Semistable fibrations over an elliptic curve with only one singular fibre
aa r X i v : . [ m a t h . AG ] M a r SEMISTABLE FIBRATIONS OVER AN ELLIPTIC CURVE WITHONLY ONE SINGULAR FIBRE.
ABEL CASTORENA, MARGARIDA MENDES LOPES, AND GIAN PIETRO PIROLA
Abstract.
In this work we describe a construction of semistable fibrationsover an elliptic curve with one unique singular fibre and we give effectiveexamples using monodromy of curves. Introduction
Let X be a compact smooth complex algebraic surface. A fibration of X is amorphism with connected fibres φ : X → B , where B is a smooth curve. It iswell known that if g ( B ) ≤ C → E of elliptic curves using monodromy and showing that under suitable hy-pothesis the surface C × C has a fibration φ : C × C → E as required. Notations and conventions.
We work over the complex numbers. ∼ denotesnumerical equivalence of divisors, whilst ≡ denotes linear equivalence. The compo-sition of permutations is done right to left.2. Generalities for a fibration over an elliptic curve with oneunique singular fibre
Let E be an elliptic curve with origin O E ∈ E , and let K E be the canonicalbundle of E . Let X be a compact smooth complex surface and φ : X → E be arelatively minimal semistable fibration with general fibre F a curve of genus g ≥ X is a minimal surface, since any rational curve must be vertical. Bythe same reason X is not birational to a ruled surface. Mathematics Subject Classification.
Primary 14H10 14D06 Secondary 114C20 14J29.
Key words and phrases. minimal number of fibres, semistable fibrations over elliptic curves,ramified covers of curves, monodromy, surfaces of general type.
Let ω X | E be the relative dualizing sheaf, and let K X be the canonical divisorof X . Denote ∆( φ ) = deg( φ ∗ ( ω X | E )). Since K E is trivial, the relative canonicaldivisor K φ is linearly equivalent to K X , that is, K φ ≡ K X − φ ∗ K E ≡ K X . Since E is of genus one, one has ∆( φ ) = χ ( O X ) and K X | E = K X . Set χ := χ ( O X )and let c := c ( X ) = χ top ( X ) be the topological Euler characteristic of X . Remarkthat for a semistable fibration over an elliptic curve c is exactly the number of nodesof the singular fibres.Assume also that the general fibres of φ are not all isomorphic (since X is minimaland the fibration is semistable this is the same as saying that the fibration is nottrivial). Then, by Arakelov’s theorem (see [2]), K X >
0. Since S is not a rationalsurface, by the classification of surfaces we conclude that S is of general type andhence χ ≥ q = h ( O X ) and p g = h ( K X ) to be respectively the irregularityand the geometric genus of X .By [2], 1 ≤ q < g + 1, where the strict inequality comes from the fact that weare assuming that the fibration is not trivial. We recall the sharpening of Vojta’sinequalities by Tan (see [11], Theorem 2 and Lemma 3.1), that in our case (i.e. φ : X → E relatively minimal, not trivial and semistable) says that when thenumber s of singular fibres is positive then χ < g s and K X < (2 g − s. Finally we recall Beauville’s formula for a reducible semistable fibre F (see theproof of Lemme 1 of [1]): n = g − g ( N ) + c − g is the genus of the general fibre, N is the normalisation of F , g ( N ) = h ( N, O N ), c is the number of components of F and n is the number of doublepoints of F . Proposition 2.1.
Let X be a compact smooth complex surface, E an elliptic curveand φ : X → E a semistable fibration with general fibre F of genus g and with oneunique singular fibre F . Then:i) K X < g − ii) χ < g ; iii) c < g + 2; iv) if φ is stable, c ≤ g − v) if φ is stable and c = 3 g − , then q = 1 and φ is the Albanese map;vi) if φ is stable, < χ < g − .In particular if φ is stable, g ≥ .Proof. If φ is semistable with one unique singular fibre, the inequalities K X < g − χ < g follow from the above mentioned sharpening of Vojta’s inequalities. Theinequality c < g + 2 is the proposition of [3].Let F be the unique singular fibre. Note that c is exactly the number n ofdouble points of F . If φ is stable, every component θ of F satisfies K X θ ≥ EMISTABLE FIBRATIONS OVER AN ELLIPTIC CURVE WITH ONLY ONE SINGULAR FIBRE.3
Since K X F = 2 g −
2, the number c of components of F is at most 2 g −
2. Usingthe above mentioned Beauville’s formula we obtain c = g − g ( N ) + c − ≤ g − g ( N ) + 2 g − − ≤ g − − g ( N )and so c ≤ g − c = 2 g − g ( N ) = 0, i.e. everycomponent of F is a rational curve. But then the Albanese map of X contracts F to a point and therefore also all fibres of φ . Hence φ is the Albanese map of X and q = 1.Also, we obtain 12 χ = c + K < g −
1) and so2 < χ < g − , where the first inequality comes from the fact that X has χ ≥ (cid:3) Remark 2.1.
Note that if φ is stable with one unique singular fibre and g = 4,then χ = 1, K ≤ c ≤
9. Now c = 9 means K X = 3. From Debarre’sinequality, [6], K ≥ p g for irregular surfaces we conclude that q = 1 and φ is theAlbanese fibration. But (see [5]) for K = 3, p g = q = 1 the genus of the generalfibre of the Albanese fibration is 2 or 3. Therefore we have 4 ≤ K ≤ χ ≥ g ≥ Covering maps and semistable fibrations.
Let E be an elliptic curve and let O E ∈ E be the origin. Let C be a smoothcurve of genus g C and let f : C → E be a covering map of degree d . We have thefollowing Construction 3.1.
Set S = C × C , and for any map f : C → E , let φ : S → Eφ ( a, b ) = f ( a ) − f ( b ) be the difference map. Despite the fact that S is a very simple surface and that the construction isvery elementary, the map φ can be interesting. We start by giving a condition thatassures the connectedness of the fibres of φ. Proposition 3.1.
Assume that the map f : C → E is primitive of degree d and g C > , (that is, f is not decomposable and not ´etale). Then the fibres of φ : S → E are connected and the general fibre F of the fibration φ has genus g ( F ) = 2( g C − d + 1 .Proof. We have to show that φ has connected fibres. Suppose by contradiction thatthe general fibre F of φ is not connected. By the Stein factorization theorem, thereexists a smooth curve Y , a finite morphism h : Y → E , deg h >
1, and a morphism e φ : S → Y with connected fibres such that the following diagram is commutative: S φ / / e φ (cid:31) (cid:31) ❅❅❅❅❅❅❅ EY. h O O Fix a point q ∈ C such that f ( q ) = O E , consider the inclusion i q : C → C × C , i q ( p ) = ( p, q ), and let C := i q ( C ). We have φ ◦ i q = f. Setting κ = e φ ◦ i q , we get a ABEL CASTORENA, MARGARIDA MENDES LOPES, AND GIAN PIETRO PIROLA decomposition f = κ ◦ h , but since f is not decomposable, deg κ = 1 and deg h = d .It follows that Y is isomorphic to C . Moreover, for the general fibre of φ one has F = d X i F i and therefore numerically F ∼ d e F , e F = F . Since F · C = d we get e F · C = 1.By symmetry we also have e F · C = 1, where C = { q } × C. Since K S ∼ (2 g C − C + (2 g C − C , and e F = 0 we obtain by the adjunctionformula 2 g ( e F ) − g C −
4, i.e. g ( e F ) = 2 g C −
1. But on the other hand e F · C j = 1implies, by considering one of the projections onto C of C × C that g ( e F ) = g C , acontradiction. (cid:3) We want to consider now the case when the map f : C → E has a unique branchpoint, which will assume to be O E ∈ E . Proposition 3.2.
Assume that f : C → E is primitive and has only O E as acritical value. Let F = φ − ( O E ) be the fibre over the origin. Then:i) The map φ is smooth outside F ;ii) F decomposes as F = Γ + △ , where △ ⊂ S is the diagonal and Γ = F − △ ;iii) △ · Γ = 2 g C − .iv) If the ramification divisor of f is reduced, then F has only nodes as singular-ities and φ is a stable fibration with only one singular fibre.Proof. The only critical value of f is the origin O E . Let η ∈ H ( E, K E ), η = 0, bea generator, then φ ∗ ( η ) = ( η , η ) = ( f ∗ ( η ) , − f ∗ ( η )). The critical locus C ( φ ) of φ is then defined by the vanishing of φ ∗ ( η ), that is, C ( φ ) := { ( p, q ) ∈ S : p, q ∈ zero locus of f ∗ ( η ) } . In particular, C ( φ ) ⊂ F = φ − ( O E ) and therefore O E is the only critical value of φ. Since △ ⊂ F , △ · F = 0 and so △ · Γ = △ · ( F − △ ) = −△ = 2 g C − . Suppose that the ramification divisor of f is reduced. Then in any ramificationpoint p j ∈ C of f , f locally is the map z → z . So, for ( p , p ) ∈ C ( φ ), we canfind local coordinates ( z i , U p i ), i = 1 ,
2, on C such that z i ( p i ) = 0, and a localcoordinate ( t, W ) on E , O E ∈ W and t ( O E ) = 0, such that locally f and φ aregiven by f ( z i ) = z i , φ ( z , z ) = z − z . Since F ∩ U × U around ( p , p ) is given by z − z = 0 , this proves that thesingularities of F are nodes. Since S does not contain any rational curve, φ is astable fibration. (cid:3) In view of Proposition 3.2, to obtain with this construction explicit examples ofsemistable fibrations over an elliptic curve with one unique singular fibre, we needto find primitive covers f : C → E such that O E is the only branch point and theramification divisor R of f is reduced. EMISTABLE FIBRATIONS OVER AN ELLIPTIC CURVE WITH ONLY ONE SINGULAR FIBRE.5
We recall (see [8, Ch. III]) that given a smooth complex irreducible projectivecurve E , a finite subset B of E and any point x ∈ E − B , there is a one-to-onecorrespondence between the set of isomorphism classes of covering maps f : C → E of degree d whose branch points lie in B and the set of group homomorphisms ρ : π ( E − B, x ) → S d with transitive image (up to conjugacy in S d ). The coveringmap is primitive if Im ( ρ ) is a primitive subgroup of S d .In the case at hand, i.e. where E is an elliptic curve E and B = O E is asingle point, the fundamental group Π := π ( E − { O E } , x ) is a free group ontwo generators x, y . The monodromy homomorphism of groups ρ : Π → S d isgiven by ρ ( x ) = α, ρ ( y ) = β and the ramification type of the covering f : C → E is encoded in the decomposition of the commutator [ α, β ] in S d as a product ofdisjoint cycles. The ramification divisor R of f is reduced if and only if every cyclein such a decomposition has length 2.So to give explicit examples it is enough to find transitive and primitive sub-groups G ⊂ S d generated by two permutations α, β, such that [ α, β ] is a productof disjoint transpositions. We give here some examples. Examples :
1) Consider the subgroup G ⊂ S generated by the permutations α, β, where: α = (1 , , , β = (2 , , . Then G is obviously a transitive subgroup and[ α, β ] = αβα − β − = (1 , , G were not primitive, we could have onlytwo blocks preserved. On the order hand α has order 3 and therefore α cannotpreserve or interchange blocks.In this case, C is a curve of genus 2 and the general fibre of φ has genus 9 (cf.Proposition 3.1).2) This example is due to Pietro Corvaja.Consider the subgroup G ⊂ S generated by the permutations α, β, where: α = (1 , , , , , , , β = (8 , , , , , , then we have [ α, β ] = (1 , , , , . Since G is isomorphic to P GL ( F ), G is transitive and primitive. In this case, C is a curve of genus 3 and the general fibre of φ has genus 33 (cf. Proposition 3.1).3) Consider the subgroup G ⊂ S d , d = 4 n + 1 , n ≥ α, β, where: α = (1 , , · · · (2 n − , n ) , β = (1 , n + 1) (2 , n + 2 , · · · n, n, n + 1)Then [ α, β ] = (1 , , · · · (4 n − , n ) . ABEL CASTORENA, MARGARIDA MENDES LOPES, AND GIAN PIETRO PIROLA
Again G is clearly a transitive subgroup of S d . Let γ = (1 , n + 1) and δ =(2 , n + 2 , · · · n, n, n + 1). Then β = γδ = δγ and furthermore β n − = γ .Hence G = < α, β > contains γ and δ .Let B be a block preserved by G containing 1. B has cardinality k >
1, because2 n + 1 ∈ B . Since k must divide 4 n + 1, k is odd and so B contains another element x , x = 1, x = 2 n + 1. Since x is a fixed element for γ , γ ( B ) = B . Similarly,since 1 is a fixed element for δ , δ ( B ) = B . So also β ( B ) = B and therefore B = { , , ...., n + 1 } , i.e. G is primitive.Here for each n , g ( C ) = n + 1 and the general fibre of φ has genus 2 nd + 1 (cf.Proposition 3.1). Remark.
We finally remark that recently, motivated by the theory of themapping class group, Pietro Corvaja and Fabrizio Catanese constructed severalnew examples.
Acknowledgments.
We want to thank Fabrizio Catanese and Pietro Corvaja forfruitful comments and for the example 2 which was suggested by Pietro Corvaja.We also thank Alexis G. Zamora for bringing to our attention the paper of A. N.Parsin cited in this work.The first author was supported by Research Grant PAPIIT IN100716 (UNAM,M´exico). The second author is a member of the CAMGSD of T´ecnico-Lisboa,University of Lisbon and she was partially supported by FCT (Portugal) throughprojects PTDC/MAT- GEO/2823/2014 and UID/MAT/ 04459/2013. The thirdauthor was supported by PRIN 2015 Moduli spaces and Lie Theory, INdAM -GNSAGA and FAR 2016 (PV) Variet`a algebriche, calcolo algebrico, grafi orientatie topologici.
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