On Lang s conjecture for some product-quotient surfaces

Transcription

1 Ann. c. Norm. uper. Pisa Cl. ci. (5) Vol. XVIII (208), On Lang s conjecture for some product-quotient surfaces JULIEN GRIVAUX, JULIANA RETREPO VELAQUEZ AND ERWAN ROUEAU Abstract. We prove effective versions of algebraic and analytic Lang s conjectures for product-quotient surfaces of general type with P g = 0 and c 2 = c 2. Mathematics ubject Classification (200): 4J29 (primary); 32Q45 (secondary).. Introduction Lang s conjecture asserts that curves of fixed geometric genus on a surface of general type form a bounded family. An effective version of this conjecture can be stated in the following way: Conjecture. (Lang-Vojta). Let a smooth projective surface of general type. Then there exist real numbers A, B, and a strict subvariety Z such that, for any holomorphic map f : C! satisfying f (C) * Z, where C is a smooth projective curve, it holds deg f (C) apple A(2g(C) 2) + B. Bogomolov proved this conjecture for minimal surfaces of general type satisfying c 2 c 2 > 0 [8]. He actually proved that such surfaces have big cotangent bundle, and that the conjecture follows from this fact. Unfortunately, this approach does not provide effective information about A and B. However, effective results for such surfaces have been obtained more recently by Miyaoka [7]. On the other hand, the analytic version of Lang s conjecture is stated as follows: Conjecture.2 (Green-Griffiths-Lang). Let be a smooth projective surface of general type. Then there exists a strict subvariety Z such that for any non constant holomorphic map f : C!, f (C) Z. The third author is partially supported by the ANR project FOLIAGE, ANR-6-CE Received October 28, 206; accepted in revised form May, 207. Published online July 208.

2 484 JULIEN GRIVAUX, JULIANA RETREPO VELAQUEZ AND ERWAN ROUEAU Bogomolov s result has been generalized to the analytic case by McQuillan in his proof of this conjecture for minimal surfaces of general type with c 2 c 2 > 0 given in [6]. Here, we are interested in product-quotient surfaces, i.e., in the minimal resolutions of quotients X := (C C 2 )/G, where C and C 2 are two smooth projective curves of respective genera g(c ), g(c 2 ) 2, and G is a finite group, acting faithfully on each of them and diagonally on the product. These surfaces generalize the so-called Beauville surfaces (the particular case where the group action is free). The classification of product-quotient surfaces with geometric genus p g = 0 was started by I. Bauer and F. Catanese in [3]; they classified the surfaces X = (C C 2 )/G with G being an abelian group acting freely and p g (X) = 0. Later in [4], both of them and F. Grunewald, extended this classification to the case of an arbitrary group G. Not long after, R. Pignatelli joined them, and in [5] they dropped the assumption that G acts freely on C C 2 ; they classified productquotient surfaces with p g = 0 whose quotient model X has at most canonical singularities. Finally in [6], I. Bauer and R. Pignatelli dropped any restriction on the singularities of X and gave a complete classification of product-quotient surfaces with p g = 0 and c 2 > 0. It turns out that all except one are in fact minimal surfaces (see [6, Tables and 2]). In this paper, we prove Conjectures. and.2, when is a product-quotient surface of general type with geometric genus p g = 0 and c 2 c 2 = 0. Note that p g = 0 implies c 2 + c 2 = 2, then the condition c 2 c 2 = 0 is equivalent to c 2 = c 2 = 6. These surfaces are a limit case not covered by Bogomolov s theorem; however, they satisfy the criterion given in [20, Theorem ], which ensures the bigness of their cotangent bundle. In order to accomplish this, we start by proving the following result for product-quotient surfaces in general. Theorem.3. Let be a product-quotient surface. If f : C! is a holomorphic map such that f (C) * E, where C is a smooth projective curve and E is the exceptional divisor on, then deg f (K E) apple 2(2g(C) 2). Note that Theorem.3 is interesting only when K E is positive (ample or big). In Conjecture., one can take deg f (C) = deg f L for L a positive line bundle on ; hence, provided that the divisor K E is big, Theorem.3 implies Conjecture. for the case of product-quotient surfaces. However, it is not known whether K E is ample or even big in general and in view of the conjecture it is foreseen to be hard to determine. We then restrict ourselves to the particular case where is a product-quotient surface of general type with geometric genus p g = 0 and c 2 = 6. In this case we prove that K E is big and we obtain Conjecture.. We also give an alternative proof for the bigness of, producing explicit symmetric tensors on coming from K E, which allows us to control rational curves on it. More precisely we prove the following theorem.

3 ON LANG CONJECTURE FOR OME PRODUCT-QUOTIENT URFACE 485 Theorem.4. Let be a product-quotient surface of general type such that p g () = 0. If c () 2 = 6, then the following facts hold () The line bundle K E and the cotangent bundle are big; (2) For any non constant holomorphic map f : P!, f P E [ B(K E), where E is the exceptional divisor on the resolution and B(K stable base locus of K E. E) is the Finally, our approach also lets us control elliptic and more generally, entire curves on. More precisely, we prove Conjecture.2 in our context. Theorem.5. Let be a product-quotient surface of general type such that p g () = 0. If c () 2 = 6, then for any non constant holomorphic map f : C!, f (C) E [ B + (K E), where E is the exceptional divisor on the resolution and B + (K augmented base locus of K E. E) is the At this point one may ask if the methods used to prove Theorems.4 and.5 could be extended to more general product-quotient surfaces than the single family with c 2 = c 2. It turns out that as c 2 gets smaller than c 2, the problem of determining whether K E is big gets more difficult. In fact, our approach still works for the case c 2 = 5 and c 2 = 7 but not for c 2 apple Preliminaries In this section we are going to recall some definitions and results that will be used throughout this paper. 2.. Product-quotient surfaces Definition 2.. A product-quotient surface is the minimal resolution of the singularities of a quotient X := (C C 2 )/G, where C and C 2 are two smooth projective curves of respective genera g(c ), g(c 2 ) 2, and G is a finite group, acting faithfully on each of them and diagonally on the product. The surface X := (C C 2 )/G is called the quotient model of [6]. Let be a product-quotient surface. Let ' :! X be the resolution morphism of the singularities of X := (C C 2 )/G, and let p : X! C /G and p 2 : X! C 2 /G be the two natural projections. Let us define :! C /G and 2 :! C 2 /G to be the compositions p ' and p 2 ' respectively. Thus, we have the following commutative diagram encoding all this information:

4 486 JULIEN GRIVAUX, JULIANA RETREPO VELAQUEZ AND ERWAN ROUEAU ϕ σ X :=(C C 2 )/G σ 2 p p 2 C /G C 2 /G The surface X := (C C 2 )/G has a finite number of singularities, since there are finitely many points on C C 2 with non trivial stabilizer. Moreover, since G is finite, the stabilizers are cyclic groups [2, III 7.7] and so, the singularities of X are cyclic quotient singularities. Thus, if (x, y) 2 C C 2 with non trivial stabilizer H (x,y), then, around the singularity (x, y) 2 X := (C C 2 )/G, X is analytically isomorphic to the quotient C 2 /(Z/nZ), where n = H (x,y) and the action of the cyclic group Z/nZ on C 2 is defined by (z, z 2 ) = ( z, a z 2 ), where n and a are coprime integers such that apple a apple n and = exp( 2 i n ) is a chosen primitive n-th root of unity. In this case, the cyclic quotient singularity is called singularity of type n (, a). Note that singular points of type n (, a) are also of type n (, a0 ) where a 0 is the multiplicative inverse of a in (Z/nZ) (see [2, III]). The exceptional fiber of a cyclic quotient singularity of X of type n (, a) on the the minimal resolution, is a Hizerbruch-Jung string (H-J string), that is to say, a connected union L = P l i=0 Z i of smooth rationals curves Z,..., Z l with selfintersection numbers less or equal than 2, and ordered linearly so that Z i Z i+ = for all i and Z i Z j = 0 if i j 2 (see [2, III Theorem 5.4]) and [3]. Then, the exceptional divisor E on the minimal resolution is the connected union of disjoint H-J strings each of them being the fiber of each singularity of X := (C C 2 )/G. The self-intersection numbers Zi 2 = b i are given by the formula n a = b b 2 lightly abusing and side notation, we denote the right of the previous formula by [b,, b l ]. Moreover, n a = [b,, b l ] if and only if n a 0 = [b l,, b ]. Note that for cyclic quotient singularities of type n (, n ) we have that all the curves Z i have self-intersection equal to 2. These singularities are then a particular case of canonical surface singularities (the latter are also known as du Val singularities or rational double points). b l

5 ON LANG CONJECTURE FOR OME PRODUCT-QUOTIENT URFACE 487 On the other hand, errano s paper [2, Proposition 2.2] tells us that the irregularity of, defined by q() := h (, O ), is given by the formula: q() = g(c /G) + g(c 2 /G). Now, if is of general type, then q() apple p g (). Therefore, we have that is a product-quotient surface of general type with p g = 0 if and only if (O ) = and C /G = C 2 /G = P. Moreover, using Noether s formula we see that the condition p g = 0 also implies that c 2 + c 2 = 2. Thanks to the work that I. Bauer, F. Catanese, F. Grunewald and R. Pignatelli, started and carried through in [3 6], we have a complete classification of productquotient surfaces of general type with geometric genus p g = 0 and c 2 > 0. Moreover, they proved that there are exactly 73 irreducible families of surfaces of this kind, and all but one of them, are in fact minimal surfaces; more precisely they proved the following result. Theorem 2.2 ([6], Theorems 4.8, 5.). The following facts hold: () Minimal product-quotient surfaces of general type with p g = 0 form exactly 72 irreducible families; (2) There is exactly one product-quotient surface with K 2 > 0 which is non minimal. It is called fake Godeaux surface. It has K 2 = and its minimal model has K 2 = 3. The irreducible families mentioned in the first part of this theorem, are listed in [6, Tables, 2] Isotrivial fibrations Definition 2.3. A fibration is a surjective morphism from a smooth projective surface into a smooth curve, with connected fibers. A fibration is called isotrivial fibration, if all its smooth fibers are mutually isomorphic. A surface is called isotrivial surface if it admits an isotrivial fibration. A product-quotient surface is an example of an isotrivial surface: it admits two natural isotrivial fibrations :! C /G and 2 :! C 2 /G whose smooth fibers are all isomorphic to C 2 and C respectively. In literature, product-quotient surfaces are also called standard isotrivial surfaces. In errano s paper [2] it is proved that any isotrivial surface is birationally equivalent to a standard one, more precisely, if : Z! C is isotrivial, then there exist a quotient (C C 2 )/G where C is isomorphic to the general fiber of and G is a finite group, acting faithfully on C and C 2 and diagonally on the product; such that Z is birational to (C C 2 )/G, the curve C is isomorphic to C 2 /G, and the following diagram commutes:

6 488 JULIEN GRIVAUX, JULIANA RETREPO VELAQUEZ AND ERWAN ROUEAU Z (C C 2 )/G γ p 2 C C 2 /G We also find in errano s paper a description of the singular fibers that can arise in a standard isotrivial surface, i.e., the possible singular fibers of its natural fibrations. Namely: Theorem 2.4 ([2], Theorem 2. ). Let a standard isotrivial surface and let consider the fibration 2 :! C 2 /G. Let y 2 C 2 and H y its stabilizer. If F is the fiber of 2 over y 2 C 2 /G, then: () The reduced structure of F is the union of an irreducible smooth curve Y, called the central component of F, and either none or at least two mutually disjoint H-J strings, each one meeting Y at one point. These strings are in one-to-one correspondence with the branch points of C! C /H y ; (2) The central component Y is isomorphic to C /H y and it has multiplicity equal to H y in F. The intersection of a string with Y is transversal, and it takes place at only one of the end components of the string; (3) If L = P n i= Z i is a H-J string on F and Y 0 is the central component of the fiber of :! C /G over (L), then L meets Y 0 and Y at opposite ends, i.e., either Z Y = Z n Y 0 = or Z n Y = Z Y 0 =. If F contains exactly r H-J strings L,, L r, where each L i is the resolution of a cyclic quotient singularity of type n i (, a i ), then we know that the central component Y satisfies rx Y 2 a i = (2.) n i [8, Proposition 2.8]. The previous theorem holds as well for the fibration. Finally, errano s paper also provides an expression for the canonical bundle of a standard isotrivial surface in terms of the fibers of the two natural fibrations. Namely, we have the following. Theorem 2.5 ([2], Theorem 4.). Let be a standard isotrivial surface with associated fibrations :! C /G and 2 :! C 2 /G. Let {n i N i } i2i and {m j M j } j2j denote the components of all singular fibers of and 2 respectively, with their multiplicities attached. Finally, let {Z t } t2t be the set of curves contracted to points by 2, i.e, the exceptional locus on. Then we have K = (K C /G) + 2 (K C 2 /G) + X (n i )N i + X (m i )M i + X Z t. i2i j2j t2t i= The fibrations :! C /G and 2 :! C 2 /G can be thought as foliations F and F 2 on, such that errano s formula can be written as follows: K = N F N F 2 O (E), (2.2)

7 ON LANG CONJECTURE FOR OME PRODUCT-QUOTIENT URFACE 489 where N F and N F 2 are the respective conormal line bundles, and E is the exceptional divisor on (see [9, page 30]). 3. Proof of Theorem.3 The purpose of this section is to prove Theorem.3. Let us begin by recalling some basic facts that will be used. Let be a product-quotient surface, C a smooth projective curve and f : C! a holomorphic map such that f (C) * E. The differential map d f : T C ( log f E)! T ( log E) induces a lifting f [] : C! P(T ( log E)). Thus we get the following diagram: P(T( loge)) f [] π C f Moreover, we have that O P(T ( log E))() ' (log E). On the other hand, recall that for each foliation F on, we have the logarithmic exact sequence 0 N F (E) Ω (log E) Ω F (log E) 0 and we can also define the divisor Z := P(T F ( log E)) on P(T ( log E)). Lemma 3.. Let F be a foliation on, C a smooth projective curve and f : C! a holomorphic map such that f (C) * E. If f is not tangent to F, then deg f NF (E) apple 2g(C) 2 where N (E) is the number of points on f + N (E), (E) counted without multiplicities. Proof. For the sake of simplicity we denote by O() the line bundle Let us consider the exact sequence O P(T ( log E))(). 0 O() [Z] O() O() Z 0 Now, taking the push-forwards we get 0 π (O() [Z]) Ω (log E) Ω F (log E) 0 and thus we obtain (O() [Z]) ' NF (E). On the other hand, since f is not tangent to F then f [] (C) * Z, thus f [] (C).Z 0 and hence deg f[] [Z] 0. Therefore, deg f NF (E) apple deg f[] O().

8 490 JULIEN GRIVAUX, JULIANA RETREPO VELAQUEZ AND ERWAN ROUEAU Moreover, the differential map d f : T C f (E)! f[] O( ) defines a non zero section of the line bundle f [] O( ) K C( f (E)) implying that this line bundle is effective. Then, deg f[] O() apple deg K C f (E) = 2g(C) 2 + N (E). Let us recall that admits two natural isotrivial fibrations :! C /G and 2 :! C 2 /G that can be thought as foliations F and F 2 on. Theorem 3.2. Let be a product-quotient surface. If f : C! is a holomorphic map such that f (C) * E, with C a smooth projective curve and E the exceptional divisor on, then deg f (K E) apple 2(2g(C) 2). Proof. First, let us suppose that f is not tangent to any of the foliations F, F 2. Then by Lemma 3. we have that for i =, 2 deg f N F i (E) apple 2g(C) 2 + N (E). Using errano s formula for the canonical bundle (see (2.2)), we get that Therefore, deg f (K + E) = 2X deg f NF i (E) i= apple 2(2g(C) 2 + N (E)) = 2(2g(C) 2) + 2N (E). deg f (K E) apple 2(2g(C) 2). Now, we suppose that f is tangent to one of the foliations, let us say to F, then f (C) is contained in a fiber F of :! C /G. Let us denote by deg f the degree of f : C! f (C). If F is a smooth fiber we know that it is isomorphic to the curve C 2, and then, deg f (K E) = (deg f )(K E).C 2 = (deg f )K C2.C 2 apple K C.C = 2g(C) 2. If F is a singular fiber, f (C) must be contained in the central component Y of the reduced structure of F and hence, deg f (K E) = (deg f )(K E).Y = (deg f ) K Y.Y Y.E + Y 2.

9 ON LANG CONJECTURE FOR OME PRODUCT-QUOTIENT URFACE 49 When F does not contain any H-J string we have Y 2 = 0 and Y.E = 0; thus, deg f (K E) = (deg f )K Y.Y apple K C.C = 2g(C) 2. On the other hand, when F contains exactly r H-J strings, L,, L r, where each L i is the resolution of a cyclic quotient singularity of type n i (, a i ), we have that Y.E + Y 2 = r rx i= a i n i 0 (see formula (2.)) and thus, deg f (K E) apple (deg f )K Y.Y apple K C.C = 2g(C) Product-quotient surfaces with p g = 0 and c 2 = c 2 In this section, we are going to study product-quotient surfaces of general type with geometric genus P g = 0 and c 2 c 2 = 0. Recall that P g = 0 implies c 2 + c 2 = 2, then the last condition is equivalent to c 2 = 6. This kind of surfaces form exactly 8 irreducible families and we have a complete description of their quotient models. In the following table we summarize some information that might be useful. Number of irreducible families Z 2 D Z A Z PL(2, 7) A c 2 () ingularities of X G G g(c ) g(c 2 ) 6 Two of type 2 (, ) For even more information see [6, Table ]. Let be a product-quotient surface and let us suppose that is of general type with P g () = 0 and c 2 = 6. We recall the following commutative diagram: σ ϕ σ 2 X:=(C C 2 )/G p p 2 C /G=P C 2 /G=P

10 492 JULIEN GRIVAUX, JULIANA RETREPO VELAQUEZ AND ERWAN ROUEAU Theorem 2.2 tells us that is minimal and since is of general type we get that K is nef. However, in this particular case we can prove that K if nef by other means, thus obtaining another argument for the minimality of. Namely, we know that the quotient model X of has only cyclic quotient singularities of type 2 (, ); since these singularities are canonical we have K = ' K X, additionally K X is nef because K C C 2 is ample, therefore K is nef and so is minimal. ince the singularities of X are of type 2 (, ), we have that around them, X is analytically isomorphic to the quotient C 2 /(Z/2Z), where the action of Z/2Z on C 2 is given by (z, z 2 )! ( z, z 2 ). This quotient is an affine subvariety of C 3, with coordinates u = z 2, v = z z 2, w = z2 2, defined by the equation uw = v2 [9, Proposition-Definition., Example.2]. Moreover, if µ, µ 2 are local coordinates on, the resolution morphism ' is locally given by '(µ, µ 2 ) = u = µ, v = µ µ 2, w = µ µ 2 2 (4.) [9, Example 3.]. Therefore, we have the following relations between the local coordinates z, z 2 and µ, µ 2 : ( z = µ /2 z 2 = µ /2 µ 2. (4.2) On the other hand, the exceptional fiber of a cyclic quotient singularity 2 (, ) on the minimal resolution, is a H-J string formed by only one smooth rational curve with self-intersection number equal to 2. Using the local coordinates µ, µ 2 on we see that it is given by the set of points (µ, µ 2 ) such that µ = 0. We denote by E the exceptional divisor on the minimal resolution of X. ince X has only two cyclic quotient singularities of type 2 (, ), then E is the disjoint union of two rational curves with self-intersection number equal to 2. Moreover, E is locally defined by the equation µ = Proofs of Theorems.4 and.5 In this section we are going to prove Theorems.4 and.5. From now on, will be a product-quotient surface of general type such that p g () = 0 and c () 2 = 6, X := (C C 2 )/G its quotient model and E the exceptional divisor. 5.. Bigness of the cotangent bundle We denote by 3 the set of points of C C 2 with non trivial stabilizer. Recall that 3 is a finite set. Let us first describe a natural way to produce sections of 2m X, from sections of K m using the following diagram:

11 ON LANG CONJECTURE FOR OME PRODUCT-QUOTIENT URFACE 493 C C 2 p ϕ X:=(C C 2 )/G Let! be a section of K m. The pushforward ' of! is a section of K m X defined on the regular part of X, that can be lifted by the pullback p to a section of (KC m C 2 ) G defined outside of 3. However, since codim C C 2 3 = 2, this section uniquely extends to a section defined on C C 2. Moreover, the canonical isomorphism between K C C 2 and C 2 C 2 where : C C 2! C and 2 : C C 2! C 2 are the projections, allows us to identify the sections of (KC m C 2 ) G with sections of ( 2m C C 2 ) G. Therefore, we get a section of ( 2m C C 2 ) G which descend by p to a section of 2m X defined on the regular part of X. We denote this section by 2(!). Let us denote by 0(!) the pullback of 2(!) by '. Note that 0(!) is, a priori, a section of 2m defined outside of the exceptional divisor E. If we start with global sections of K m, this process is summarized in the following commutative diagram: ϕ H 0 (, K m ) H0 (X reg, K m X ) H0 ((C C 2 ) Λ, K m p C C 2 ) G Γ Θ H 0 (C C 2, K m C C 2 ) G H 0 (C C 2, Ω m C Ω m C 2 ) G ϕ H 0 ( E, 2m Ω ) H 0 (X reg, 2m Ω X ) H 0 (C C 2, 2m Ω C C 2 ) G p The following proposition ensures that taking global sections of K m vanishing along E, at least with multiplicity m, is a sufficient condition to obtain global sections of 2m. Proposition 5.. If! is a global section of O(m(K extends to a well-defined global section of 2m. E)), then 0(!) naturally Proof. Let! 2 H 0 (, O(m(K E))). Following the previous diagram, we get 2(!) 2 H 0 (X reg, 2m X ). By definition, the corresponding section on C C 2 can be written locally, let us say around a fixed point, as a(z, z 2 )dz m dz 2 m.

12 494 JULIEN GRIVAUX, JULIANA RETREPO VELAQUEZ AND ERWAN ROUEAU Using the change of coordinates z = µ /2 and z 2 = µ /2 µ 2 given by ' at singular points of X (see Formulas (4.) and (4.2)), we get that the pullback by ' of 2(!), which is nothing else than 0(!), can be written locally as mx j=0 m j µ m j 2 (a ')(µ, µ 2 ) 2 2m j µ m j dµ 2m j dµ j 2 and it naturally extends to a well-defined global section of 2m, since a vanishes along E at least with multiplicity m. ' The two following theorems constitute the proof of the first part of Theorem.4. Theorem 5.2. The line bundle O(K E) is big. Proof. We know the canonical bundle K is nef and big. Then, as a consequence of Riemman-Roch theorem for surfaces and Mumford vanishing theorem [2, VI Theorem 2.], we get that for any m > h 0, K m = m2 m c () 2 + (O ). 2 Now, we have that c () 2 = 6 and (O ) =, then we obtain h 0, K m = 3m 2 + 3m +. On the other hand, let! be a section of K m. The corresponding section on C C 2 can be written locally, around a fixed point, as a(z, z 2 )(dz ^ dz 2 ) m, where a is a holomorphic function defined as a(z, z 2 ) = X i, j a i j z i z j 2. Using the change of coordinates z = µ /2 and z 2 = µ /2 µ 2 given by ' at singular points of X, we see that! vanishes along E, at least with multiplicity m if a i, j = 0, for every i, j such that i + j < 2m, and this gives us, m sufficient conditions. However, the section is invariant by the action of G, which means that around a singular point, a(z, z 2 ) is invariant by the action of its stabilizer H ' Z 2, therefore a i j = 0 for all i, j such that i + j is odd; thus, we just need to consider half of the conditions. Finally, since these conditions are given around one singular point, we have to multiply them by the number of singularities.

13 ON LANG CONJECTURE FOR OME PRODUCT-QUOTIENT URFACE 495 Thus, for any m we have that h 0 (, O(m(K E))) h 0, K m 2( m) 2 = 3m 2 3m + 2m 2 + m = m 2 4m +. Note for any 0 < C <, we have that m 2 4m + Cm 2 for m large enough. Therefore, h 0 (, O(m(K E))) Cm 2 for m large enough, which means that O(K E) is big [5, Lemma 2.2.3]. Theorem 5.3. The cotangent bundle is big. Proof. For the sake of simplicity, we are going to use the same notation to refer to a divisor and its associated line bundle. In order to prove that is big, we show that the line bundle O P(T )() is big, which is equivalent to see that O P(T )(k) is linearly equivalent to the sum of an ample divisor and an effective divisor, for a k large enough [5, Corollary 2.2.7]. Theorem 5.2 tells us that O(K E) is big, then there exists an ample line bundle A and a positive integer m such that H 0, O(m(K E)) A 6= 0 and hence H 0, 2m A 6= 0. However, 2m ' O P(T )(2m) where : P(T )! is the projective bundle associated to the tangent bundle T, and so we obtain that H 0 P( ), O P(T )(2m) A 6= 0, which means that O P(T )(2m) A is an effective divisor. On the other hand, since O() is relatively ample [5, Proposition.2.7], we know that there exists a large enough positive integer l such that O P(T )() + l A is an ample divisor on P(T ). Finally, taking k = 2ml + we get O P(T )(2ml + ) = l(o P(T )(2m) A) + O P(T )() + l A {z } {z } effective divisor ample divisor and so O P(T )() is big.

14 496 JULIEN GRIVAUX, JULIANA RETREPO VELAQUEZ AND ERWAN ROUEAU 5.2. Rational curves We already know that is big and then, by Bogomolov s argument, there is only a finite number of rational curves on ; now we want to get more constraints. Lemma 5.4. The central component Y of any singular fiber on, defined in proposition 2.4, is not rational. Proof. Recall that the only singularities of X are two ordinary double points, i.e., two cyclic quotient singularities of type 2 (, ). Then, the singular fibers on are the union of a central component Y and either none or exactly two mutually disjoint rational curves (the exceptional divisor E) which correspond to the resolution of the two singularities. In the first case we have that Y 2 = 0. In the second case, we use the formula (2.) and we obtain that Y 2 =. On the other hand, we have that 2g(Y ) 2 = K Y.Y = (K + Y ).Y = K.Y + Y 2, where K.Y 0 since K is nef. Therefore, we obtain in both cases g(y ), which means that Y is not rational. Now, we are going to prove the second part of Theorem.4. We will present two proofs: the first one is an immediate consequence of Theorem 3.2 and the second one uses a local argument that will be extended to entire curves in Theorem 5.0. Definition 5.5. Let D be a divisor on and let Bs(D) be the base locus of the linear system D. The stable base locus of D is defined as B(D) := \ m>0 Bs(m D). Theorem 5.6. Let f : P! be a non constant holomorphic map. Then f P E [ B(K E). First proof. Let us suppose that f (P ) * E. We already know that K E is big (Theorem 5.2), so if f (P ) * B(K E), it holds deg f (K E) 0. However, from Theorem 3.2 we obtain deg f (K E) apple 4, a contradiction. Thus f (P ) B(K E) and the proof is complete. econd proof. Let f : P! be a non constant holomorphic map. Then, either f is tangent to one of the foliations F, F 2 given by the fibrations, 2 respectively, or f is not.

15 ON LANG CONJECTURE FOR OME PRODUCT-QUOTIENT URFACE 497 If f is tangent to one of the foliations, let us say to F, then f (P ) must be contained in a singular fiber of :! C /G. Otherwise f (P ) would be contained in a smooth fiber, but smooth fibers are hyperbolic, since they are isomorphic to C 2 and g(c 2 ) 2, and this is contradiction. Therefore, f P Y [ E, where Y is the central component and E is the exceptional divisor; however, Lemma 5.4 tells us that g(y ) and hence f (P ) E. Now, let us suppose that f is not tangent to any of the foliations and let us consider the composition bf := ' f. o we have the following diagram: P f f ϕ X := (C C 2 )/G By Theorem 5.2 there exists a positive integer m 0 such that for all m m 0 we have a non zero section! 2 H 0 (, O(m(K E))). Recall the section 2(!) 2 H 0 (X reg, 2m X ) obtained via 2; the section bf 2(!) = f 0(!) vanishes because H 0 (P, n ) = 0 for all n. Moreover, since 2(!) is locally written as P a(z, z 2 )dz m dz m 2 and bf is locally given by bf = (bf, bf 2 ) = '( f, f 2 ) where f, f 2, bf, bf 2 are holomorphic functions, the section bf 2(!) is locally given as a b f, bf 2 b f 0 m b f 2 0 m = 0. Thus we obtain that a(bf, bf 2 ) = (a ')( f, f 2 ) = 0 since by hypothesis the other factors are not always equal to zero. This last equation means that the section! vanishes on f (P ), but this is true for any section of O(m(K E)), then f (P ) Bs(m(K E)); besides this is true for all m m 0, therefore f (P ) B(K E) Entire curves We have already seen that the central components of singular fibers on are not rational, but we do not know yet if they can be elliptic. In the following example we will see that, in fact, for any product-quotient surface, the central components that do not intersect with the exceptional divisor, have genus bigger than one; however, in the case where the central components do intersect with the exceptional divisor, we give an example of a surface with a central component that is elliptic.

16 498 JULIEN GRIVAUX, JULIANA RETREPO VELAQUEZ AND ERWAN ROUEAU Example 5.7. Let be a quotient-product surface and let us consider the natural fibration :! C /G and a point x 2 C /G with non trivial stabilizer H x. Recall that the fiber F of over x is the union of a central component Y ' C /H x and either none or at least two mutually disjoint H-J strings which are in one-to-one correspondence with the branch points of C 2! C 2 /H x. In the first case, using the Riemann-Hurwitz formula, we obtain 2g(C ) 2 = H x (2g(Y ) 2), but 2g(C ) 2 > 0, then g(y ) 2. For the second case, we suppose belongs to the first family of the table given in section 3. ince X = (C C 2 )/G has only two singularities of type 2 (, ), then C 2! C 2 /H x has two branch points with multiplicity equal to 2. Thus, using the Riemann-Hurwitz formula, we get 2g(C ) 2 = H x (2g(Y ) ), but g(c ) = 3, then, 4 = H x (2g(Y ) ). We easily conclude that H x must be equal to 4 and hence g(y ) =. Now, we recall a well known theorem asserting that entire curves satisfy an algebraic differential equation. Namely: Theorem 5.8 ([0, Corollary 7.9], [4]). If there exists a non zero section s 2 H 0 (, m A ) with A an ample line bundle and m an integer, then for every entire curve f : C!, it holdds f s = 0. Using this, we can follow the same argument used in the second proof of Theorem 5.6 to prove an analogous result for entire curves. Definition 5.9. Let D be a divisor on and let Bs(D) be the base locus of the linear system D. The augmented base locus of D is defined as B + (D) := \ m>0 Bs(m D A) for any ample divisor A. Proposition 5.0. Let f : C! be a non constant holomorphic map. Then f (C) Y [ E [ B + (K E), where Y is the union of all central components.

17 ON LANG CONJECTURE FOR OME PRODUCT-QUOTIENT URFACE 499 Proof. For any entire curve f : C! we also have the following two possibilities: either f is tangent to one of the foliations F, F 2, or f is not. In the first case we have again that f (C) must be contained in the singular fibers because the smooth ones are Brody hyperbolic. Thus, f (C) Y [ E. In the second case, let A be an ample divisor; since K E is big (Theorem 5.2), there exists an infinite number of integers m such that H 0 (, O(m(K s E)) A ) 6= 0. o, we can consider a non-zero section! 2 H 0 (, O(m(K s E)) A ) and via 0 we obtain a nonzero section 0(!) 2 H 0 (, 2m A ). By Theorem 5.8, we have that f 0(!) = 0 and then, following the same argument than in the case of rational curves we obtain f (C) B + (K s E). Note that Example 5.7 shows that, a priori, we can not avoid the central components because they could be elliptic. However, using Theorem 3.2, we will prove that elliptic curves are contained in the the augmented base locus of K E. Proposition 5.. If is a product-quotient surface of general type such that p g = 0 and c 2 = 6, and f : C! is a holomorphic map where C is a smooth projective curve of genus g(c) =, then f (C) B + (K E). Proof. ince K E is big, then it can be written as the sum of an ample divisor A and an effective divisor D. Moreover, the augmented base locus can be given in terms of all these possible sums as \ B + (K E) = upp D K E=A+D [, Remark.3]. Now, if f (C) * B + (K E) then there is a D such that f (C) * D, thus deg f D 0 and hence, deg f (K E) = deg f A + deg f D > 0, but note that f (C) * E, thus from Theorem 3.2 we have that deg f (K E) apple 0, a contradiction. Therefore f (C) B + (K E). Finally, as a consequence of Propositions 5.0 and 5. we obtain Theorem.5. Theorem 5.2. If is a product-quotient surface of general type such that p g = 0 and c 2 = 6, then for any non constant holomorphic map f : C!, f (C) E [ B + (K E).

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19 ON LANG CONJECTURE FOR OME PRODUCT-QUOTIENT URFACE 50 [8] F. POLIZZI, Numerical properties of isotrivial fibrations, Geom. Dedicata 47 (200), [9] M. REID, urface cyclic quotient singularities and hirzebruch-jung resolutions, manuscript 202. [20] X. ROULLEAU and E. ROUEAU, Canonical surfaces with big cotangent bundle, Duke Math. J. (7) 63 (204), [2] F. ERRANO, Isotrivial fibred surfaces, Ann. Mat. Pura Appl. (4) 7 (996), UMR-CNR 7373 CMI, Université d Aix-Marseille 39, Rue Frédéric Joliot-Curie 3453 Marseille Cedex 3, France jgrivaux@math.cnrs.fr CMI, Université d Aix-Marseille 39, Rue Frédéric Joliot-Curie 3453 Marseille Cedex 3, France juliana.restrepo-velasquez@univ-amu.fr erwan.rousseau@univ-amu.fr