo-lines. This trigonal curve has as singularities 6k non real ordinary double points coming from the above 6k non real roots. A small perturbation gives a nonsingular trigonal curve whose real part consists of three pseudo-lines.

Assume now that C(R) does not consist of three pseudo-lines. Let H = 0 be some bi-homogeneous polynomial equation for C ∪ E. We have [C] • E = 0, hence C does not intersect E and C ∪ E is a real nonsingular curve of bidegree (3, 0) + (1, -2k) = (4, -2k). Since this bidegree is even, the polynomial H has a well-defined sign on F 2k (R). Hence the ovals of C can be divided into two groups according to the sign of H in the interior of each oval. It follows that, with the exception of C(R) consisting of three pseudo-lines, the real scheme of C, that is the pair (F 2k (R), C(R)) up to homeomorphism, is uniquely determined by the numbers a and b of ovals in each of these two groups and will be encoded by a | b , see Figure 2. Note that these two numbers are defined only up to permutation. Let ρ : X → F 2k be the double covering of F 2k ramified over C ∪ E (it is welldefined since the bidegree of the ramification curve is even). The complex surface X is a nonsingular jacobian elliptic surface, which can be equipped with two real structures lifting that of F 2k . According to the chosen real structure, the real part of X projects via ρ onto the subset of F 2k (R) where either H ≥ 0, or H ≤ 0.

Lemma 3.3. If C(R) consists of three pseudo-lines, then X(R) is homeomorphic to S 1 ⊔ S 1 if k even, or to V 2 ⊔ V 2 if k is odd.
If C has real scheme a | b , then, up to permutation of a and b which corresponds to a permutation between the two real structures for X, the surface [START_REF] Persson | Chern Invariants of Surfaces of General Type[END_REF]. Thus, when k is odd, the square α 2 of the fundamental class of S(R) in H 1 (M, Z/2) does not vanish whence M is non orientable.

X(R) is homeomeomorphic to a S ⊔ S b+1 if k is even, or to a S ⊔ V 2b+2 if k is odd. Proof. A direct computation shows that χ(X(R)) = 0 if C(R)
[S] = 1 2 B • B = k, see e.g.
Conversely, suppose k even, the canonical class of X is given by

K X = ρ * (K F 2k + L) where K F 2k = -2B + (2k -2)F and 2L = [C ∪ E].
Thus we recover the formula used in the proof of Lemma 2.10 from K X = ρ * ((k -2)F ). We therefore conclude that w 1 (X(R)) = 0 hence X(R) is orientable. x 3 + p(u)x + q(u) = 0

where deg p = 4k and deg q = 6k. We will assume that C is generic in the sense that the discriminant ∆ = 4p 3 + 27q 2 has only simple roots and p, q, ∆ have distinct roots. The roots of ∆ being simple roots, the curve C is a nonsingular curve and any root of ∆ corresponds to a vertical line tangent to C. Following Orevkov [START_REF] Orevkov | Riemann existence theorem and construction of real algebraic curves[END_REF], consider the real rational function P 1 (C) → P 1 (C) defined by

f = ∆ q 2 = 4p 3 q 2 +
27 . This a rational function of degree 12k. The roots of ∆, p and q are mapped to 0, 27 and ∞, respectively. These roots as well as the values 0, 27 and ∞ will be called special.

Remark 4.2. Recall that the functional invariant J : P 1 (C) → P 1 (C) associated with the elliptic surface given by the affine equation

y 2 = x 3 + p(u)x + q(u) is J = 4p 3 ∆ .
The real rational functions f and J are related via J = γ • f , where γ is the real automorphism of P 1 (C) sending 0, 27, and ∞ to ∞, 0, and 1, respectively.

Color the real line on the target space in three distinct colors as shown in Figure 3 (one color for each interval delimited by two special values, and special symbols for special values) and define a colored graph on the source space P 1 (C) as the graph Γ = f -1 (P 1 (R)) colored via the pull-back of the previous coloring of P 1 (R). Such a graph is sometimes called a dessin d'enfant. The graph Γ is invariant under the complex conjugation and contains P 1 (R). Each vertex of Γ has even valency and corresponds to a critical point of f with multiplicity half the valency (if the valency is 2 then the point is in fact not a critical point of f ). Due to the special form of the rational function f , the vertices given by the roots of ∆, p and q should have valencies which are multiple of 2, 6 and 4, respectively. In fact, since we have assumed that ∆ has only simple roots, the roots of ∆ are not critical points of f and have valency 2. Hence, to a real trigonal curve C corresponds a colored graph on P 1 (C) verifying the above properties. Conversely, drawing dessins d'enfants allows to construct real trigonal curves with a given real scheme (see, for example, [START_REF] Orevkov | Riemann existence theorem and construction of real algebraic curves[END_REF], [START_REF] Bertrand | A Viro theorem without convexity hypothesis for trigonal curves[END_REF]). Here we will only use a local description of the graph Γ that will allow us to collapse independently the ovals. Proof. Consider one oval of the curve (if the curve has no oval, there is nothing to prove). Its image under the ruling π : (u, x) → u is a closed interval whose endpoints are roots of ∆. Denote by I the interior of this interval. For each u 0 ∈ I, the vertical line u = u 0 intersects the oval in three distinct real points. This means that f is negative on I. In other words, the interval I is uniquely colored as a part of Γ like (∞, 0) on the target space P 1 (R) (see Figure 3). Conversely, any such interval corresponds to an oval of the curve. The critical points of f which belong to I are either roots of q (critical points with value ∞), or non special critical points. Let U be a small neighbourhood of I ⊂ P 1 (C) which is invariant under the complex conjugation and which contains only real critical points of f . Let H be one of the two complex conjugate parts of U \ P 1 (R). Recall that the valency of a vertex of Γ is twice its multiplicity as a critical point of f . The multiplicity of each root of q contained in I should be even for otherwise f would be positive inside I. This means that the number of branches of Γ contained in H and which start from a real root of q is odd. Moreover, since the derivative f ′ take opposite (non zero) values at the endpoints of I, it follows that the sum of the multiplicities as roots of f ′ of the non special critical points contained in I should be an odd integer m. We note that this m coincides with the total number of branches of Γ contained in H and which start from these non special critical points. We perform a continuous deformation of colored graphs (Γ t ) t∈[0,1] , Γ = Γ 0 , which is concentrated in the neighbourhood U, as it is shown in Figure 4 and 5 (the corresponding deformation for the graph of f |P 1 (R) is depicted at the bottom). In these pictures, horizontal segments are parts of P 1 (R) while the complex conjugation is given by the reflection about P 1 (R).

First we deform Γ around each real root of q as depicted in Figure 4. The real root is transformed into pairs of complex conjugate simple roots, while a non special real critical point (of the same mulitplicity) is created. Shrinking U if necessary, we can then assume that it contains only (real) non special critical points of f . This means that, with the exception of the complement of I in P 1 (R) ∩ U, the part of Γ contained in U is now entirely colored like (∞, 0) and the total number of branches contained in H is the odd number m. This gives a new graph at time t = t 1 . We then pursue the deformation as shown in Figure 5 (the disk represents U).

It remains to show that along the deformation any graph Γ t corresponds to a real trigonal curve C t on F 2k and that the real scheme of C 1 is obtained from that of C 0 by collapsing the oval we started with.

From the topological point of view, the map f : P 1 (C) → P 1 (C) is a symmetric (by which we mean that complex conjugate points are mapped to complex conjugate points) branched covering of degree 12k which respects the colorings. It is not difficult to see that the deformation (Γ t ) t∈[0,1] comes from a continous deformation (f t ) t∈[0,1] , with f 0 = f , of maps f t sharing the same properties and such that Γ t = f -1 t (P 1 (R)). Indeed, there exists a continuous map G :

P 1 (C) × [0, 1] → P 1 (C)
such that Γ t = G(Γ 0 , t) for each t and the restriction of G to Γ 0 ×{0} is given by the identity map. Moreover, there is a continous family (ϕ t ) t∈[0,1] of maps ϕ t : Γ t → P 1 (R) starting at the restriction ϕ 0 of f to Γ = Γ 0 and which respect the colorings. Consider now one connected component D 0 of P 1 (C) \ Γ 0 and let D t = G(D 0 , t) be the corresponding connected component of P 1 (C) \ Γ t . Denote by Dt and D0 their closures. Looking at Figure 5 shows that Dt and D0 are homeomorphic for each t. Moreover, one can choose homeomorphisms ψ D,t : Dt → D0 coming into a continuous family such that ϕ t and ϕ 0 • ψ D,t coincide on the boundary of D t . This allows to extend the restriction of ϕ t to the boundary of D t into a continuous map f D,t : Dt → f 0 ( D0 ) by setting f D,t := f 0 • ψ D,t . Doing that for each D 0 in one given connected component of P 1 (C) \ P 1 (R), and extending the resulting map into a symmetric map, one obtains a continuous family of symmetric continuous maps f t : P 1 (C) → P 1 (C). It is then easy to see that each f t has the desired properties. By Riemann's uniformization Theorem, each map f t becomes a real rational map of degree 12k for the standard complex structure on the target space and its pull-back by f t on the source space. For any t ∈ [0, 1] the (special) points of Γ t where f t take values 0, ∞ and 27 have all a valency which is a multiple of 2, 4 and 6, respectively. This gives the existence of real polynomials p t and q t of degree 4k and 6k, respectively 2 , such that f t =

4p 3 t q 2 t + 27 = ∆t q 2 t
where ∆ t := 4p 3 t + 27q 2 t . Let C t denotes the associated trigonal curve with affine equation x 3 + p t (u)x + q t (u) = 0. For any t = t 4 , the curve C t is a nonsingular curve since all roots of ∆ t have valency 2 in Γ t . The curve C t4 has an unique singular point which corresponds to the real root of ∆ t4 contained in U whose valency is 4. This singular point is an ordinary double point which is isolated in F 2k (R) since f t4 , hence ∆ t4 , is positive around it. To finish, it suffices to note that the number of intervals uniquely colored like (0, ∞) on the target space decreases by 1 along the deformation. 4.2. Combinatorial patchworking of trigonal curves. We first recall how the combinatorial patchworking works for the construction of real trigonal curves on F 2k Let k be any positive integer and denote by T the triangle with vertices (0, 0), (6k, 0) and (0, 3). The input of the combinatorial patchworking are

• A triangulation of T whose vertices have integer coordinates,

• a distribution of signs ±1 at the vertices of this triangulation. Assume that there exists a convex piecewise-linear function ν : T → R linear on each triangle of the triangulation, but not linear on the union of any two triangles. Such a triangulation is called convex (or sometimes coherent).

The so-called T-polynomial associated with these data is the polynomial

r t ∈ R[u, x] defined by r t (u, x) = s(i 1 , i 2 )t ν(i1,i2) u i1 x i2 ,
where the sum is taken over all the vertices (i 1 , i 2 ) of the triangulation and s(i 1 , i 2 ) is the sign of (i 1 , i 2 ). Consider now the quadrangle Q with vertices (±6k, 0) and (0, ±3). Extend the triangulation of T to a triangulation of Q which is symmetric with respect to the coordinate axes. Extend also the initial sign distribution to a sign distribution at the vertices of the triangulation of Q following the rule

s(ǫ 1 i 1 , ǫ 2 i 2 ) := ǫ i1 1 • ǫ i2 2 • s(i 1 , i 2 )
2 A priori we can only conclude that deg pt ≤ 4k and deg qt ≤ 6k with at least one equality occuring, but Figure 5 shows that pt and p 0 = p (resp., qt and q 0 ) have the same number of roots, counted with multiplicities, hence the same degree. for any (ǫ 1 , ǫ 2 ) ∈ {±1} 2 .

We are now able to perform the combinatorial patchworking. For any triangle of the triangulation of Q whose vertices do not all have the same sign, select the segment joigning the middle points of the two edges whose endpoints have different signs. Denote by C the piecewise linear curve in Q obtained by taking the union of all these selected segments. We will use the following version of the combinatorial patchworking theorem, see [START_REF] Viro | Gluing of algebraic hypersurfaces, smoothing of singularities and construction of curves[END_REF][START_REF] Viro | Gluing of plane algebraic curves and construction of curves of degree 6 and 7[END_REF][START_REF] Viro | Patchworking real algebraic varieties[END_REF]. Proposition 4.4. There exists t 0 > 0 such that for any positive t < t 0 the polynomial r t defines a real nonsingular trigonal curve C. Moreover, there exists an homeomorphism from R 2 to the interior of Q sending the affine curve defined by r t to C.

We recall the Harnack's theorem for real curves, which is a particular case of Proposition 2.1. If C is a real nonsingular algebraic curve, then the number of connected components of C(R) is no more than g(C) + 1, where g(C) is the genus of (1) For any integer λ such that 0 ≤ λ ≤ k, there exists a real nonsingular trigonal M -curve C with real scheme k -1 + 4λ | 5k -1 -4λ . (2) For any integer λ such that 0 ≤ λ ≤ k -1, there exists a real nonsingular trigonal (M -2)-curve C on F 2k with real scheme k + 4λ | 5k -4 -4λ .

C. If C(R) has g(C) + 1 connected components, then C is called an M -curve, while it is called an (M -d)-curve if C(R) has g(C) + 1 -d connected components.
Proof. Use the segments [(0, 3), (6ℓ ± 1)] in order to triangulate T into the two triangles [(0, 0), (1, 0), (0, 3)], [(6k -1, 0), (6k, 0), (0, 3)] and the triangles T ℓ := [(6ℓ + 1, 0), (6ℓ + 5, 0), (0, 3)], ℓ = 0, . . . , k -1

t ℓ := [(6ℓ -1, 0), (6ℓ + 1, 0), (0, 3)], ℓ = 1, . . . , k -1.
Then triangulate each T ℓ and t ℓ as depicted in Figure 6. It is easy to see that the resulting triangulation of T is convex, see Remark 4.6.

Let us now choose a sign distribution at the vertices of this triangulation. Put the sign -1 to the points (0, 0), (6k, 0) and take the sign distribution shown in Figure 6 for the vertices of the triangulation which are contained in triangles t ℓ . For the vertices of the triangulation which are contained in triangles T ℓ , take the sign distribution shown in Figure 6 with ǫ = + for λ triangles among T 0 , . . . , T k-1 and with ǫ =for the remaining kλ triangles.

Let r t ∈ R[u, x] be a T-polynomial associated with these data. Let C be the trigonal curve on F 2k defined by r t for t > 0 sufficiently small and let C ⊂ Q be the piecewise linear curve constructed via the combinatorial patchworking from the previous triangulation and sign distribution. Proposition 4.4 implies that C is a nonsingular curve whose desired properties can be read off the picture of C ⊂ Q.

One can easily check that the union of the four symmetric copies of the interior of each triangle t ℓ contains exactly two ovals of C, one of them surrounding a vertex with the sign + while the other surrounds a vertex with the sign -. It is also easy to see that the union of the four symmetric copies of the interior of each triangle T ℓ contains four ovals of C, all of them surrounding a vertex with the sign ǫ (see Figure 7 for ǫ = -1). This gives 2(k -1) + 4k = 6k -2 ovals of C(R). Note that C has one connected component more which intersects the segment with vertices (0, ±3). This gives an account of 6k -1 connected components of C(R). But 6k -2 is the genus of the complex curve C. Hence C(R) has exactly 6k -1 connected components by Harnack's theorem, that is C is an M -curve, and the last component we have obtained is the pseudo line. Finally, it remains to compute the real scheme a | b of C. Clearly, up to permutation of a and b, the number a is the number of ovals of C which surround a vertex with the sign +, while b is the number of ovals of C which surround a vertex with the sign -. Since we have choosen the sign ǫ = + for λ triangles among T 0 , . . . , T k-1 , we obtain a = k -1 + 4λ and b = k -1 + 4(kλ) = 5k -1 -4λ. Remark 4.6. A classical argument explained below shows that all triangulations we used are convex. Note that this is in fact unnecessary due to [START_REF] Bertrand | A Viro theorem without convexity hypothesis for trigonal curves[END_REF] which proves that the convexity hypothesis can be dropped in the combinatorial patchworking theorem when starting from a triangulation of [(0, 0), (3n, 0), (0, 3)]. Take a triangulation of T by segments starting from (0, 3) and ending to points of [(0, 0), (6k, 0)]. Such a triangulation is evidently convex. Now subdivide one its triangles into three triangles using an interior point. Let ν be a function which certifies the convexity of the starting triangulation. We can construct a function ν ′ certifying the convexity of the refined triangulation in the following way: let ν ′ coincide with ν at the vertices of the starting triangulation, take the same value than ν but diminished by ι with 0 < ι << 1 at the new vertex, and be linear on each triangle of the new triangulation. The triangulations used in this paper are all obtained by successive refinements of the previous type, and are thus convex. 4.3. Proof of Theorem 1.1. We organize all the results obtained so far to give a proof of Theorem 1.1. The fact that the topological types of real jacobian elliptic surfaces of given holomorphic characteristic k belong to the list in Theorem 1.1 has been proven in Section 2.

By Lemma 3.3, the extremal topological type S 1 ⊔ S 1 if k is even, or V 2 ⊔ V 2 if k is odd, is obtained via a double covering starting with a trigonal curve constructed in Example 3.2.

Applying Lemma 3.3 to the curves in parts 1) and 2) of Proposition 4.5 produces via a double covering real jacobian elliptic surfaces whose topological types form the

Figure 1 .

 1 Figure 1. Diagram of possible values for (χ(X(R)), h * (X(R)) The diagram is the right one for k = 8. The black points correspond to the k + 1 M-surfaces (which are automatically extremal) and the extremal (M -2)-surfaces. The small circle surrounds the point (0, 8) which is the only point in the diagram corresponding to two possible topological types comprising an extremal one: S1 ⊔ S1 if k is even or V2 ⊔ V2 if k is odd. Note that for any k the diagram is symmetric with respect to the vertical axis χ(X(R)) = 0 while this axis contains a point corresponding to an M-surface (as it is the case in the picture) only when k is even.

  or equivalently, if a = [C] • F and b = [C] • E. The intersection number of two curves of bidegree (a, b) and (c, d) is then equal to acn + ad + bc. The exceptional divisor E has bidegree (1, -n) and self-intersection -n.

Figure 2 .

 2 Figure 2. Real scheme 2 | 3

  consists of three pseudo-lines and χ(X(R)) = 2(ab) if C has real scheme a | b . It suffices then to determine if each non spherical component is orientable or not. Let M be such a component, and let S be a real section whose real part S(R) is contained in M . The linear class of the effective divisor S is the (reduced part) of the pull-back of the class B in Pic(F 2k ). The class B is a part of the branch class of the double covering ρ. Hence [S] •

4 . Constructions 4 . 1 .

 441 Collapsing ovals of trigonal curves via dessins d'enfants. Let C be a real nonsingular trigonal curve on F 2k with affine reduced equation (4.1)

Figure 3 .

 3 Figure 3. Coloring of P 1 (R)

Proposition 4 . 3 .

 43 The ovals of C can be collapsed independently. In other words, if C has real scheme a | b , then for any pair of integers (a ′ , b ′ ) such that 0 ≤ a ′ ≤ a and 0 ≤ b ′ ≤ b there exists a real nonsingular trigonal curve C ′ on F 2k with real scheme a ′ | b ′ .

Figure 4 .

 4 Figure 4. Deformation around a real root of q (with multiplicity 4)

Figure 5 .

 5 Figure 5.

  Proposition 4.5.

Figure 6 .

 6 Figure 6. Triangulation and signs For T ℓ and t ℓ .Points with integer coordinates are represented by black points.

Figure 7 .

 7 Figure 7. Modification of the triangulation and sign distribution

list (1) and (2) of Theorem 1.1, respectively. Now, Proposition 4.3 applies to each of these curves, which in turn implies via Lemma 3.3 that all desired topological types can be realized by real jacobian elliptic surfaces with holomorphic Euler characteristic k.


émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

When Hawthorne undertook the writing of his children books, it was also, and perhaps above all, to face a delicate financial situation. In this respect, it seems that these books, namely A Wonder Book for Boys and Girls and Tanglewood Tales, respectively published in 1851 and 1852, that is right after the success of The Scarlet letter, were to be considered as a brief literary parenthesis in Hawthorne's career. Yet, they are rightly to be described as adroit pieces of literary rewriting. Greek mythology is being transmuted into Gothicized, Americanized and, what is of most interest to me, feminized stories. Traces of Puritan thought are to be found under this palimpsestic artwork. They fall into the politicized crucible of Hawthorne's nascent feminism which, to me, is one of the main aspects of his fiction, and more particularly his tales and short stories. Indeed, Hawthorne, under the pretext of offering to the public his version of classical Greek mythology, is creating a new text where women are to be given the possibility, enacted or not, of delineating the borders of a new female territory. I here refer to Deleuze and Guatarri's concept of minor literature, and am using it to refer to women, and the female characters of Hawthorne's fiction, as a political minority. Indeed, even in the realm of Greek mythology, Hawthorne purposes to denounce and address the wrongs done to this minorized portion of humanity. I'd like to hold on for a minute on what I mean by "minority." In Thousand Plateaux, Deleuze and Guatarri define it as an entity of individuals whose status of inferiority will, paradoxically allow a process of reterritorialization following, almost logically and naturally, the other process of deterritorializaion due to that inferior, "minor" as Deleuze and Guatarri choose to call it, status. This development should produce a new status, what they call a "becoming". In Hawthorne's case, the "becoming-woman" is what the female character tends towards. As Hawthorne rewrites the Biblical founding myth of Genesis, primarily with his New Eve "type," he also enterprises to rewrite classical Greek myths, not afraid to introduce, when needed, a touch of what I call his "proto-feminism" like when he takes the liberty to give life to a charming daughter for King Midas, a child aptly called Marygold in whose hands lie metaphorically the redemption of the greedy monarch. In Hawthorne's fiction, women are found to be the main actor in man's regenerating process, and, more symbolically, the main actor in America's renaissance.

Here, I will concentrate my efforts on a few couple of Hawthorne's mythological tales, namely "The Gorgon Head," "The Golden Touch," and "The Paradise of Children." In these stories, the character of Primrose, one of the children in Eustace Bright's audience, or, to use Genette's terminology, one of the implied readers, stands out as a responsive actor in the literary process of rewriting and retelling. It will be seen that this Fuller-like girl child of almost thirteen is one of Hawthorne's female speaker and defender of women's right to write and respond to literary works. The characters of Marygold of "The Golden Touch" and Pandora of "The Paradise of Children" will also allow me to decline Deleuze and Guatarri's concepts of minority and becoming-woman.

I.

Primrose is part and parcel of the complex network of diegetic layers, and is indeed the most interesting children of the attentive audience. She incessantly disparages Eustace's literary talents, and even questions his authority and legitimacy as story-teller. Here I'd like to quote some revealing remarks by "saucy" Primrose:

"[…]
Primrose, who was a bright girl of twelve, with laughing eyes, and a nose that turned up a little, 'the morning is certainly the best time for the stories with which you so often tire out our patience. We shall be in less danger of hurting your feelings, by falling asleep at the most interesting points, -as little Cowslip and I did last night!'" ("Tanglewood Porch: Introductory to "The Gorgon's Head," p.15). "'As to their one tooth [speaking about the Three Grey Women], which they shifted about,' observed Primrose, 'there was nothing so very wonderful in that. I suppose it was a false tooth. But think of your turning Mercury into Quicksilver, and talking about his sister! You are too ridiculous!'" ("Tanglewood Porch: After the Story of "The Gorgon's Head," p.42) "'Why, as to the story f King Midas, ' said saucy Primrose, 'it was a famous one thousands of years before Mr Eustace Bright came into the world, and will continue to be so as long after he quits it. But some people have what we may call 'The Leaden Touch,' and make everything dull and heavy that they lay their fingers upon.'" ("Shadow Brook," after the story of "The Golden Touch," p.64) Her interaction with Eustace, and even her interference in Eustace's glory, is at once a source of motivation for him and a way for Hawthorne to defend criticism of literary works by women. She seems to be a miniature of Hawthorne's friend and female writer Margaret Fuller, whom he admired as one of his scholarly friendships. Primrose may also be read as a deviant and early version of those "scribbling women" whom Hawthorne himself famously disparaged in this often commented passage from his letter to his friend and publisher William Ticknor dated January 1855:

"America is now wholly given over to a damned mob of scribbling women, and I should have no chance of success while the public taste is occupied with their trash -and should be ashamed of myself if I did succeed." [cited in John T. Frederick, "Hawthorne's Scribbling Women," in The New England Quarterly 48-2 (June 1975): 231] Primrose is one who does not know how to "hold her tongue" ("Tanglewood Porch", Introductory to "The Gorgon Head," p.17), regrettably enough for Eustace, who thus betrays a certain lack of confidence in the sensible realism and even fictional legitimacy of his tales. She takes after her father, Mr. Pringle, who strongly discourages Eustace to "put any of [his] travesties on paper" and, most of all, "never more to meddle with a classical myth." His imagination is, according to this sharp connoisseur of classical works, "altogether Gothic, and will inevitably Gothicize everything that [he] touch [es]. The effect [would systematically be] like bedaubing a marble statue with paint." ("Tanglewood Fireside: Atfer the Story" of "The Three Golden Apples," p.115). Mr. Pringle seems to have been an effective teacher of literary taste to his daughter Primrose who, like Margaret Fuller, displays an independent mind as to what a good story is, or should be. Hawthorne thus endows one of his female characters, a child at it, with a literary education and intelligence which were supposed to be man's precious treasure.

After the story, this portion of the tales which offers a meta-analysis of the unfolded narrative, Primrose comes back on certain elements of the tale, thus delivering her opinion on the quality of the textual fabric. A piece of literary work, whether it be a written text or an oral delivering, becomes valuable only for the reactions, negative or positive, that it excites. "Saucy Primrose," as she is justly nicknamed by the narrator of the "Shadow Brook" section of "The Golden Touch" tale (p.64), evinces a straightforward judgement of Eustace's story.

To me, Primrose is like a surrogate author figure, another one of Hawthorne's personae, or masks, that serve to look at his fiction with a sharply critical eye. That Hawthorne gives this difficult role to a girl is in itself relevant in his treatment of women in the field of literature.

She is also a Sophia-like figure who, more than once, gave her husband opinions about his work. Primrose, with both Fuller and Sophia Peabody as spiritual and literary mothers, is thus to be read as a positive character in that she constitutes an invaluable help in the creative process of story-writing and story-telling. She provides the authors, both Eustace and Hawthorne himself, with the necessary metatextual comments to perfect themselves in the art of literary re-creation. These women prevent the authors from enjoying a delusive selfsatisfaction in their talents which then need constant sharping.

One such intervention by Primrose deals with the literary clichés that Eustace seems to take pleasure in displaying in his tales. Following our interest in minor literature, these clichés portray female characters as somewhat lacking originality of personality. For example, Primrose denounces the flatness of Mercury-turned-Quicksilver's sister who is a precious help for Perseus in his vanquishing the Gorgon Medusa. Eustace sarcastically defends himself against this creative flaw saying that "If I had thought of it sooner, I would have described her as a maiden lady, who kept a pet owl!" ("The Gorgon's Head," p.42). He thus further proves his lack of originality when it comes to depicting female characters since he merely follows a stock tradition of beautiful maidens whom a pet animal contributes to domesticate still further.

Where he does innovate is not so much in the physical description as in the role attributed to those female characters. Indeed, it is with her help, together with that of Quicksilver, that Perseus manages to cut the head of the terrifying Medusa. She is thus given an active role in the hero's quest. What Primrose does criticize then is the regrettably aborted quality of this active role. Yet, in Eustace's defense, it seems that his remark is also a veiled criticism of this female passivity which literary tradition had erected as the canonical and ultimate quality as it was the outward sign of self-control and virtue. In fact, Eustace is subtly voicing a judgement of his own against the reader's expectations in the depiction of female (secondary) characters. As for Pandora, I'd like to return, if time will allow me, to the underlying biblical tones that pave the way for a reevaluation of women' place in the fall of Man.

Literalization of figurative images: towards a becoming-woman

In his fiction in general, and in his mythological tales in particular, Hawthorne resorts to irony and a sarcastic kind of humor to convey some of his revolutionary ideas, especially when it comes to denounce the tragic destiny of women. This irony can take different forms and one of them is the process that consists in literalizing what is supposed to be, and what should remain at a figurative level. A didactic purpose can also be discovered under that veil of humorous speech. The most telling example that can be found in the corpus of mythological texts is that of the daughter of King Midas, amusingly named "Marygold" by the narrator. I have to point out the spelling variation of that name. A "-y" has come to replace the "-i" of the original flower. This manipulation is in itself revealing of Hawthorne's purpose to lament the killing, both physical and metaphorical, of the female identity, and to wonder why such treatment should be tolerated both by society at large, and by women themselves.

The underlying, palimpsestic religious tone also transpires through this superficial modification. We can clearly hear Virgin Mary's name, and the purity of Marygold's heart validates this interpretation, as well as the purifying tears that King Midas sheds on account of her. This side of Marygold's character corresponds to Hawthorne's adherence to traditional symbolism and conventional feminine virtue as being the healer of men's sins.

Yet, the process of literalization intervenes in the second half of Marygold's name. The melted semantic reference to the original flower and the overt mention of the precious metal constitute the drama of the tale and the tragedy of the little girl. The love for beautiful and colorful flowers that so much characterizes Marygold triggers off her successive deterritorializations. From little girl, she becomes-flower through the passive influence of her name. From natural flower, she becomes a golden flower: the precious touch by her father transmutes her into a golden marigold (with an "-i" this time), a sister for all her other flowery friends likewise transformed into gold plants. As she is a human being, the final deterritorialization that she undergoes is that of becoming-statue of gold under the fatherly embrace. The gold contained in the little girl's name should have remained figurative, but Midas's greed has turned it into literal gold, and has thus killed the little human flower that so resembles Beatrice Rappaccini, another one of those women-flowers that seemed to intrigue Hawthorne. Their fragility, their ephemerality and their beauty condense those qualities that are reputed to be essential in and to women, namely paleness, modesty and virtue. The becoming-statue undergone by Marygold is essential for Hawthorne to convey his point: these "natural" qualities in fact are mortifying to women who are thus imprisoned in unchanging molds that dictate assumed identities that cannot be wholly livable in the outer world. That Marygold turns into a statue of gold under the touch of her father, a man, is in itself a denunciation of the patriarchal system that is at the origin of such mortification. Hawthorne manages to blend a more religious reading with a political defense of women's right to an identity of their own. The different becomings open to women are thus entangled, perpetually succeeding and following one another, so much so that the becoming-flower and the becoming-statue finally fuse to allow for the becoming-woman: they are then the necessary steps to be taken to achieve their female identities.

Pandora: biblical and mythological intertextuality at the service of Hawthorne's feminist perspective

In the literary economy developed by Gérard Genette, intertextuality is one out of the five of the fundamental "types of transtextual relations" and partakes of the "textual transcendence of the text" defined by Genette himself as being "everything that institutes a manifest or discreet relation with other texts." (Palimpsests, p.8) As I have already pointed out in my introduction, the biblical text and the mythological one are superimposed on one another so that Hawthorne's legendary tales offer a multi-layered creation. Hawthorne's personal reading is inseparable from a necessary subjective filtering, a long and complex decantation which eventually give out a clean product. To paraphrase Pierre Arnaud, Hawthorne's deconstructivistic approach consists in these three steps: "alteration, displacement, and alienation" (Pierre Arnaud, "Au service de Lucifer: le diable, la magie et le pacte dans Le Moine de M. G. Lewis," in Maître et serviteur dans le monde anglo-américain aux XVIIe et XVIIIe siècles: [actes du Colloque Société d'études anglo-américaines des XVIIe et XVIIIe siècles], 1985: 143). By alienating the original myths, this process makes it possible to distort the spirit of those myths. This allows Hawthorne to elaborate still further his protofeminist ideology, and to direct it to a privileged public that are young children. Indeed, he hopes to make young people, boys and girls alike, just as the title of his Wonder Book clearly indicates, realize how deeply and strongly preconceptions and received ideas about the sexes are anchored in the minds and hearts of men and women. The author, through the voice of Eustace Bright, but also that of "naughty Primrose", wishes to contribute to an evolution in the mentality of the American society. A return to the founding myth of humanity, namely the Fall of Man, is therefore crucial to bring a new light to this story that has put women in an awkward situation for centuries. The pagan version of Genesis is to be found in the story of Pandora's box.

Unfortunately, I can't elaborate on this point, and I have now to conclude my talk.

CONCLUSION

To bring Deleuze and Guatarri a bit further, I'd say that the becoming-flower of those of Hawthorne's female characters can also be read as a way for Hawthorne to indicate the possibility of a "ligne de fuite" for those women entrapped in hermetically defined categories.

10

In fact, they are the targets of male desires, and their bodies become the object of male idealization. Hawthorne's fiction will be the locus of a dismantling of all those (sometimes) contradictory constructions of the feminine, those same definitions which allowed the holders of political, religious, social and moral authority to further strengthen their power over girls and women. Hawthorne deftly uses female stereotypes to his advantage, and turns them into a meaningful weapon to denounce what they are meant to represent. In other words, the literalization of figurative discourse make it possible to implement those "lignes de fuite" "from the outside," as Deleuze and Guatarri say, by defining "dimensions" or qualities on the top of which Hawthorne will add other challenging qualities without the former ones being impacted by the new ones. This makes for the ambiguous position held by Hawthorne towards the generic question. The metaphor of the statue, that we find in "The Birthmark" (1843) ("But it would be as reasonable to say that one of those small blue stains which sometimes occur in the purest statuary marble would convert the Eve of [Hiram] Powers to a monster." (Twice-Told Tales, 1987, p.119), is somehow a double-sided literary weapon as it both serves to subtly reinforce the esthetic canons of female beauty and to condemn those same norms.

The female body is that marble or gold that Midas-like and Powers-like men carve according to a pre-established pattern. The artistic metaphor plays the tragedy of the feminine figures, trapped in what Bourdieu calls the "invisible enclosure" of male domination. Greek mythology, as much as the Bible or other literary classical works like Milton's Paradise Lost, Lewis's The Monk or Bunyan's Pilgrim's Progress, thus appear to provide Hawthorne with the indispensable authoritarian references to feed his inspiration and his proto-feminist politics: this literary legacy is, under Hawthorne's pen, first Americanized, to be, finally, feminized. Women can, with his enlightened help, come of age, and break free from the ideal of the eternal feminine.

  'd like to focus on two female characters who, to me, epitomize Hawthorne's proto-feminist vision as they portray both concepts of minority and becoming-woman, the first by falling victim to male domination, both on a physical and a symbolical level; the second by trying to overcome this minorized situation. With Pandora and Marygold, Hawthorne manages to create an entangled piece of writing as it condenses a rewriting of the Greek myth of the Pandora Box and of the Biblical Myth of the Fall, itself being the original Ur-text of both the Greek myth and Hawthorne's version of it. Entanglement is then the right word when it comes to qualify this mythological patchwork that is the tale of "The Paradise of Children." The Christian connotation lurks behind this title which betrays a subtle process of Puritanization of the Greek fabric. Concerning Marygold, I'd like to analyze what I call the literary process of literalization of the figurative, which is the founding stone of Hawthorne's claim of the "becoming-woman."

MISCELLANEOUS

 Pearl, the minor woman par excellence;  Woman as the instrument of the regeneration of America; // "The Great Carbuncle," "The New Adam and Eve;"


Antigoni P Anninou
email: anninou@ece.upatras.gr

Peter P Groumpos
email: groumpos@ece.upatras.gr

Polychronopoulos Panagiotis

Modeling Health Diseases using Competitive Fuzzy Cognitive Maps

Keywords: Fuzzy cognitive maps, medical decision support system, competitive fuzzy cognitive maps, knee injuries

This paper presents the medical decision support systems (MDSS) and their architecture. The aim of this paper is to present a new approach in modeling knee injuries using Competitive Fuzzy Cognitive Maps (CFCMs). Basic theories of CFCMs are reviewed and presented. Decision Support Systems (DSS) for Medical problems are considered. Finally, it illustrates the development of an MDSS for finding knee injury with the architecture of CFCMs.

INTRODUCTION

A fuzzy cognitive map (FCM) is a soft computing technique, which is capable of dealing with complex systems in situations exactly as a human does using a reasoning process that can include uncertain and ambiguity descriptions [START_REF] Stylios | Modeling complex systems using fuzzy cognitive maps[END_REF], [START_REF] Groumpos | Modeling supervisory control systems using fuzzy cognitive maps[END_REF]. FCM is a promising modeling method for describing particular domains showing the concepts (variables) and the relationships between them (weights) while it encompasses advantageous features [START_REF] Papageorgiou | Brain tumor characterization using the soft computing technique of fuzzy cognitive maps[END_REF]. Fuzzy Cognitive Maps (FCM) are fuzzy-graph structures for representing causal reasoning. Their fuzziness allows hazy degrees of causality between causal objects (concepts) [START_REF] Anninou | Nonlinear Hebbian Learning Techniques and Fuzzy Cognitive Maps in Modeling the Parkinson's Disease[END_REF].

In this paper a decision support system will be implemented. Decision Support System (DSS) is defined as any interactive computerbased support system for making decisions in any complex system, when individuals or a team of people are trying to solve unstructured problems on an uncertain environment. Medical Decision Systems have to consider a high amount of data and information from interdisciplinary sources (patient's records and information, doctors' physical examination and evaluation, laboratory tests, imaging tests) and, in addition to this, medical information may be vague, or missing [START_REF] Groumpos | A theoretical mathematical modeling of Parkinson's disease using Fuzzy Cognitive Maps[END_REF][START_REF] Anninou | Nonlinear Hebbian Learning Techniques and Fuzzy Cognitive Maps in Modeling the Parkinson's Disease[END_REF]. Furthermore the Medical Diagnosis procedure is complex, taking into consideration a variety of inputs in order to infer the final diagnosis. Medical Decision Systems are complex ones, consisting of irrelevant and relevant subsystems and elements, taking into consideration many factors that may be complementary, contradictory and competitive; these factors influence each other and determine the overall diagnosis with a different degree. It is apparent that these systems require a modeling tool that can handle all these challenges and at the same time to be able to infer a decision. Thus, FCMs are suitable to model a Medical Decision Support Systems (MDSS) [START_REF] Anninou | Nonlinear Hebbian Learning Techniques and Fuzzy Cognitive Maps in Modeling the Parkinson's Disease[END_REF].

A special type of FCM has been introduced for Medical Diagnosis systems, with advanced capabilities, the Competitive Fuzzy Cognitive Map (CFCM) [START_REF] Georgopoulos | Augmented fuzzy cognitive maps supplemented with case based reasoning for advanced medical decision support[END_REF][START_REF] Georgopoulos | A fuzzy cognitive map approach to differential diagnosis of specific language impairment[END_REF]. Each decision concept represents a single decision, which means that these concepts must compete against each other so that only one of them dominates and is considered as the correct decision. This is the case of most medical applications, where, the symptoms have to conclude to one diagnosis. The factor-concepts take values from patient data. These concepts represent symptoms, experimental and laboratory tests, and their values are dynamically updated. The decision concepts are considered as outputs in which their calculated values outline the possible diagnosis for the patient [START_REF] Georgopoulos | Competitive Fuzzy Cognitive Maps Combined with Case Based Reasoning for Medical Decision Support[END_REF].

In this paper we present in a simple but illustrative way how useful the FCMs can be in medical problems. In section 2 the basic theories of CFCM, while the decision making support system in knee injuries is described in Section 3. The paper concludes in section 4.

Competitive Fuzzy Cognitive Maps

The algorithm that describes the Competitive Fuzzy Cognitive Maps consists of the following steps [START_REF] Georgopoulos | Competitive Fuzzy Cognitive Maps Combined with Case Based Reasoning for Medical Decision Support[END_REF]:

Step 1: Set values A i of nodes according to values of the factors involved in the decision process. The values Ai are described using linguistic variables.

Step 2: Read the weight matrix W. The interrelationships between nodes may be positive, negative or zero. The existing relationships between the concepts are described firstly, as ''negative'' or ''positive'' and secondly, as a degree of influence using a linguistic variable, such as ''low'', ''medium'', ''high'', etc. A positive weight means that the specific factor increases the possibility of diagnosis the interconnected disease. A negative one reduces the possibility of diagnosis the interconnected disease. Lack of interconnection indicates that there is no impact of the factor to the disease. By using a defuzzification method [START_REF] Saade | A unifying approach to defuzzification and comparison of the outputs of fuzzy controller[END_REF], weights between the factor-concepts and the decision-concepts are converted to initial values, which for the current research are between 0 and 1. These are placed in matrix W of size (n+m) x (n+m). The values in the first n columns correspond to the weighted connections from all the concepts towards the n decision-concepts, and the values in the remaining m columns correspond to the weighted connections from all the concepts towards the factor-concepts. This matrix also includes the -1 weight values for competition between output decision-concepts.

Step 3: Update the values: A new =A old *W.

Step 4: A=f (A new ), where the sigmoid function f belongs to the family of squeezing functions, and the following function is usually used to describe it:

1 1 x f e     (1)
This is the unipolar sigmoid function, in which λ>0 determines the steepness of the continuous function f(x).

Step 5: Repeat steps until equilibrium has been reached and the values of the concepts no longer change.

Step 6: The procedure stops and the final values of the decision-concepts are found, the maximum of which is the chosen decision.

Although the model FCMs-MDSS designed to include all possible symptoms, the causal factors and their relationships, there is a particular situation in which very few of the symptoms are available and taken into account. Thus, in such a diagnostic or prognostic model FCMs-MDSS, the decision will be based using only a small part of the nodes of the entire system, leading to a wrong decision or difficulty convergence since weights of active nodes reflect a small part of the knowledge of specialists.

By using a FCM augmented with a Case-Based-Reasoning (CBR), in such situations, the DSS would draw upon cases that are similar according to distance measures and would use the CBR subsystem to generate a sub-FCM emphasizing the nodes activated by the patient data and thus redistributing the causal weights between the concept-nodes.

Fig. 1. FCM-CBR

Competitive Fuzzy Cognitive Maps for Knee Injuries

Knee pain is a common complaint for many people. There are several factors that can cause knee pain. Awareness and knowledge of the causes of knee pain lead to a more accurate diagnosis. Management of knee pain is in the accurate diagnosis and effective treatment for that diagnosis. Knee pain can be either referred pain or related to the knee joint itself [START_REF] Majewski | Epidemiology of athletic knee injuries: A 10-year study[END_REF].

CFCM distinguish two main types of nodes: decision-nodes (decision-concepts) and factor-nodes [START_REF] Georgopoulos | A fuzzy cognitive map approach to differential diagnosis of specific language impairment[END_REF]. The following figure presents an example model CFC M that is used in order to determine the medical diagnosis, and includes types of FCMs concepts and causal relationships between them [START_REF] Georgopoulos | Competitive Fuzzy Cognitive Maps Combined with Case Based Reasoning for Medical Decision Support[END_REF].

Fig. 2. A General Medical Decision Support CFCM

All concepts may interact with each other and decide the value of diagnosis-nodes that are mutually exclusive in order to show one single diagnosis. In most medical applications according to symptoms, physicians need to complete only one diagnosis and then to determine the appropriate treatment.

In the specific model that describes knee injuries, there are two kinds of nodes:

─ n decision-nodes (as outputs): It's obvious that in order to achieve the competition, the relationships between each of these nodes with the rest should have a very large negative weight (-1). This implies that the higher price of a given node, should lead to a reduction in the values of competing nodes.

After defuzzification method Table 1 is converted to the following table :   Table 2. Numerical values of the Concepts Relationships After using the recursive algorithm each node converges to a final value and the decision node with the maximum value is the most possible diagnosis based on the model. In any case, if the model is correct, even with incomplete data, the result will confirm the experts' diagnosis.

Fig. 4. Subsequent values of concepts till convergence

We observe that in the specific case there is more than one output that has also high convergence values. This means that these types of injuries are very similar because of the adjacent symptoms. So patellofemoral pain syndrome, osteochondritis dissecans of the femoral condyles, and tibial tuberosity apophysitis are the predominant types of injuries that the patient of the example may suffer.

Conclusions

There is a need for software tools for Medical Decision Support Systems using theories of FCMs. In addition learning algorithms, such as Hebbian and non-linear, should be considered to further investigate Medical problems. The challenging research is the development of new suitable software tools. An interesting research topic is the development of recursive dynamic state equations for FCMs and the use of them for mathematical modeling of medical problems. This paper deals with a promising future research area. This approach that exploits the experts' knowledge and experience in a modern medical automation system can be used both for diagnosing diseases and also proposing a treatment. More simulation results and more examples are necessary in order to compute the system's reliability in the diagnosis of a specific injury. The model requires further research. To implement this, the cooperation of both physicians and engineers is necessary. The first ones should determine the right concepts and the rules and the second ones are responsible for the implementation of this problem.

C 1 : 2 : 3 : 4 : 5 : 6 : 7 :

 1234567 Fig.3shows the representation of the 15 factor-nodes (C8-C21), 7 decision-nodes (C1-C7) and the relationships between them.

Fig. 3 .

 3 Fig. 3. Competitive Fuzzy Cognitive Map for Knee Injuries

Table 1 .

 1 Linguistic Variables of the Concepts Relationships

	where:
	VVH: Very Very High
	VH: Very High
	H: High
	M: Medium
	L: Low
	VL: Very Low
	VVL: Very Very Low