Flora Koukiou
email: flora.koukiou@u-cergy.fr

On the entropy of the mean field spin glass model

From the study of a functional equation relating the Gibbs measures at two different tempratures we prove that the specific entropy of the Gibbs measure of the Sherrington-Kirkpatrick Spin Glass Model vanishes at the inverse temperature β = 4 log 2.

Introduction and main results

Over the last decade, mean field models of spin glasses have motivated increasingly many studies by physicists and mathematicians [START_REF] Aizenman | Some Rigorous Results on the Sherrington-Kirkpatrick Spin Glass Model[END_REF][START_REF] Guerra | Broken replica symmetry bounds in the mean field spin glass model[END_REF][START_REF] Guerra | The thermodynamic limit in mean field spin glass models[END_REF][START_REF] Koukiou | The low temperature free energy of the Sherrington-Kirkpatrick spin glass model[END_REF][START_REF] Parisi | A sequence of approximated solutions to the Sherrington-Kirkpatrick model for spin glasses[END_REF][START_REF] Pastur | Absence of self-averaging of the order parameter in the Sherrington-Kirkpatrick model[END_REF][START_REF] Talagrand | The Parisi formula[END_REF]. The existence of infinite volume limit of thermodynamic quantities is now rigorously established thanks to the development of numerous remarkable analytical techniques. For the Sherrington-Kirkpatrick model, the first major results of Guerra and Toninelli [START_REF] Guerra | The thermodynamic limit in mean field spin glass models[END_REF] on existence and uniqueness of the free energy, are generalized by Aizenman, Sims and Starr [START_REF] Aizenman | An extended variational principle for the SK spin-glass model[END_REF] in a scheme giving variational upper bounds on the free energy. Talagrand [START_REF] Talagrand | The Parisi formula[END_REF], under some conditions on the overlap function, contributed to the entirely rigorous account of the original formulae proposed by Parisi [START_REF] Parisi | A sequence of approximated solutions to the Sherrington-Kirkpatrick model for spin glasses[END_REF].

An interesting question, related to the behaviour of Gibbs measures, is the study of their specific entropy. Despite the numerous developments achieved lately on this model, the study of the properties of the entropy is still missing in the literature. The specific entropy decreases with the temperature and the high temperature entropy can easily be estimated. By lowering the temperature the entropy should eventually vanish and an early result, given in [START_REF] Aizenman | Some Rigorous Results on the Sherrington-Kirkpatrick Spin Glass Model[END_REF], corroborates the idea that the entropy does not vanish very fast. In this note we estimate the value of the (low) temperature at which the mean entropy of the Gibbs measure vanishes.

The approach we use here is totally self-contained. From the low-temperature results, we need solely the existence of the thermodynamic limit of the quenched specific free energy and its self-averaging property.

We first recall some basic definitions. Suppose that a finite set of n sites is given. With each site we associate the one-spin space Σ := {1, -1}. The natural configuration space is then the product space Σ n = {-1, 1} n = Σ n , with cardΣ n = 2 n equipped with the uniform probability measure ν n . For each σ ∈ Σ n , the finite volume Hamiltonian of the model is given by the following real-valued function on

Σ n H n (σ) = - 1 n 1≤i < j ≤n J i j σ i σ j ,
where the family of couplings J = (J i j ) 1≤i < j ≤n are independent centred Gaussian random variables of variance 1.

At the inverse temperature β = 1 T > 0, the disorder dependent partition function Z n (β, J ), is given by the sum of the Boltzmann factors

Z n (β, J ) = σ e -βH n (σ,J ) .
Moreover, if E J denotes the expectation with respect to the randomness J , it is very simple to show that E J Z n (β, J ) = 2 n e β 2 4 (n-1) . When the randomness J is fixed, the corresponding conditional Gibbs probability measure is denoted by µ n,β (σ|J ) and given by:

µ n,β (σ|J ) = e -βH n (σ,J ) Z n (β, J ) .
The entropy of µ n,β , is defined as usual by S(µ n,β,J ) =σ µ n,β (σ|J ) log µ n,β (σ|J ). The real functions

f n (β) = 1 n E J log Z n (β, J )
and

f n (β) = 1 n log E J Z n (β, J ),
define the quenched average of the specific free energy and the annealed specific free energy respectively. The ground state energy densityn (J ) is given by

-n (J ) = 1 n inf σ∈Σ n H n (σ, J ).
At the low temperature region (β > 1), the following two infinite volume limits

lim n→∞ f n (β, J ) = f ∞ (β),
and,

-lim n→∞ n (J ) = lim β→∞ f ∞ (β) β = -0
exist for almost all J and are non random; this result has been rigorously proved by Guerra and Toninelli [START_REF] Guerra | The thermodynamic limit in mean field spin glass models[END_REF].

The main results of this note are stated in the following and proved in the next section. Proposition: Almost surely, at the inverse temperature β * = 4 log 2 = 2.77258 • • •, the thermodynamic limit of the quenched free energy is given by

f ∞ (β * ) = lim n→∞ 1 n E J log Z n (β * , J ) = f ∞ (1) + (β * -1) log 2 = β 2 * 4 + β 2 1 4 = β * log 2 + 1 4 .
The Parisi formula provides with the expression of the free energy for the entire low temperature region in terms of a functional equation; the pertinence of the precise calculation of the limit at a particular value of the temperature stems from its usefulness in determining the point where the entropy vanishes. This gives new insight to the behaviour of the model and is summarised in the following Theorem: At the inverse temperature β * = 4 log 2 = 2.77258 • • •, the specific entropy s(µ β * ) of the Gibbs measure vanishes almost surely:

s(µ β * ) := lim n→∞ 1 n S(µ n,β * ,J ) = -lim n→∞ 1 n σ µ n,β * (σ|J ) log µ n,β * (σ|J ) = 0.
Remark: The formulation of the above statement assumes that the limit lim n→∞ 1 n S(µ n,β * ,J ) exists and is independent of J . This follows from general principles and can immediately be obtained from the existence and self-averaging of the low temperature specific free energy.

Proof of the main results

Notice first, that for all β > 0, the quenched limit f ∞ (β) exists and is a convex function of β [START_REF] Guerra | The thermodynamic limit in mean field spin glass models[END_REF]. Let β 1 ≡ 1. From the high temperature results [START_REF] Aizenman | Some Rigorous Results on the Sherrington-Kirkpatrick Spin Glass Model[END_REF], we have, almost surely, that

f ∞ (β 1 ) = lim n→∞ 1 n E J log Z n (β 1 , J ) = lim n→∞ 1 n log E J Z n (β 1 , J ) = f ∞ (β 1 ) = log 2 + β 2 1 4 = log 2 + 1 4 .
The following figure 1 illustrates the definition of the inverse temperature β * ; the an-

nealed free energy f ∞ (β) = log 2 + β 2
4 is plotted as a function of β and the straight line is defined by

β β 1 f ∞ (β 1 ) ≡ β f ∞ (β 1
). The two graphs intersect at β 1 = 1 and β * = 4 log 2 = 2, 77258 • • •. One can now easily check that, at β = β * , the annealed free energy f ∞ (β * ) is simply related to f ∞ (β 1 ) by the following relationship

f ∞ (β * ) = β 2 * 4 + log 2 = β * β 1 ( β * β 1 4 + β 1 β * log 2) = β * β 1 (log 2 + 1 4 ) = β * β 1 f ∞ (β 1 ).
We denote by T the mapping T : µ n,β 1 (σ|J ) → µ n,β (σ|J ) defining, for all β > β 1 , the Gibbs probability measure µ n,β (σ|J ) via the functional equation

µ n,β (σ|J ) := exp(-βH n (σ, J )) Z n (β, J ) = µ β/β 1 n,β 1 (σ|J ) Z β/β 1 n (β 1 , J ) Z n (β, J ) .
Notice that β/β 1 is a non dimensional quantity. Moreover the value β 1 fixes the temperature scale i.e. the temperature β > β 1 is expressed in units where β 1 ≡ 1.

β 1 β * β → f ∞ (β) = ln 2 + β 2 4 β → β f ∞ (1)
Figure 1: The value β * = 4 log 2, is given by the intersection of the graph of the annealed free energy f ∞ (β) with the straight line β f ∞ (1).

Since µ n,β is a probability on the configuration space, summing up over the configurations σ and taking the thermodynamic limit, we have indeed

lim n→∞ 1 n log σ µ n,β (σ|J ) = lim n→∞ 1 n log σ µ β/β 1 n,β 1 (σ|J ) + α ∞ (β 1 , β) = 0,
where the limit α ∞ (β 1 , β) is given by

α ∞ (β 1 , β) = lim n→∞ β β 1 1 n log Z n (β 1 , J ) -lim n→∞ 1 n log Z n (β, J ) = lim n→∞ β β 1 1 n E J log Z n (β 1 , J ) -lim n→∞ 1 n E J log Z n (β, J ) (due to the self-averaging) = β β 1 f ∞ (β 1 ) -f ∞ (β).
The existence, for all β > β 1 , of the limit α ∞ (β 1 , β) follows immediately from the existence of the two limits f ∞ (β 1 ) and f ∞ (β). Now, by making use of the relation between the limits f ∞ (β * ) and f ∞ (β 1 ), one can check, that for β = β * , the limit α ∞ (β 1 , β * ) gives the deviation of the free energy f ∞ (β * ) from its mean value :

a ∞ := α ∞ (β 1 , β * ) = β * β 1 f ∞ (β 1 ) -f ∞ (β * ) = f ∞ (β * ) -f ∞ (β * ).
The proof of the proposition reduces thus in determining the value a ∞ .

Proof of the Proposition: At β = β 1 , the quenched limit f ∞ (β 1 ) equals the annealed one f ∞ (β 1 ) = β 2 1 /4 + log 2, where the term β 2 1 /4 comes from the mean value of the Boltzmann factor (i.e. the typical behaviour and the mean behaviour coincide at this temperature). Since for β > β 1 , the typical and the average behaviour are no longer the same, we use the standard large deviations argument in order to make the deviant behaviour at β * look like the typical behaviour at β 1 .

The affine mapping T on measures induces a transformation on the free energies

reading f ∞ (β * ) = β 2 * 4 + log 2 = β * β 1 f ∞ (β 1 )
. It follows that the pre-image of the term β 2 * /4 -coming from the average of the Boltzmann factor -(point C of the figure 2), is β 1 β * /4 = β 1 log 2 = log 2 (point C ); one gets the value of the free energy f ∞ (β 1 ) if the term β 2 1 /4 = 1/4 is added to this pre-image. We remark that the sheer particularity of the two temperatures β 1 and β * is that the pre-image of log 2 is 1/4! Therefore, to obtain the quenched limit at β * is enough to add to the image of log 2 (i.e. to the segment OC ) the value 1/4 (segment C B ).

One can now easily check that the difference of the two limits f ∞ (β * ) and f ∞ (β 1 ), is simply given by the segment O A:

f ∞ (β * ) -f ∞ (β 1 ) = (β * -β 1 ) log 2. β 1 β * β → f ∞ (β) = ln 2 + β 2 4 β → β f ∞ (1)
β → β log 2 A C B O A' C' B'
f ∞ (β * ) = (β * -β 1 ) log 2 + f ∞ (β 1 ) = β * log 2 + β 2 1 4 = β 2 * 4 + 1 4 = 2.1718 • • • ,
and, moreover

a ∞ = f ∞ (β * ) -f ∞ (β * ) = β * β 1 f ∞ (β 1 ) -f ∞ (β * ) = β * β 1 4 - β 2 1 4 .
One can check that the value of f ∞ (β * ) is slightly lower than the bound one can obtain by making use of the spherical model (2.2058 • • •).

Proof of the Theorem: For β * , we have

s(µ β * ) = lim n→∞ 1 n S(µ n,β * ,J ) = -lim n→∞ 1 n σ µ n,β * (σ|J ) log µ β * /β 1 n,β 1 (σ|J ) Z β * /β 1 n (β 1 , J ) Z n (β * , J ) = -lim n→∞ 1 n σ µ n,β * (σ|J ) log µ β * /β 1 n,β 1 (σ|J ) -α ∞ ,
and, by the positivity of the entropy one checks readily that

lim n→∞ 1 n σ µ n,β * (σ|J ) log µ n,β 1 (σ|J ) ≤ - β 1 β * α ∞ = 1 4β * - 1 4 .
In the following, we shall show that this inequality is saturated. For this we introduce a slightly different notation. Let W n,β 1 (σ|J ) = e -β 1 H n (σ,J ) /2 nβ 1 be the random weight associated with each configuration σ ∈ Σ n . The Gibbs measures µ n,β 1 (σ|J ) and µ n,β * (σ|J ) are now given by

µ n,β 1 (σ|J ) = W n,β 1 (σ|J ) σ W n,β 1 (σ|J )
, and,

µ n,β * (σ|J ) = W β * /β 1 n,β 1 (σ|J ) σ W β * /β 1 n,β 1 (σ|J )
.

We have indeed, from the high temperature results,

lim n→∞ 1 n E J log σ W n,β 1 (σ|J ) = lim n→∞ 1 n log E J σ e -β 1 H n (σ,J ) 2 nβ 1 = β 2 1 4 + log 2 -β 1 log 2 = 1 4 ,
and, from the previous proposition,

lim n→∞ 1 n E J log σ W β * /β 1 n,β 1 (σ|J ) = lim n→∞ 1 n E J log σ e -β 1 H n (σ,J ) 2 nβ 1 β * β 1 = β 2 * 4 -β * log 2 + β 2 1 4 = 1

Figure 2 :

 2 Figure 2: The affine map T maps C to C . The length of the segment A B corresponds to the value f ∞ (β 1 ) that is parallel transported to the segment AB . The segment OB equals f ∞ (β * ). The dashed lines A A and B B are parallel to C C .

  4 , i.e. the behaviour of the sums σ W n,β 1 (σ|J ) and σ W β * /β 1 n,β 1 (σ|J ) is the same. Thus, for the comparison of the two measures, namely for distinguishing between the behaviour of the summands W n,β 1 (σ|J ) and W β * /β 1 n,β 1 (σ|J ) we need additional information. We introduce the relative entropy density s(µ β * |µ β 1 ) of the measure µ β * w.r.t. the measure µ β 1 which gives the extend to which the measure µ β * "differs" from the measure µ β 1 :β 1 )where the equality of the limsup and the limit is a consequence of the positivity of s(µ β * |µ β 1 ).Using now the functional definition of the measure one gets s(µ β * |µ β 1 ) = lim

	s(µ β * |µ β 1 ) := lim n→∞	1 n	S(µ n,β * |µ n,β 1 ) = lim n→∞	1 n σ	µ n,β * (σ|J ) log	µ n,β * (σ|J ) µ n,β 1 (σ|J )	.
	This limit exists and it is a non-negative function vanishing in the case the two mea-
	sures are equal. We notice moreover that
			s(µ β * |µ β 1 ) = lim n→∞	1 n σ	µ n,β * (σ|J ) log	µ n,β * (σ|J ) µ n,β 1 (σ|J )
											= lim n→∞	1 n σ	µ n,β * (σ|J ) log	W W n,β 1 (σ|J ) β * /β 1 (σ|J ) n,β 1
											= lim n→∞	1 n σ	µ n,β β * β 1 n,β 1 -1	(σ|J ).
	Obviously, W	β * /β 1 n,β 1	(σ|J ) ≤ σ W	β * /β 1 n,β 1	(σ|J ). Hence,
	lim sup n→∞	1 n σ	µ n,β * (σ|J ) logW	β * β 1 n,β 1 -1	(σ|J ) = lim n→∞	1 n	log	σ	W	β * /β 1 n,β 1	(σ|J )	1-	β 1 β *	.
	and, consequently,
	s(µ β β * /β 1 n,β 1 4β n→∞ (σ|J ) 1-β 1 β * 1 = 1 n σ µ n,β * (σ|J ) µ n,β * (σ|J ) log µ n,β 1 (σ|J )
										= lim n→∞	1 n σ	µ n,β * (σ|J ) log µ n,β 1 β * β 1 -1	(σ|J ) + α ∞
										=	1 4β *	(β * -β 1 ).
	Recalling that α ∞ =	β 1 4 (β * -β 1 ), it follows immediately that
		lim n→∞	1 n σ	µ n,β 1 4β *	-	1 4	= -	β 1 β

* (σ|J ) logW * |µ β 1 ) = lim n→∞ 1 n log σ W * (β * * (σ|J ) log µ n,β 1 (σ|J ) = * α ∞

which proves the theorem. Remarks: Another interesting quantity is the relative entropy density s(µ β * |ν) of the measure µ n,β * w.r.t. the uniform measure ν n (σ) :

(We recall that s(µ

). One can also easily check that the value of the limit α ∞ corresponds to the entropy difference α ∞ = s(µ β 1 )s(µ β * ).

Concluding remarks

In this note we showed that the mean entropy of the Gibbs measure vanishes at the inverse temperature β * = 4 log 2. A related question concerns the Hausdorff dimension of the support of the Gibbs measure. From our result on the entropy one can easily show that this dimension vanishes at β * .

A last observation concerns the value of the temperature β * : it is obtained from the relationship between the free energies f∞ (β * ) and f ∞ (1); moreover, one can readily check that β * = β 2 c , where β c = 2 log 2 is the critical temperature of the Random Energy Model (REM). The REM is defined by 2 n energy levels E i (i = 1, • • • , n), a family of random, independent, identically distributed random variables; many results are qualitatively the same as those of the SK model. It would be interesting to clarify this relationship in order to obtain some information on the behaviour and properties of the Gibbs measure at low temperatures. Both β c and β * are to be compared with the value at β 1 ≡ 1, i.e. the maximum value of β where the free energies of the two models coincide. What we learn by the comparison of the two models is that the Gibbs measure of the SK has seemingly a richer structure than for the REM. As a matter of fact, the entropy of the REM vanishes at β c while the entropy of the SK model is still strictly positive at this point.


Xavier Roulleau

Erwan Rousseau

ON THE HYPERBOLICITY OF SURFACES OF GENERAL TYPE WITH SMALL c 2 1

Keywords: 2000 Mathematics Subject Classification. 14J25, 32Q45, 14J29 Hyperbolicity, Green-Griffiths-Lang conjecture, Horikawa surfaces, Orbifolds

Surfaces of general type with positive second Segre number s2 := c 2 1 -c2 > 0 are known by results of Bogomolov to be algebraically quasihyperbolic i.e. with finitely many rational and elliptic curves. These results were extended by McQuillan in his proof of the Green-Griffiths conjecture for entire curves on such surfaces. In this work, we study hyperbolic properties of minimal surfaces of general type with minimal c 2 1 , known as Horikawa surfaces. In principle these surfaces should be the most difficult case for the above conjecture as illustrate the quintic surfaces in P 3 . Using orbifold techniques, we exhibit infinitely many irreducible components of the moduli of Horikawa surfaces whose very generic member has no rational curves or even is algebraically hyperbolic. Moreover, we construct explicit examples of algebraically hyperbolic and quasi-hyperbolic orbifold Horikawa surfaces.

Introduction

Our motivations are the following conjectures of Green-Griffiths and Lang:

Conjecture 1 (Lang [START_REF] Lang | Hyperbolic and Diophantine analysis[END_REF]). Let X be a complex variety of general type. Then a proper Zariski closed subset Z of X contains all its subvarieties not of general type. In particular, X has only a finite number of codimension-one subvarieties not of general type.

Even in the case of surfaces, this conjecture is still open, except in some specific cases such as surfaces of general type with irregularity at least two [START_REF] Lu | On surfaces of general type with maximal Albanese dimension[END_REF]. It has attracted a lot of attention because of its important conjectural links with arithmetic: according to the Bombieri-Lang conjecture, the rational points on a variety of general type defined over a number field should not be Zariski dense.

A surface satisfying Conjecture 1, i.e. with finitely many rational and elliptic curves, is said to be algebraically quasi-hyperbolic. Following the terminology of Demailly [START_REF] Demailly | Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials[END_REF], a projective variety X ⊂ P n is called algebraically hyperbolic if there exists a positive real number ε such that 2g(C)-2 ≥ ε deg C for each reduced irreducible curve C ⊂ X, where g(C) and deg C are the geometric genus and the degree of the curve C ֒→ X respectively.

The analytic version of Conjecture 1 is:

Conjecture 2 (Green-Griffiths [START_REF] Green | Two applications of algebraic geometry to entire holomorphic mappings[END_REF], Lang [START_REF] Lang | Hyperbolic and Diophantine analysis[END_REF]). Let X be a variety of general type. Then a proper Zariski closed subset of X contains all images of entire curves f : C → X.

A projective variety satisfying Conjecture 2 is said to be quasi-hyperbolic and a variety that contains no non-constant entire curve is said to be hyperbolic.

Let us recall that minimal surfaces of general type X are classified according to their Chern numbers. These Chern numbers c 2 1 , c 2 satisfy the two famous inequalities of Noether and Bogomolov-Miyaoka-Yau (see for example [START_REF] Barth | Compact complex surfaces, volume 4 of Ergebnisse der Mathematik und ihrer Grenzgebiete[END_REF]):

1 5 (c 2 -36) ≤ c 2 1 ≤ 3c 2 .
If c 2 1 = 3c 2 then, by Yau [START_REF] Yau | On Calabi's conjecture and some new results in algebraic geometry[END_REF], X is a quotient of the unit ball. Therefore X is hyperbolic.

A striking overhang toward Conjecture 1 is its proof by Bogomolov for surfaces whose Segre number s 2 = c 2 1 -c 2 is positive [START_REF] Bogomolov | Families of curves on a surface of general type[END_REF]. The key point in Bogomolov's proof is that for surfaces with positive Segre number, the sufficiently high symmetric differentials have global sections.

An extension of Bogomolov's result to its analytic version was obtained by Lu and Yau [START_REF] Lu | Holomorphic curves in surfaces of general type[END_REF] proving conjecture 2 for surfaces with c 2 1 > 2c 2 . Then McQuillan [START_REF] Mcquillan | Diophantine approximations and foliations[END_REF] proved Conjecture 2 for surfaces with positive second Segre number.

For surfaces with s 2 ≤ 0 there is no such a good result. Demailly and El Goul [START_REF] Demailly | Hyperbolicity of generic surfaces of high degree in projective 3-space[END_REF] proved Conjecture 2 for some surfaces with 13c 2 1 > 9c 2 . In the extreme case, a surface that reaches the equality c 2 = 5c 2 1 + 36 if c 2 1 is even and c 2 = 5c 2 1 + 30 otherwise is called a Horikawa surface. Thus the Horikawa surfaces are the one for which the Segre number s 2 = c 2 1c 2 is the most negative as possible in the geography of (minimal) surfaces of general type.

As the previous list of results suggests, one is naturally lead to believe that it is for these surfaces that the above conjectures will be the most difficult to establish.

Even for hypersurfaces in P 3 , (which have s 2 < 0), we are far from complete results. The case of quintics (c 2 1 = 5, c 2 = 55) is particularly difficult to treat and this may be explained by the fact that they are Horikawa surfaces, thus with minimal second Segre number. Indeed, a conjecture by Kobayashi predicts that generic quintics should be hyperbolic. We know by Geng Xu [START_REF] Xu | Subvarieties of general hypersurfaces in projective space[END_REF] that a very generic surface of degree d ≥ 5 in P 3 contains no curve of geometric genus 0 or 1. But we still do not know that they are hyperbolic, even algebraically hyperbolic.

Moreover, we have very few examples of hyperbolic surfaces of low degree. A striking fact is that no example of hyperbolic surface is known for d = 5.

In this paper we investigate the hyperbolic properties of Horikawa surfaces using their very interesting geometric properties. Indeed, we know that most Horikawa surfaces realizing the equality c 2 = 5c 2 1 + 36 appear as ramified coverings [START_REF] Horikawa | Algebraic surfaces of general type with small c 2 1[END_REF]. Therefore it is very natural to associate to such surface an orbifold. Our philosophy is to use this orbifold to study the geometry of curves in Horikawa surfaces.

We will see that "orbifold" techniques as systematically introduced by Campana ( [START_REF] Campana | Orbifolds, special varieties and classification theory[END_REF], [START_REF] Campana | Orbifoldes spéciales et classification biméromorphe des variétés kählériennes compactes[END_REF], see also [START_REF] Rousseau | Hyperbolicity of geometric orbifolds[END_REF]) are useful in this context.

Let N ≥ 0 be an integer and let F N be the N th Hirzebruch surface. The surface F N has a natural fibration F N → P 1 . We denote by F a fiber, and by T the section of this fibration such that T 2 = N . Any divisor D on F N is numerically equivalent to aT + bF for a, b integers. We denote by (a, b) the equivalence class of D. The holomorphic Euler characteristic of a surface is χ := c 2 1 +c 2 12 . Horikawa obtained the following classification [START_REF] Horikawa | Algebraic surfaces of general type with small c 2 1[END_REF]: a Horikawa surface with even c 2 1 is either (1) a double covering of P 2 branched along an octic (χ = 4), (2) a double covering of P 2 branched along a curve of degree 10 (χ = 7), (3) a double covering of F N branched along a curve of type (6, 2a), (2a ≥ -N ) (χ = 3N + 2a -1). Here the branch curve has at most ADE singularities. We show: Theorem 3.

(1) Let X be a very generic Horikawa surface of type [START_REF] Barth | Compact complex surfaces, volume 4 of Ergebnisse der Mathematik und ihrer Grenzgebiete[END_REF]. Then X has no rational curves.

(2) Let X be a very generic Horikawa surface of type (2). Then X is algebraically hyperbolic, in particular X has no rational or elliptic curves. (3) Let X be a very generic Horikawa surface of type (3) with a = 3 and χ = 3N + 5. Then X has no rational curves.

Here, the terminology "very generic" is used to indicate that the property is satisfied outside a countable union of algebraic subsets in the moduli space. This result is obtained as a consequence of more general results on the algebraic hyperbolicity of branched covers which may be of independent interest.

One of the most natural examples of surfaces with s 2 < 0 are hypersurfaces in P 3 . In [START_REF] Bogomolov | Hyperbolicity of nodal hypersurfaces[END_REF], Bogomolov and De Oliveira studied the hyperbolicity of hypersurfaces of degree d > 5 with sufficiently many nodes. Translated into the langue of orbifold, their key observation is that for such a surface X, the natural structure of orbifold X has positive orbifold Segre number s 2 (X ) > 0. We investigate hyperbolic properties of the quintic surfaces that are degree 5 cyclic covers of P 2 branched over 5 lines in general position. In this study, we develop the theory of jet differentials in the orbifold setting and, as an application, prove Theorem 4. Let X be a quintic surface that is a degree 5 cyclic cover of P 2 branched over 5 lines in general position. Then any entire orbifold curve f : C → X satisfies a differential equation of order 2.

Using Nevanlinna theory and constructions of Persson [START_REF] Persson | Horikawa surfaces with maximal Picard numbers[END_REF] of Horikawa surfaces with maximal Picard number, we exhibit some explicit examples of quasihyperbolic Horikawa orbifold surfaces.

Theorem 5. For χ equal to 4 and 2k -1 (for any integer k > 2), there exists quasi-hyperbolic orbifold Horikawa surfaces whose minimal resolutions have Euler characteristic χ.

To finish this introduction, let us remark that in characteristic p > 0 the analog of the Lang conjecture is false. In [START_REF] Liedtke | Unirational Surfaces on the Noether Line[END_REF], unirational Horikawa surfaces are constructed over fields of positive characteristic, these unirational surfaces have maximal Picard number.

The paper is structured as follows. In section 2 we recall the definition of orbifolds and the classical results we will use. In section 3 we prove our main Theorem 3 and some further results on surfaces that are cyclic ramified covers of the plane. In section 4 we prove Theorem 4. Section 5 is devoted to the construction of quasi-hyperbolic Horikawa orbifold surfaces.

Acknowledgements. Part of this research was done during the authors stay in Strasbourg University. We thank Matthias Schütt for many discussions on singularities of quintic surfaces, Julien Grivaux for discussions on orbifolds, Benoît Claudon and Jevgenija Pavlova for their drawings. We also thank the referee for his remarks and suggestions.

2. Orbifold set-up 2.1. Orbifolds. As in [START_REF] Ghigi | Kähler-Einstein metrics on orbifolds and Einstein metrics on spheres[END_REF], we define orbifolds as a particular type of log pairs. The data (X, ∆) is a log pair if X is a normal algebraic variety (or a normal complex space) and ∆ = i d i D i is an effective Q-divisor where the D i are distinct, irreducible divisors and d i ∈ Q.

For orbifolds, we need to consider only pairs (X, ∆) such that ∆ has the form

∆ = i 1 - 1 m i D i ,
where the D i are prime divisors and m i ∈ N.

Definition 6. An orbifold chart on X compatible with ∆ is a Galois covering ϕ : U → φ(U ) ⊂ X such that (1) U is a domain in C n and ϕ(U ) is open in X, (2)
the branch locus of ϕ is ⌈∆⌉ ∩ ϕ(U ), (3) for any x ∈ U ′′ := U \ ϕ -1 (X sing ∪ ∆ sing ) such that ϕ(x) ∈ D i , the ramification order of ϕ at x verifies ord ϕ (x) = m i .

Definition 7. An orbifold X is a log pair (X, ∆) such that X is covered by orbifold charts compatible with ∆.

Remark 8.

(1) In the language of stacks, we have a smooth Deligne-Mumford stack π : X → X, with coarse moduli space X.

(2) More generally, Campana introduces geometric orbifolds in [START_REF] Campana | Orbifolds, special varieties and classification theory[END_REF] as pairs of this type but which are not necessarily locally uniformizable.

Example 9. Let X be a complex manifold and ∆ = i (1 -1 m i )D i with a support ⌈∆⌉ which is a normal crossing divisor, i.e. for any point x ∈ X there is a holomorphic coordinate system (V, z 1 , . . . , z n ) such that ∆ has equation

z (1-1 m 1 ) 1 . . . z (1-1 mn ) n = 0.
Then (X, ∆) is an orbifold. Indeed, fix a coordinate system as above. Set

ϕ : U → V, φ(x 1 , . . . , x n ) = (x m 1
1 , . . . , x mn n ). Then (U, φ) is an orbifold chart on X compatible with ∆.

In the case of surfaces we have more examples of orbifolds looking at singularities that naturally appear in the logarithmic Mori program. Definition 10. Let (X, ∆), ∆ = i 1 -1 m i C i , be a pair where X is a normal surface and K X +∆ is Q-Cartier. Let π : X → X be a resolution of the singularities of (X, ∆), so that the exceptional divisors, E i and the components of ∆, the strict transform of ∆, have normal crossings and

K X + ∆ + i E i = π * (K X + ∆) + i a i E i .
We say that (X, ∆) is klt (Kawamata log terminal) if m i < ∞ and a i > 0 for every exceptional curve E i .

Thanks to a result of [START_REF] Nakamura | Classification and uniformisation of log-cannonical singularities in the presence of branch loci[END_REF] (see also the appendix of [START_REF] Campana | Abelianity conjecture for special compact Kahler threefolds[END_REF]), we have Example 11. Let (X, ∆), ∆ = i 1 -1 m i C i , be a klt pair with X a surface, then (X, ∆) is an orbifold. 2.2. Chern classes. Let π : X → (X, ∆) be a two dimensional orbifold. Let S be the singular points of X and of the divisor ⌈∆⌉, ∆ = (1 -1 m i )D i . The orbifold canonical line bundle is: K X = π * (K X + ∆), therefore:

c 1 (X ) = -π * (K X + (1 - 1 m i )D i ).
To each point p of S, there is a well defined integer β(p), the order of the isotropy group, such that by the orbifold Gauss-Bonnet formula [START_REF] Töen | Théorèmes de Riemann-Roch pour les champs de Deligne Mumford[END_REF]:

c 2 (X ) = e(X) - (1 - 1 m i )e(D i \ S) - p∈S (1 - 1 β(p)
).

Let p ∈ S be a smooth point of X.

Lemma 12. Suppose that p is an ADE singularity of ⌈∆⌉. We have : Type

β(p) A 1 m i m j A 2n 2 2n+1 ( 1 m i + 1 2n+1 -1 2 ) -2 A 2n-1 4 n ( 1 m i + 1 m j + 1 n -1) -2 , n ≥ 2 D 2n+2 4 n ( 1 m i + 1 m j + 1 nm k -1) -2 D 2n+3 2(2n + 1)m 2 i , n ≥ 1 E 7
96 where the branches are as follows: Proof. See [START_REF] Uludag | Galois coverings of the plane by K3 surfaces[END_REF] Table 2.3, [START_REF] Uludag | Covering relations between ball-quotient orbifolds Math[END_REF], [START_REF] Campana | Abelianity conjecture for special compact Kahler threefolds[END_REF] and [START_REF] Kobayashi | Uniformization of complex surfaces, Kähler metric and moduli spaces[END_REF].

r m i m j r m i r m k m i m j r m i 2 r 2 2 A 2n+1 A 2n D 2n+2 D 2n+1 E 7
Let C 1 , . . . be disjoint reduced divisors on X whose irreducible components are (-2)-curves. There exist a surface X ′ and map X → X ′ such that the C i 's are contracted onto ADE-singularities and that is an isomorphism outside. Let a n (resp d n , e n ) be the number of A n (resp. D n , E n ) singularities on X ′ . Proposition 13. ([27], 1.8). The surface X ′ has a natural structure of orbifold X and its Chern numbers are c 2 1 (X ) = c 2 1 (X) and:

(2.1)

c 2 (X ) = c 2 (X) - (n + 1)(a n + d n + e n ) + a n n + 1 + d n 4(n -2) + e 6 24 + e 7 48 + e 8 120 .
The denominators 4(n -2), 24, 48, 120 are the order of the binary dihedral BD 4(n-2) , the binary tetrahedral, the binary octahedral, and the binary icosahedral group respectively.

2.3.

Orbifold Riemann-Roch. Let L be an orbifold line bundle on the orbifold X of dimension n. We will use Kawazaki's orbifold Riemann-Roch theorem [START_REF] Kawasaki | The Riemann-Roch theorem for complex V-manifolds[END_REF] or Toën's for Deligne-Mumford stacks [START_REF] Töen | Théorèmes de Riemann-Roch pour les champs de Deligne Mumford[END_REF] using intersection theory on stacks. Theorem 14 ([37]). Let X be a Deligne-Mumford stack with quasi-projective coarse moduli space and which has the resolution property (i.e every coherent sheaf is a quotient of a vector bundle). Let E be a coherent sheaf on X then

χ(X , E) = X ch(E) T d(T X ).
From this formula, we obtain the asymptotic:

χ(X , L k ) = c 1 (L) n n! k n + O(k n-1 ),
using orbifold Chern classes.

It should be remarked that we use here the fact that we work only with effective orbifolds: the stabilizers are generically trivial therefore the correction terms of the orbifold Riemann-Roch do not affect the leading term.

We will apply this result to the case of orbifold surfaces X of general type with big orbifold cotangent bundle Ω X . Indeed, let T X be the orbifold tangent bundle. Then P(T X ) is naturally an orbifold and we can apply the previous Riemann-Roch formula to the tautological line bundle O P(T X ) [START_REF] Barth | Compact complex surfaces, volume 4 of Ergebnisse der Mathematik und ihrer Grenzgebiete[END_REF]. Orbifold Serre duality gives

h 2 (X , S m Ω X ) = h 0 (X , S m Ω X ⊗ K 1-m X ) ֒→ h 0 (X , S m Ω X ). As a corollary, if c 2 1 (X ) -c 2 (X ) > 0 then H 0 (S m Ω X ) ≥ c.m 3
for a positive constant c.

Orbifold hyperbolicity.

In the context of manifolds, complex hyperbolicity is concerned with the geometry of rational, elliptic curves and more generally entire curves i.e. holomorphic maps from C. For orbifold hyperbolicity, algebraic curves are replaced by orbifold Riemann surfaces C i.e. the data of a Riemann surface C of genus g ≥ 0 and r points p 1 , . . . , p r ∈ X marked by orders of stabilizer groups m 1 , . . . , m r ≥ 2 corresponding to the

Q-divisor ∆ = i 1 -1 m i p i . Definition 15. An orbifold Riemann surface C is rational (resp. elliptic) if deg(K C ) < 0 i.e. 2g -2 + i 1 -1 m i < 0 (resp. deg(K C ) = 0).
Example 16. A case-by-case check gives that (C, ∆) is elliptic if it is isomorphic to one of the following orbifold curves: [START_REF] Bloch | Sur les systèmes de fonctions uniformes satisfaisant à l'équation d'une variété algébrique dont l'irrégularité dépasse la dimension[END_REF][START_REF] Bogomolov | Symmetric tensors and geometry of P N subvarieties[END_REF][START_REF] Bogomolov | Symmetric tensors and geometry of P N subvarieties[END_REF], [START_REF] Bogomolov | Families of curves on a surface of general type[END_REF][START_REF] Bogomolov | Families of curves on a surface of general type[END_REF][START_REF] Bogomolov | Families of curves on a surface of general type[END_REF].

✷ (E, ∅) where E is an elliptic curve. ✷ (P 1 , 1 -1 m 1 {0} + 1 -1 m 2 {1} + 1 -1 m 3 {∞} where (m 1 , m 2 , m 3 ) is either (2, 3, 6),
✷ (P 1 , 1 -1 2 {0} + 1 -1 2 {1} + 1 -1 2 {p 3 } + 1 -1 2 {∞} with p 3 ∈ C \ {0, 1}.
Recall that an orbifold map between orbifolds f : X 1 → X 2 is a map between the underlying spaces X 1 and X 2 which lifts to an equivariant map in orbifold charts.

Definition 17. An orbifold map p : X 1 → X 2 is an orbifold covering if every x ∈ X 2 is in some U ⊂ X 2 such that for every component V of p -1 (U ) the corresponding orbifold chart φ 1 : Ṽ → V verifies that p • φ 1 : Ṽ → U is an orbifold chart for U .
Definition 18. Let X be an orbifold. An orbifold rational (resp. elliptic) curve in X is the image of an orbifold map f : C → X where C is rational (resp. elliptic).

A hyperbolic orbifold should not have rational and elliptic curves and more generally: Definition 19. An orbifold X is (Brody-)hyperbolic if there is no non-constant orbifold map f : C → X . An orbifold X is quasi-hyperbolic if the Zariski closure of the union of the images of orbifold maps f : C → X is a proper sub-variety.

Example 20. From the uniformization theorem available in the orbifold setting (see for example [START_REF] Farkas | Riemann Surfaces[END_REF]), we have that an orbifold curve C is hyperbolic if and only if deg(K C ) > 0.

In the algebraic setting, we can generalize the notion of algebraic hyperbolicity to orbifolds: Definition 21. A compact orbifold X , with ω an orbifold hermitian metric, is algebraically hyperbolic if there exists ǫ > 0 such that for any orbifold morphism

f : C → X with C compact, deg(K C ) ≥ ǫ C f * ω.
We suggest the following generalization of Green-Griffiths-Lang conjecture in the orbifold context Conjecture 22 (Orbifold Green-Griffiths-Lang conjecture). Let X be an orbifold of general type. Then there exists a proper sub-orbifold Z ⊂ X which contains the image of all holomorphic maps f : C → X .

Even in the setting of manifolds, this conjecture is widely open. An important result was obtained by McQuillan [START_REF] Mcquillan | Diophantine approximations and foliations[END_REF] with the confirmation of the conjecture for surfaces of general type with c 2 1c 2 > 0. The positivity of the second Segre number ensures by a Riemann-Roch computation as already explained above that H 0 (S m Ω X ) > cm 3 . McQuillan has extended his result to 2-dimensional Deligne-Mumford stacks with projective moduli [START_REF] Mcquillan | Bloch hyperbolicity[END_REF] (see also [START_REF] Rousseau | Hyperbolicity of geometric orbifolds[END_REF] and [START_REF] Rousseau | Degeneracy of holomorphic maps via orbifolds[END_REF]).

Since orbifolds are examples of DM stacks, a consequence of his result that we shall use is Theorem 23. Let X be an orbifold surface of general type with positive orbifold second Segre number c 2 1 (X )c 2 (X ) > 0. Then there exists a proper sub-orbifold Z ⊂ X which contains the image of all orbifold maps f : C → X , in other terms: the orbifold X is quasi-hyperbolic.

Let C → X be an orbifold rational or elliptic curve. The subjacent space to C is P 1 or a smooth elliptic curve, thus such a curve gives rise to an entire orbifold curve C → X . Therefore a quasi-hyperbolic orbifold is algebraically quasi-hyperbolic in the sense that it has only finitely many rational and elliptic orbifold curves.

Example 24. Following [START_REF] Bogomolov | Hyperbolicity of nodal hypersurfaces[END_REF] (see also [START_REF] Rousseau | Hyperbolicity of geometric orbifolds[END_REF]), we see that nodal surfaces in P 3 of degree d with l nodes provide examples of orbifolds with positive orbifold second Segre number if

l > 8 3 d 2 - 5 2 d .
It should be remarked that this numerical condition can be satisfied only for d ≥ 6. Moreover, in [START_REF] Bogomolov | Hyperbolicity of nodal hypersurfaces[END_REF], the authors claim that a subspace H 0 (X , S m Ω X ) of symmetric orbifold differentials can be extended to the resolutions of such surfaces but it seems to us that there is a gap in the argument of [START_REF] Bogomolov | Hyperbolicity of nodal hypersurfaces[END_REF] Lemma 2.2.

2.5. Orbifold jet differentials. Since Bloch [START_REF] Bloch | Sur les systèmes de fonctions uniformes satisfaisant à l'équation d'une variété algébrique dont l'irrégularité dépasse la dimension[END_REF], jet differentials have turned out to be a useful tool to study hyperbolic properties of complex manifolds. We would like to use these technics in the orbifold setting extending the construction of Demailly [START_REF] Demailly | Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials[END_REF] for manifolds to this situation.

Start with an orbifold X and an orbifold subbundle V ⊂ T X . Then we consider the orbifold X := P(V) with its natural orbifold line bundle O X (-1).

We have an orbifold map π : X → X . We define Ṽ := π -1 * O X (-1) ⊂ T X , here π * is the differential map between T X and π * T X (see [START_REF] Diverio | A survey on hyperbolicity of projective hypersurfaces[END_REF]).

Therefore, as in the case of manifolds, we have an inductive process which gives the orbifold Demailly-Semple jet bundles:

(X 0 , V 0 ) = (X , V), (X k , V k ) = ( Xk-1 , Ṽk-1 ). Taking V 0 := T X 0 , one can define jet differentials E k,m as the direct image sheaf E k,m := (π 0,k ) * O X k (m),
where π 0,k := X k → X is the natural projection map.

Remark 25. Alternatively, following [START_REF] Demailly | Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials[END_REF] one could construct Green-Griffiths jet differentials E GG k,m on orbifolds and define E k,m as the subbundle of orbifold jet differentials which are invariant under the reparametrization of germs of orbifold curves.

One can use the orbifold Riemann-Roch to compute the Euler characteristic. In the case of orbifold surfaces, one obtains

χ(X , E k,m ) = m k+2 (α k c 2 1 -β k c 2 ) + O(m k+1
), where c 1 and c 2 denote the orbifold Chern classes and α k , β k are just real numbers depending on k.

Using the semi-stability of T X with respect to K X , which is also true for orbifolds ( [START_REF] Kobayashi | Uniformization of complex surfaces, Kähler metric and moduli spaces[END_REF] and [START_REF] Tian | Existence of Kähler-Einstein metrics on complete Kähler manifolds and their applications to algebraic geometry[END_REF]), one can control the higher cohomology as in [START_REF] Green | Two applications of algebraic geometry to entire holomorphic mappings[END_REF]. In particular, one obtains for an orbifold surface of general type [START_REF] Demailly | Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials[END_REF] (2.2)

h 0 (X , E 2,m ) ≥ m 4 648 (13c 2 1 -9c 2 ) + O(m 3 ).
In the case of smooth hypersurfaces of degree d of P 3 , one obtains that

h 0 (X, E 2,m ) ≥ C.m 4
for d ≥ 15 (and C > 0). In general, the bigger c 2 1 /c 2 is, the easier it is to obtain global jet differentials. This is another illustration that Horikawa surfaces are the most difficult cases to deal with. Therefore, for smooth surfaces in P 3 of degree d = 5, one has to take k much larger. In contrast, we will provide below an example of an orbifold surface of P 3 of degree 5 where h 0 (X , E 2,m ) ≥ C.m 4 (for C > 0).

As in the case of manifolds, one obtains as a consequence of Ahlfors-Schwarz lemma:

Theorem 26 ([29], [START_REF] Rousseau | Degeneracy of holomorphic maps via orbifolds[END_REF]). Let π : X → X be a projective orbifold and an ample line bundle A on X such that

H 0 (X k , O Xk (m) ⊗ π * 0,k A -1 ) ≃ H 0 (X , E k,m ⊗ A -1
) has a non-zero section P . Then every orbifold entire curve f : C → X must satisfy the algebraic differential equation P (f ) = 0.

The generalization of jet differentials to orbifolds can be applied to classical problems which are not yet solved by Nevanlinna theory. Among these problems, we suggest the following one. Consider f : C → P n a holomorphic curve which ramifies with multiplicity divisible by m i over H 1 , . . . , H q , hypersurfaces of degrees d i of P n in general position. Then it defines an orbifold entire curve f : C → (P n , ∆) where ∆ :

= i 1 -1 m i H i . The condition K P n + ∆ > 0, is equivalent to i 1 - 1 m i d i > n + 1.
In this setting, Conjecture 22 takes the following form Conjecture 27. Let H 1 , . . . , H q be hypersurfaces of degrees d i of P n in general position. Assume that f : C → P n is a holomorphic curve which ramifies over H i with multiplicity divisible by m i and that

i 1 - 1 m i d i > n + 1,
then f is algebraically degenerate i.e. its image is contained in a proper algebraic hypersurface.

2.6. Horikawa orbifolds. Let us recall that for minimal surfaces of general type, we have the inequalities

5c 2 1 + 36 ≥ c 2 if c 2 1 is even, 5c 2 1 + 30 ≥ c 2 if c 2
1 is odd. Surfaces realizing the equalities are called Horikawa surfaces. Horikawa gave a classification of these surfaces [START_REF] Horikawa | Algebraic surfaces of general type with small c 2 1[END_REF]:

Theorem 28.
Let Z be a Horikawa surface with c 2 1 even, then there is a birational map Z → X ′ where X ′ is a surface branched along a curve with ADE singularities, more precisely X ′ is either

(1) a double covering of P 2 branched along an octic (χ = 4),

(2) a double covering of P 2 branched along a curve of degree 10 (χ = 7),

(3) a double covering of F N branched along a curve of type (6, 2a), (2a ≥ -N ) (χ = 3N + 2a -1).

In the sequel, we will say that a Horikawa surface is of type (i) according to the above classification.

It is therefore natural to associate orbifolds to Horikawa surfaces.

Definition 29. We say that an orbifold X = (X, ∆) is Horikawa if there is a Horikawa surface Z with a birational map Z → X ′ where X ′ → X is a branched covering with ramification divisor ∆.

Example 30. From Horikawa's theorem (P 2 , 1 -1 2 C) where C is a curve of degree 8 or 10 with at most ADE singularities is a Horikawa orbifold.

It is possible to construct Horikawa surfaces with c 2 1 = 5 as the (resolutions of) 5-fold covers of P 2 branched along five lines. More generally, examples of Horikawa orbifolds are provided by (P 2 , 1 -1 5 C) where C is a curve of degree 5 with at most nodes.

Algebraic hyperbolicity

3.1. Horikawa surfaces with even first Chern number. Having in mind Lang's conjecture claiming that there are only a finite number of curves of genus 0 or 1 on surfaces of general type, the aim of this section is to provide a lower bound on the genus of a curve on some explicit surfaces.

To study curves in Horikawa surfaces, one strategy is to study orbifold curves in Horikawa orbifolds. Using this philosophy, we will prove Theorem 31.

(1) Let X be a very generic Horikawa surface of type [START_REF] Barth | Compact complex surfaces, volume 4 of Ergebnisse der Mathematik und ihrer Grenzgebiete[END_REF]. Then X has no rational curves.

(2) Let X be a very generic Horikawa surface of type (2). Then X is algebraically hyperbolic, in particular X has no rational or elliptic curves.

(3) Let X be a very generic Horikawa surface of type (3) with a = 3 and χ = 3N + 5. Then X has no rational curves. This theorem will be a consequence of more general results which we establish now.

3.2.

Covers of the plane. Let D = ∪D k ⊂ P 2 , where D k is a very general curve of degree d k . For n > 1 dividing d = d k , let us consider p : X → P 2 the n-cyclic covering branched along D (for the construction of cyclic cover see [START_REF] Barth | Compact complex surfaces, volume 4 of Ergebnisse der Mathematik und ihrer Grenzgebiete[END_REF]).

The singularities of D are the intersection points of the D k 's. For such a singularity, there exist local coordinates z 1 , z 2 such that ∆ has equation z 1 z 2 = 0 ; the singularity on the cover is therefore

{t n = z 1 z 2 } it is a A n-1 .
The map p is an orbifold covering between X and (P 2 , ∆) where

∆ = 1 -1 n D i .
Theorem 32. Let f : C → X be an orbifold compact curve in X not contained in the branch locus D. Then

deg(K C ) ≥ (d - d n -4) deg C,
where deg C is the degree of C computed as deg p(f (C)).
For d > 4 and (d, n) = (5, 5), (6, 2), (6, 3), (8, 2), the surface X is algebraically hyperbolic modulo the rational and elliptic curves in the branch locus of p : X → P 2 . For (d, n) = (5, 5), [START_REF] Brieskorn | Singular elements of semi-simple algebraic groups[END_REF][START_REF] Bogomolov | Families of curves on a surface of general type[END_REF] or (8, 2), X has no rational curves except the rational curves in the branch locus.

Remark 33. Of course, if X is smooth, every curve f : C → X is an orbifold curve.

Remark 34. If there are no rational or elliptic curves in ⌈∆⌉, then there are no rational or elliptic curves in the branch locus.

Remark 35. The Chern numbers of the desingularisation of X are:

c 2 1 = n(-3 + (1 -1 n )d) 2 c 2 = 3n + (n -1)(d 2 -3d).
For (d, n) = (6, 2), the surface X is K3 : such a surface contains an infinite number of elliptic curves [START_REF] Green | Two applications of algebraic geometry to entire holomorphic mappings[END_REF]. The three other cases (d, n) = (5, 5), [START_REF] Brieskorn | Singular elements of semi-simple algebraic groups[END_REF][START_REF] Bogomolov | Families of curves on a surface of general type[END_REF] or (8, 2) are Horikawa surfaces. The case (5, 5) is a quintic surface. For (d, n) = (6, 3), we have c For the proof of Theorem 32, we use the main Theorem of Xi Chen in [START_REF] Xi | On Algebraic Hyperbolicity of Log Varieties[END_REF] (see also [START_REF] Pacienza | On the logarithmic Kobayashi conjecture[END_REF]): Theorem 36. Let D = ∪D k ⊂ P 2 be as above. For all reduced curve C ⊂ P 2 , we have: Proof. (Of Theorem 32). Let f : C → X be an orbifold map. We have the following commuting diagram:

C f / / h X p (C ′ 1 , ∆ ′ ) g / / (P 2 , ∆)
where

C 1 = p(f (C)), C ′ 1 is the desingularisation of C 1 , and g is an orbifold mor- phism. Let g * (D j ) = i(C 1 ,D) i=1 t i,j p i , g * (D) = i(C 1 ,D) i=1 t i p i . Now, let ∆ = i(C 1 ,D) i=1 1 -1 mi p i be the minimal orbifold structure on C ′ 1 such that g : (C ′ 1 , ∆) → (P 2 , ∆
) is an orbifold morphism. The conditions for g to be an orbifold morphism are n| mi .t i,j for all j. Therefore the minimal orbifold structure is given by mi = lcm j n gcd(n, t i,j ) .

As t i = j t ij , we have gcd(n, t ij ) ≤ t i for all i, j, thus:

n t i ≤ n gcd(n, t i,j ) ≤ lcm j n gcd(n, t i,j ) = mi .
So, we have:

i(C 1 ,D) i=1 1 - 1 mi ≥ i(C 1 ,D) i=1 1 - t i n = i(C 1 , D) - d n deg C 1 .
Now, inequality 3.1 of [START_REF] Xi | On Algebraic Hyperbolicity of Log Varieties[END_REF] gives:

2g(C ′ 1 ) -2 + i(C 1 , D) ≥ (d -4) deg C 1 , therefore: 2g(C ′ 1 ) -2 + i(C,D) i=1 1 - 1 mi ≥ (d - d n -4) deg C 1 .
To conclude, we observe that

deg(K C ) ≥ 2g(C ′ 1 ) -2 + i(C,D) i=1 1 - 1 mi
because C is a uniformisation of C ′ 1 . This implies (1) and (2) of Theorem 31. When n = d, the surface X is the degree d hypersurface in P 3 defined by:

X = {x d 4 -G(x 1 , x 2 , x 3 ) = 0}
, where G is a homogeneous polynomial in x 1 , x 2 , x 3 of degree d defining the curve C ⊂ P 2 . We can apply the above results to X: Corollary 37. If C is very generic and d ≥ 6, then the smooth algebraic surface X is algebraically hyperbolic. If C is very generic and d = 5, then X contains no curves of geometric genus 0. This result should be compared to results of Clemens, Ein, Pacienza, Voisin establishing the algebraic hyperbolicity of very general hypersurfaces of P n of large degree (see [START_REF] Diverio | A survey on hyperbolicity of projective hypersurfaces[END_REF] for a survey).

Moreover let us mention that, even if here we are interested in surfaces, since inequality 3.1 of Chen generalizes to higher dimension (see [START_REF] Xi | On Algebraic Hyperbolicity of Log Varieties[END_REF] and [START_REF] Pacienza | On the logarithmic Kobayashi conjecture[END_REF]), the proof above immediately generalizes to Theorem 38. If D ⊂ P n is a very generic hypersurface of degree d and d ≥ 2n + 2, then the degree d branched cover X ⊂ P n+1 ramified over D is algebraically hyperbolic. Theorem 39. (Chen, [START_REF] Xi | On Algebraic Hyperbolicity of Log surfaces[END_REF] Corollary 1.12). We have:

Covers of

(3.2) 2g(C) -2 + i(C, D) ≥ min(a -3, b -2) deg C for all reduced irreducible curves C ⊂ F N with C ⊂ D where deg C = (T +F ).C.
As before, for n dividing a and b, we consider p : X → F N a n-cyclic covering branched along D. Then we have Theorem 40. Let f : C → X be an orbifold compact curve in X not contained in the branch locus. Let c, d be the integers such that p(f (C)) = C 1 ∼ cT + dF , then:

deg(K C ) ≥ min(a -3, b -2)(c(N + 1) + d) - 1 n (bc + ad + acN ).
Proof. We follow the notations and proof of theorem 32 :

C ′ 1 is the desingularisation of C 1 . First we remark that deg C 1 = c(1 + N ) + d, C 1 .D = bc + ad + acN.
Then, using the inequality 3.2, we obtain:

2g(C ′ 1 ) -2 + i(C,D) i=1 1 - 1 mi ≥ min (a -3, b -2) (deg C 1 ) - C 1 .D n .
We wish to apply Theorem 40 to the Horikawa surfaces (for which a = 6, n = 2). To obtain a non-trivial result, we need b ≥ 3 if N ≥ 1 and b ≥ 6 if N = 0 (recall that a(N -1) + b ≥ 0). Recall moreover that b must be divisible by 2. The case b = 4 gives :

deg(K C ) ≥ c(N + 1) + d - 1 2 (bc + 6d + 6cN ) = c(-1 -3N ) -2d,
and therefore is not interesting. If b ≥ 6, we have:

deg(K C ) ≥ 3(c(N + 1) + d) - 1 2 (bc + 6d + 6cN ) = c(3 - b 2 ),
thus we obtain deg(K C ) ≥ 0 for a = 6, b = 6.

Let us suppose that D ∼ 6T + 6F on F N (N ≥ 0) is a very general member of a base point free complete linear series. The double cover of F N ramified over D is a Horikawa surface X with χ = 3N + 5. That implies (3) of Theorem 31.

The proof of Theorem 31 is now complete. We would like to emphasize here the fact that this result is valid for a very generic surface in some irreducible component of the moduli space of Horikawa surfaces.

3.4. Quintic surfaces. Our starting point was the fact that we do not know quasi-hyperbolic quintics. Consider 5 lines L 1 , . . . , L 5 in general position on P 2 . The degree 5 ramified cover of P 2 branched over the L i 's is a quintic surface X in

P 3 . If ℓ i = ℓ i (x 1 , x 2 , x 3 ) is an equation of L i , an equation of X is : X = {x 5 4 -ℓ 1 ℓ 2 ℓ 3 ℓ 4 ℓ 5 = 0}.
As the five lines are in general position, their union is a degree 5 curve D with 10 nodal singularities. Over such a singularity p, there exists local coordinates x, y such that the equation of D is xy = 0. The equation of the singularity of X over p is then z 5xy = 0 and the surface X contains therefore 10 A 4 . We consider this surface X as an orbifold. A corollary of Theorem 32 is Corollary 41. X has no orbifold rational curves except over the 5 lines of the ramification locus.

As X is an orbifold of general type, a natural question is Problem 42. Show that X has only finitely many (orbifold) elliptic curves.

A first result toward this is

Theorem 43. The image in P 2 of any orbifold elliptic curve f : C → X is not rational. More precisely, it is mapped in P 2 to a singular genus one curve which intersects every line L i with multiplicity divisible by 5.

Proof. As above, we have the following diagram:

C f / / h X p (C ′ 1 , ∆ ′ ) g / / (P 2 , ∆)
where

C 1 = p(f (C)), C ′ 1 is the desingularisation of C 1 , and g is an orbifold morphism. Therefore (C ′ 1 , ∆ ′ = 1 -1 m i p i ) is an orbifold elliptic curve i.e. K C ′ 1 + ∆ ′ = 0. A case-by-case check gives that (C ′ 1 , ∆ ′
) is isomorphic to one of the following orbifold curve:

✷ (E, ∅) where E is an elliptic curve [START_REF] Bloch | Sur les systèmes de fonctions uniformes satisfaisant à l'équation d'une variété algébrique dont l'irrégularité dépasse la dimension[END_REF][START_REF] Bogomolov | Symmetric tensors and geometry of P N subvarieties[END_REF][START_REF] Bogomolov | Symmetric tensors and geometry of P N subvarieties[END_REF], [START_REF] Bogomolov | Families of curves on a surface of general type[END_REF][START_REF] Bogomolov | Families of curves on a surface of general type[END_REF][START_REF] Bogomolov | Families of curves on a surface of general type[END_REF])

✷ (P 1 , 1 -1 m 1 {0} + 1 -1 m 2 {1} + 1 -1 m 3 {∞} where (m 1 , m 2 , m 3 ) is either (2, 3, 6),
✷ (P 1 , 1 -1 2 {0} + 1 -1 2 {1} + 1 -1 2 {p 3 } + 1 -1 2 {∞} with p 3 ∈ C \ {0, 1}. Let g * (L j ) = i(C 1 ,L) i=1 t i,j p i . g * (L) = i(C 1 ,L) i=1 t i p i .
The conditions for g to be an orbifold morphism are 5|m i .t i,j for all j. As gcd(5, m i ) = 1 by the preceding case-by-case analysis, we conclude that 5|t i,j . Therefore g : C ′ 1 → (P 2 , ∆) is an orbifold morphism. From theorem 32, we deduce that this excludes the case C ′ 1 = P 1 . Therefore in the above list, only the first case one can occur. So C 1 is a genus one curve. The condition 5|t i,j implies that deg(C 1 ) = j t ij is divisible by 5 and C 1 cannot be a smooth elliptic curve.

Therefore we have reduced the problem of counting orbifold elliptic curves in these quintics to counting genus one curves in P 2 which intersects every line L i with multiplicity divisible by 5.

Remark 44. The following computation shows that it is natural to believe that there are finitely many of these curves.

The Severi variety V d,δ of plane curves of degree d and δ nodes has dimension 3d + g -1. Let C ∈ V d,δ be a curve which intersects L i in j β j points with multiplicity 5j. The total number of conditions is

D := 5 i=1 l i j=1 β j (5j -1), with j 5jβ j = d. Then D = i (d -j β j ), et β j ≤ d 5 . Therefore D ≥ i d(1 -1 5 ) = 4d
. Thus for g ≤ 1 the number of conditions is greater than the number of parameters.

Analytic hyperbolicity

4.1. Nevanlinna theory. Nevanlinna theory can be used to study entire curves in ramified covers. We will briefly recall in this context the truncated defect relation of Cartan following the notations of [START_REF] Kobayashi | Hyperbolic complex spaces[END_REF] to which we refer for details.

Let f : C → P n be a linearly-non degenerate entire curve and D 1 , . . . , D q be q hyperplanes in general position. Let us denote as usual N [n] (f, r, D i ) and T (f, r, D i ), the truncated counting function and the characteristic function.

The defect is defined by

δ [n] (f, D i ) := lim inf r->∞ 1 - N [n] (f, r, D i ) T (f, r, D i ) .
Then we have the truncated defect relation of Cartan:

Theorem 45. δ [n] (f, D i ) ≤ n + 1.
We can apply this to ramified covers of the plane. Let X ⊂ P 3 be the degree d cover of P 2 ramified over d lines D i .

Theorem 46. If d ≥ 6 then X is a quasi-hyperbolic orbifold.

Proof. Composing with the projection, we obtain an orbifold map g : C → (P 2 , ∆). If g is linearly degenerate then from Theorem 32, its image lies in the branch locus. Let us suppose that it is linearly non-degenerate. By the First Main Theorem of Nevanlinna theory, we have

d 2 N [2] (f, r, D i ) ≤ N (f, r, D i ) ≤ T (f, r, D i ).
Therefore δ [2] (f,

D i ) ≥ 1 - 2 d , which implies δ [2] (f, D i ) ≥ d -2, contradicting Cartan's relation if d ≥ 6.
Unfortunately we see in the previous proof that we cannot say anything for d = 5, the case of the quintics we studied. An interesting question is to prove that we have again algebraic degeneracy of entire curves in this case. Note that, according to Conjecture 27, this should be true since K P 2 +∆ = O(1). Surprisingly, it seems that this problem is still open.

Problem 47. Prove that any entire curve f : C → P 2 intersecting 5 lines in general position with multiplicity at least 5 is algebraically degenerate.

Nevertheless, we can use Nevanlinna theory to construct some hyperbolic orbifold Horikawa surfaces.

Theorem 48. Let a ≥ 3 be an integer. Then there exists a quasi-hyperbolic orbifold Horikawa surface whose minimal resolution has χ = 2a -1.

Proof. Let F 0 = P 1 × P 1 . Consider the divisor

∆ = 1 - 1 2 6 i=1 G i + 1 - 1 2 2a i=1 H i ,
where the G i are fibers of the first projection p 1 , and the H i are fibers of the second projection p 2 . Then (F 0 , ∆) is an Horikawa orbifold. The corresponding Horikawa surface has χ = 2a -1.

Let f : C → (F 0 , ∆) be an orbifold entire curve. Then h := p 1 • f : C → P 1 is an entire curve with multiplicity at least 2 over the 6 points a i := p 1 (G i ). Then δ [1] (h, a i ) ≥ 6 1 -1 2 .

The truncated defect relation (i.e. Nevanlinna Second Main Theorem in this case), which gives the upper bound 2 for the sum of the defects, implies that p 1 • f is constant since . Therefore f has its image contained in a fiber of p 2 and has multiplicity at least 2 over 2a points of this fiber. This implies again that f is constant since 2a ≥ 6.

We can apply this kind of construction to numerical quintics. A quintic surface has c 2 1 = 5 and c 2 = 55. Horikawa [START_REF] Horikawa | On deformations of quintic surfaces[END_REF] proved that a surface with c 2 1 = 5 and c 2 = 55, called a numerical quintic, is either a quintic or a double cover.

Let us consider surfaces of the second type with their attached natural geometric orbifolds (X, ∆).

Theorem 49. There exists a numerical quintic whose geometric orbifold is quasihyperbolic.

Proof. Let F 0 = P 1 × P 1 . Consider the divisor

∆ = 1 - 1 2 5 i=1 G i + 1 - 1 2 (H 1 + H 2 + C + D 1 + D 2 ),
where the G i are fibers of the first projection p 1 , the H i are fibers of the second projection p 2 , C is a curve of type (2, 1), and the D i are curves of type (1, 1) such that:

(1) G 1 , H 1 , C and D 1 meet in a point a.

(2) G 1 , H 2 , C and D 2 meet in a point b. Then ⌈∆⌉ is a curve of type [START_REF] Brieskorn | Singular elements of semi-simple algebraic groups[END_REF][START_REF] Campana | Orbifoldes spéciales et classification biméromorphe des variétés kählériennes compactes[END_REF] with two quadruple points lying on a single fiber of F 0 .

(F 0 , ∆) is a numerical quintic orbifold. Let f : C → (F 0 , ∆) be an orbifold entire curve. Then p 1 • f : C → P 1 is an entire curve with multiplicity at least 2 over 5 points. The truncated defect relation (i.e. Nevanlinna Second Main Theorem in this case) implies that p 1 • f is constant. Therefore f has its image contained in a fiber of p 2 . For a generic fiber i.e. not G 1 , f has multiplicity at least 2 over at least 5 points of this fiber. This implies again that f is constant. Therefore the image of f is contained in G 1 .

One can remark that Cartan's theorem unfortunately says nothing in the case of Horikawa surfaces which are degree 2 covers of the plane P 2 . 4.2. Jet differentials. As we have just seen, applications of Nevanlinna theory are quite limited. This is one motivation to develop the tool of orbifold jet differentials.

Let us illustrate it in the case of quintics X that are Z/5Z-covers of the plane ramified over 5 lines in general position. First one could think of using jets of order to apply Theorem 23.

As already mentioned in paragraph 3.4, the surface X has a natural structure of orbifold X = (X, 0). Proposition 50. The Chern numbers of X are:

c 2 1 (X ) = 5 c 2 (X ) = 7,
Proof. By paragraph 3.4, X contains 10A 4 singularities. Let us denote by X ′ → X the minimal resolution of these singularities. The Chern numbers of X ′ are equals to the Chern numbers of a smooth quintic (Brieskorn resolution Theorem [START_REF] Brieskorn | Singular elements of semi-simple algebraic groups[END_REF]) i.e. K 2 X ′ = 5, e(X ′ ) = 55. We have K 2 X = K 2 X = 5 and c 2 (X ) = e(X ′ )-10•(5-1 5 ) = 7 (see Proposition 13).

Therefore the orbifold second Segre number is negative and we cannot apply Theorem 23.

Turning to jet differentials of order 2, one sees that 13c 2 1 (X ) -9c 2 (X ) = 2 > 0. Therefore by inequality (2.2):

Corollary 51. We have:

h 0 (X, E 2,m ⊗ L -1 ) ≥ 2. m 4 648 + O(m 3 ),
for L an ample line bundle on X. As a consequence (see (2.2) and Theorem 26), any entire orbifold curve f : C → X satisfies a differential equation of order 2.

There are many known quintic surfaces with quotient singularities (see e.g. [START_REF] Schütt | Quintic Surfaces with maximum and other Picard numbers[END_REF]), but as far as we know, the surfaces that are Z/5Z-covers of the plane ramified over 5 lines in general position are the only one such that the associated orbifold satisfies 13c 2 1 (X ) -9c 2 (X ) > 0.

This result can also be seen as a first step toward problem 47.

Persson-Horikawa surfaces

Let us recall some notations. For N a positive integer, the N th Hirzebruch surface is F N = P(O P 1 ⊕ O P 1 (N )). We denote by F denote a fiber of the P 1 -bundle F N → P 1 and by T a section s.t. T 2 = N . We denote by (a, b) a divisor equivalent to aT + bF in the Néron-Severi group.

Recall (see Theorem 28) that a Horikawa surface is the double cover of the plane branched along an octic (case χ = 4), or a curve of degree 10 (case χ = 7) or a double cover of F N branched over a curve D of type (6, 2a), with 2a ≥ -N . In this section we will exhibit quasi-hyperbolic Horikawa orbifolds of each type. Proof. The divisor -2(F + T ) is the canonical divisor on Hirzebruch surface F 0 = P 1 × P 1 . Following Persson's construction in [START_REF] Persson | Horikawa surfaces with maximal Picard numbers[END_REF] Lemma 4.5, there exists on F 0 a Q-divisor:

∆ = (1 - 1 k )(F 1 + F 2 ) + (1 - 1
2 )(C 1 + C 2 + E 0 + E 1 + E 2 + E 3 )
such that the orbifold (F 0 , ∆) is uniformizable for all k ≥ 2 and the desingularisation of the uniformization is a Horikawa surface with Chern invariants χ = 2k -1 and c 2 1 = 2χ -6. The curves C 1 , C 2 are (1, 1)-curves meeting with multiplicity 2 into one point. The curves F 1 , F 2 are fibers of the first projection P 1 ×P 1 → P 1 , the curves E i are sections of this fibration. The singularities of ⌈∆⌉ are 6A 1 +4D 4 +D 6 .

Let us use the results of subsection 2.2. Two singularities of type A 1 are on branches with multiplicities m i equal to 2 (thus β(p) = 4), 4A 1 are on branches with multiplicities 2 and k (with β(p) = 2k), the branches of the 4D 4 have the same multiplicities : 2, 2, k (giving β(p) = 4k 2 ) and the branches of the D 6 point have multiplicities 2, thus β(p) = 32 by Lemma 12.

E 3 E 2 E 0 E 1 F 1 F 2 C 1 C 2
Let us compute the orbifold Chern numbers of the associated orbifold X . We have:

(K + ∆) 2 = (T + (1 - 1 k )F ) 2 = T 2 + 2(1 - 1 k )T F + (1 - 1 k ) 2 F 2 = 2(1 - 2 k )
because F 2 = 0, F T = 1, T 2 = N = 0 (see section 3.3). For the second Chern number of X , (2.1) gives:

c 2 (X ) = 4 -2(1 -1 k )(2 -4) -1 2 (3(2 -3) + (2 -2) + 2(2 -4)) -(2(1 -1 4 ) + 4(1 -1 2k ) + 4(1 -1 4k 2 ) + (1 -1 32 )) and c 2 (X ) = 33 32 - 2 k + 1 k 2 . Therefore : c 1 (X ) 2 -c 2 (X ) = 31 32 - 2 k - 1 k 2 ,
and the orbifold X is quasi-hyperbolic for k > 2 by Theorem 23.

Remark 53. Let X → F 0 be the Z/2Z × Z/kZ-cover of F 0 branched over ∆. The desingularisation of X is a Horikawa surface. The surface X has only ADE singularities and has a natural structure of orbifold X ′ . The map X ′ → X is an orbifold covering, in particular c 1 (X ′ ) 2 = 2kc 1 (X ) 2 and c 2 (X ′ ) = 2kc 2 (X ). )) Proposition 54. For every integer k > 1, there exists a quasi-hyperbolic orbifold Horikawa surface X with c 2 1 (X ) = 4(1 -1 k ) c 2 (X ) = 17 12 -2 k -1 k 2 . The desingularisation of the Z/2Z × Z/kZ-cover of the subjacent space to X is a Horikawa surface with χ = 4k -1.

E 3 E 2 E 0 E 1 F 1 F 2 C 1 C 2
Proof. The canonical divisor on F 2 is K F 2 = -2T . Let k > 1 be an integer. Persson ([32] Proposition 4.7) constructed on F 2 a divisor : Proof. Let C be the degree 10 curve that is the union of the degree 8 curve in Proposition 55 and two lines in general position. It has singularities:

∆ = (1 - 1 k )(F 1 + F 2 ) + (1 - 1
23A 1 + 2A 3 + d 4 + 3E 7 .
The orbifold Chern classes of X = (P 2 , C) are :

c 2 1 (X ) = (-3 + 10

Figure 2 . 1 .

 21 Figure 2.1. ADE singularities and conditions on the multiplicities

2 1 = 3 and 5c 2 1 + 30 =

 130 c 2 = 45. The case (d, n) = (8, 2) is a Horikawa surface of type (1).

( 3 . 1 )

 31 2g(C) -2 + i(C, D) ≥ (d -4) deg C, where d = d k and i(C, D) is the number of distinct points of ν * D if ν : C ′ → C is the normalization.

  Hirzebruch surfaces F N , Algebraically hyperbolic Horikawa surfaces. In order to prove Theorem 31 (3), we study covers of Hirzebruch surfaces. For N ≥ 0, let F N := P(O ⊕ O(N )) be the Hirzebruch surface and let T and F the divisors on F N generating P ic(F N ) with T 2 = N , T.F = 1 and F 2 = 0. Let D = ∪D k ⊂ F N where each D k is a very general member of a base point free complete linear series. And let D ∼ aT + bF with a(N -1) + b ≥ 0.

5. 1 .

 1 Horikawa surfaces with χ = 2k-1. Let us prove the following Proposition: Proposition 52. For any integer k > 2, there exists a quasi-hyperbolic Horikawa orbifold X whose minimal resolution has Euler characteristic equals to χ = 2k -1.

5. 1 . 1 .

 11 Generalization : a family of quasi-hyperbolic Horikawa orbifold with χ = 2k -1, k ≥ 4. Let us consider the configuration of curves on F 0 in figure 5.1. The two (1, 1)-curves C 1 , C 2 in F 0 are in general position. The singularities of ∆ are 7A 1 + 5D 4 . One D 4 with multiplicities (2, 2, 2), 4D 4 with multiplicities (2, 2, k), 4A 1 with (2, k) and 3A 1 with (2, 2).Let X be the associated orbifold. By subsection 2.2, its Chern numbers are:c 1 (X ) 2 = 2(1 -2 k ) and c 2 (X ) = 4 -2(1 -1 k )(2 -4) -1 2 (3(2 -3) + (2 -2) + 2(2 -5)) -(3(1 -1 4 ) + 4(1 -1 2k ) + 4(1 -1 4k 2 ) + (1 -1 16

Figure 5 . 1 .-2 k + 2 k 2 and c 1 (X ) 2 -c 2 ( 2 . 5 . 2 .

 51122252 Figure 5.1. The divisor ∆ on F 0 .

2 ) 3 Figure 5 . 3 . 2 ( 2 - 5 . 3 . 2 .

 235322532 Figure 5.3. The Steiner quartic, its bitangent and the 3 tangents to its cusps.

  c 2 (X ) > 0 and by Theorem 23, X is quasi-hyperbolic. The moduli of 2 lines in P 2 is 4 dimensional.

			2	) 2 = 4
	and
	c 2 (X ) = 3 -	1 2	(2 -16 + 3(2 -6) + (2 -7) + 2(2 -9)) -29 +	23 4	+	2 8	+	1 16	+	3 96	=	83 32	,
	thus : c 2 1 (X )

The first author is supported by grant FCT SFRH/BPD/72719/2010 and project Geometria Algébrica PTDC/MAT/099275/2008. The second author is supported by the Agence Nationale de la Recherche (ANR) through projects POSITIVE (ANR-10-BLAN-0119) and COMPLEXE (ANR-08-JCJC-0130-01).

Let us compute the orbifold Chern numbers of X = (F 2 , ∆): Proof. Let X be the orbifold whose subjacent variety is P 2 and with ∆ =

where Q is the Steiner quartic (the unique quartic curve with 3 cusps, see [START_REF] Persson | Horikawa surfaces with maximal Picard numbers[END_REF]), L 1 , L 2 , L 3 are the tangents to the 3 cusps and L 4 is the unique bitangent of Q (see figure 5.3).

The 2 points where L 4 is tangent to Q are A 3 singularities of ⌈∆⌉. The singularity of ⌈∆⌉ at the cusp Q have type E 7 . The three lines L 1 , . . . , L 3 meet at