Elliptic fibrations on covers of the elliptic modular surface of level 5
Francesca Balastrieri, Julie Desjardins, Alice Garbagnati, Céline Maistret, Cecília Salgado, Isabel Vogt
aa r X i v : . [ m a t h . AG ] M a y ELLIPTIC FIBRATIONS ON COVERS OF THE ELLIPTICMODULAR SURFACE OF LEVEL 5
FRANCESCA BALESTRIERI, JULIE DESJARDINS, ALICE GARBAGNATI,C´ELINE MAISTRET, CEC´ILIA SALGADO, AND ISABEL VOGT
Abstract.
We consider the K3 surfaces that arise as double covers of theelliptic modular surface of level 5, R , . Such surfaces have a natural ellipticfibration induced by the fibration on R , . Moreover, they admit several otherelliptic fibrations. We describe such fibrations in terms of linear systems ofcurves on R , . This has a major advantage over other methods of classificationof elliptic fibrations, namely, a simple algorithm that has as input equations oflinear systems of curves in the projective plane yields a Weierstrass equationfor each elliptic fibration. We deal in detail with the cases for which the doublecover is branched over the two reducible fibers of type I and for which it isbranched over two smooth fibers, giving a complete list of elliptic fibrationsfor these two scenarios. Introduction
Let S/ C be a smooth complex surface and B/ C be a smooth complex curve. Wesay that a proper flat map E : S → B is a elliptic fibration if the generic fiber S b is a smooth genus 1 curve and a section O : B → S is given. Given a section, weregard the generic fibre of E as an elliptic curve over the function field k ( B ) andso we can work with a Weierstrass form of E . We will say that an elliptic fibrationis relatively minimal if there are no contractible curves contained in its fibers. Forthe remainder of this paper all elliptic fibrations will be assumed to be relativelyminimal and not of product type.Not all surfaces S admit elliptic fibrations and if S admits an elliptic fibration,a lot is known about the base curve and about the maximal number of ellipticfibrations on it. More precisely if S is of general type, then S admits no ellipticfibrations; if the Kodaira dimension of S is non-positive, then the curve B is ra-tional; if a surface S admits more than one elliptic fibration as above, then it isa K3 surface (a surface with trivial canonical bundle and trivial irregularity). Inparticular if S is either a K3 surface or a rational surface, then B ≃ P . Everyrelatively minimal rational elliptic surface is the total space of a pencil of planecubics; such surfaces admit only the obvious elliptic fibration. We refer to [5] andto [8] for more on the theory of elliptic fibrations on surfaces.In this paper we will consider K3 surfaces S which are obtained by a base changeof order 2 from rational elliptic surfaces. Such K3 surfaces are the minimal resolu-tion of the fiber product of a rational elliptic surface E R : R → P and a degree twomap P → P , which is necessarily branched over two points. This immediatelyimplies that S admits a non-symplectic involution ι , which is the cover involution ofthe generically 2 : 1 map S R , and that this involution fixes the inverse image Last Updated: May 11, 2017 of two fibers of the fibration E R : R → P . We recall that an involution ι on a K3surface is non–symplectic if it acts by − H ( S, K S ) ≃ C .The fixed locus of any non-symplectic involution ι on S , Fix( ι ) is known to besmooth and, if it is non-empty, it consists of the disjoint union of smooth curves.In our setting there are the following possibilities for Fix( ι ): it is the union of twosmooth genus 1 curves (which are the inverse images of smooth fibers of E R ), it isthe union of one smooth genus 1 curve and k rational curves (the smooth genus 1curve is the inverse image of a smooth fiber of E R and the other rational curves arethe inverse images of the irreducible components of a reducible fiber of E R ), it isthe union of k + 1 rational curves (which are the inverse image of the components oftwo reducible fibers of E R ). Vice versa, it was proved by Zhang (see [12]) that everyK3 surface S admitting a non–symplectic involution ι whose fixed locus containscurves of genus at most 1 arises by a base change of order two from a rationalelliptic fibration E R : R → P as described above.The general setting of this paper is the following: E R : R → P is a relativelyminimal elliptic fibration on a rational surface R . The K3 surface S is obtained bya base change of order 2 from E R and so it is naturally equipped with an inducedelliptic fibration E S : S → P and with a non-symplectic involution ι . The surface S/ι is rational and is either R (if ι fixes two smooth curves of genus 1) or a blow-upof R (denoted by e R in what follows). In [3] the relations among the elliptic fibrationson K3 surfaces S constructed as above and linear systems on the rational surface R are studied. The elliptic fibrations on S fall into one of the three categories belowaccording to the action of ι on the fibers. Note that E S belongs to the second one: • if ι preserves each fiber of the fibration, then, because it is non-symplectic, itacts on the fibers as the elliptic involution. The elliptic fibration on S is thereforeinduced by fibrations in rational curves on e R . We will call these pencils “conicbundles” if they are rational fibrations on R and “generalized conic bundles” ifthey are rational fibrations on e R (but not on R ); • if ι preserves the fibration, but not each fiber of the fibration, this implies that ι acts on the base of the fibration (with two fixed points). In this case the ellipticfibration on S is induced by a pencil of genus 1 curves on R , whose members splitin the double cover. We call these pencils “splitting genus 1 pencils”; • if ι does not preserve the elliptic fibration, we call the fibration of type 3 . In thiscase the image of the linear system defining the fibration on S is a non-completelinear system of genus 1 curves on R . A fibration is of type 3 if and only if the classof the fiber of the fibration is not preserved by ι .As a result of this classification and of the technique introduced in [3], in goodcases one may classify the singular fibers of all elliptic fibrations E : S → P in termsof the singular fibers of more tractable linear series on S/ι . Our focus here is oneven finer information: obtaining explicit Weierstrass equations of elliptic fibrationson such K3 surfaces. In Sections 5.2 and 6.1.2 we give methods and algorithms fordetermining Weierstrass equations coming from conic bundles and splitting genus1 pencil on rational elliptic surfaces, under some assumptions.We focus particularly on K3 surfaces arising as double covers of R , , the ellipticmodular surface of level 5. This is the universal elliptic curve over the modularcurve X (5) and the evident map E R , : R , → X (5) ≃ P is the unique ellipticfibration. The fibers of E R , are smooth except for two nodal rational curves (type I ) and two 5-gons (type I ); this property also determines R , and implies that LLIPTIC FIBRATIONS ON COVERS OF THE ELLIPTIC MODULAR SURFACE 3 the Mordell-Weil group MW( E R ) = Z / Z . The geometry of this surface as the totalspace of a pencil of plane cubics is described in Section 2.We then move on to describe elliptic fibrations arising naturally from linear serieson (a blow-up of) R , both in terms of classes in the N´eron-Severi group, and frompencils of plane curves. In Section 3 we classify the conic bundles on R , up toautomorphisms.In the remainder of the paper we study elliptic fibrations on different K3 surfacesarising from R , by choosing different base changes. In Section 4 we describe suchK3 surfaces, writing their equations as double covers of P , as well as giving theWeierstrass form of the elliptic fibrations induced by E R .In Section 5 we consider the elliptic fibrations induced on K3 surfaces by theconic bundles classified in Section 3. We focus on different K3 surfaces, arisingfrom E R , : R , → P by different base changes, and observe directly that the sameconic bundle induces elliptic fibrations with very different properties according tothe choice of the branch curves. Note that the choice of a certain base change isessentially equivalent to choosing the type of the branch fibers of the 2 : 1 cover S R .In Sections 6 and 7 we restrict our attention to two K3 surfaces obtained bychoosing maximally different branch fibers: in Section 6 we consider the very specialcase where the branch fibers are 2 I and in Section 7 we consider the generic casewhere the branch fibers are 2 I .When the double cover is branched over the two 5-gons, the K3 surface is called S , and the involution ι fixes the union of 10 rational curves. There is a unique suchK3 surface possessing such a non–symplectic involution. This K3 surface admits 13types of elliptic fibrations, classified by Nishiyama in [7]. In this special case we areable to determine equations for all elliptic fibrations on S , using our techniquesand algorithms. We observe that in this case there are no fibrations of type 3.When the double cover is branched on two smooth fibers, the K3 surface movesin the 2-dimensional family of the K3 surfaces admitting an elliptic fibration witha 5-torsion section. We call a generic member of this family X , and using thelattice-theoretic technique of [7] we list all the admissible configurations of fibers ofan elliptic fibration on X , in Table 7.1. In this case the elliptic fibrations cannotbe induced by splitting genus 1 pencils or by generalized conic bundles. On theother hand there are plenty of elliptic fibrations of type 3 and we describe one ofthem in detail. 2. The surface R , The surface R , is the blow-up of P in the base points of the pencil P of cubics(2.1) λx x ( x − x ) + µx ( x − x − x )( x − x ) = 0 . We will denote this blow-up morphism by β : R , → P . The cubic correspond-ing to λ = 0 is reducible and consists of the three lines m : x = 0, m : x − x − x = 0, m : x − x = 0; the cubic corresponding to µ = 0 is reducibleand consists of the three lines ℓ : x − x = 0, ℓ : x = 0, ℓ : x = 0.The nine base points of this pencil of cubics are: Q := (0 : 0 : 1); the point Q ′ ,infinitely near to Q and corresponding to the tangent direction m ; Q := (1 : 1 :0); the point Q ′ , infinitely near to Q and corresponding to the tangent direction m ; Q := (0 : 1 : 0); the point Q ′ , infinitely near to Q and corresponding to ℓ ℓ ℓ m m m • T • T • Q • Q • Q • Q • Q Figure 1.
The reducible cubics and base points of the pencil P .the tangent direction ℓ ; Q := (1 : 0 : 1); the point Q ′ , infinitely near to Q andcorresponding to the tangent direction ℓ ; Q := (1 : 1 : 1).In the following we will denote by T the point (1 : 0 : 0), which is not a basepoint of the pencil and corresponds to the intersection of the lines ℓ and ℓ , andby T the point (0 : 1 : − m and m .Let h denote the preimage of the class of a line; then NS( R , ) is spanned by h and the components of the exceptional divisors of the blow up β : R , → P . Wewill denote the (irreducible) exceptional divisor corresponding to Q i (resp. Q ′ i ) by E i (resp. F i ) for i = 1 , , ,
4. At Q there is only E . Note that F i = − E i = − i = 1 , , , E = − E i E j = 0 if i = j , F i F j = 0 if i = j , E i F i = 1. Byslight abuse of notation, let ℓ , ℓ , ℓ and m , m , m denote the proper transformson R , of the corresponding lines in P . We have the following realtions: ℓ = h − E − F − E − F − E m = h − E − F − E − F ℓ = h − E − F − E − F m = h − E − F − E − F ℓ = h − E − F − E − F m = h − E − F − E − F − E . (2.2)The Weierstrass equation of the elliptic fibration of R , is obtained by 2.1putting x = λ = 1 and applying standard transformation. It is(2.3) y = x + A ( µ ) x + B ( µ ) , where A ( µ ) := − µ − µ − µ + 14 µ − , and B ( µ ) := 1864 µ + 148 µ + 25288 µ + 25288 µ − µ + 1864 . The discriminant is µ ( − µ + µ ), so there are, as expected, two fibers oftype I over µ = 0 and µ = ∞ . Moreover there are two fibers of type I over µ = − ± √ LLIPTIC FIBRATIONS ON COVERS OF THE ELLIPTIC MODULAR SURFACE 5
Now the function µ ( x ( µ ); y ( µ )) = (cid:18) µ + 6 µ + 12 ; µ (cid:19) is a 5-torsion section of this fibration. It is known, see e.g. [8, Section 9.5] that theelliptic fibration on R , has Mordell–Weil group equal to Z / Z .The negative curves on R , are(1) the ten components of the two fibers of type I denoted by Θ (1)0 , Θ (1)1 , Θ (1)2 ,Θ (1)3 , Θ (1)4 on the first fiber and Θ (2)0 , Θ (2)1 , Θ (2)2 , Θ (2)3 , Θ (2)4 on the secondfiber (these are all ( − P , P , P , P , and P , where P meets the componentsΘ (1)0 and Θ (2)0 , P meets the components Θ (1)1 and Θ (2)2 , P meets the com-ponents Θ (1)2 and Θ (2)4 , P meets the components Θ (1)3 and Θ (2)1 and P meets the components Θ (1)4 and Θ (2)3 (these sections are all ( − • •••• ◦ ◦◦◦◦ • •••• Θ (1)0 P Θ (2)0 Θ (1)1 P Θ (2)2 Θ (1)2 P Θ (2)4 Θ (1)3 P Θ (2)1 Θ (1)4 P Θ (2)3 Figure 2.
Dual graph of negative curves on R , . The symbol • denotes a ( − ◦ denotes a ( − (1)0 = m Θ (1)1 = E Θ (1)2 = m Θ (1)3 = E Θ (1)4 = m Θ (2)0 = E Θ (2)1 = ℓ Θ (2)2 = ℓ Θ (2)3 = E Θ (2)4 = ℓ P = F P = F P = E P = F P = F . (2.4)We observe that there is an automorphism σ on R , of order 5, which is thetranslation by the section P . It acts on the negative curves as follows: σ (Θ (1) i ) =Θ (1) i +1 and σ (Θ (2) i ) = Θ (2) i +2 , where i + 1 and i + 2 are considered modulo 5; σ ( P k ) = P k +1 , where k + 1 is considered modulo 5.There is also an automorphism σ of order 2 on R , which is the elliptic involu-tion on the elliptic curve (2.3) over the function field k ( µ ). Note that σ restrictsto the elliptic involution on each smooth fiber of the fibration (2.3). It acts on thenegative curves as follows: σ (Θ ( j ) i ) = Θ ( j ) − i , where i, j ∈ Z / Z and σ ( P k ) = P − k ,where k ∈ Z / Z . Conic bundles on R , In this section we classify the conic bundles on R , by considering their reduciblefibers proving the following result. Proposition 3.1.
There are four conic bundles on R , up to automorphisms, i.e.the conic bundles B , B , B and B induced by the pencils of plane rational curveswith equations 3.1, 3.2, 3.3, 3.4 respectively. The key result we use is that on a rational elliptic surface, every conic bundle hasat least one reducible fiber. Further, any reducible fiber must be of type A n or D m ,as shown in Figure 3, see e.g. [3]. As R , (and in fact any extremal rational ellipticsurface) has only finitely many curves of negative self-intersection, one simply mustfind all possible A n and D m configurations among them. ◦ • • • ◦ A n •• • • • ◦ D m Figure 3.
Possible reducible fibers of conic bundles on (minimal)rational elliptic surfaces. The number n and m refer to the numberof components. Multiplicity of a component is indicated above thecorresponding vertex if it is not 1. The symbol • denotes a ( − ◦ denotes a ( − R , :(1) | B | = | P + Θ (1)0 + Θ (1)1 + P | . The components of one reducible fiber are P ,Θ (1)0 , Θ (1)1 , P . This fiber is of type A . The curves Θ (1)2 , Θ (1)4 , Θ (2)0 and Θ (2)2 aresections of the bundle. There is another reducible fiber of type A which is madeup by the curves P , Θ (2)4 , Θ (2)3 , P , and one of type D which is made of P (withmultiplicity 2), Θ (1)3 , Θ (2)1 .Using the identifications made earlier, this class can also be written as: B = F + m + E + F = F + ( h − E − F − E − F ) + E + F = h − E − F . Unwinding what this means geometrically: | B | comes via proper transform fromthe pencil of lines through Q in P which has equation(3.1) x = τ x . Under this description we can also understand the singular fibers: they correspondto the special lines m , ℓ , and ℓ . For example the line ℓ corresponds to reduciblefiber β ∗ ( ℓ ) − E − F = ℓ + E + 2 F = Θ (2)1 + Θ (1)3 + 2 P .The conic bundle | B | is sent to other conic bundles by σ , by σ and by theirpowers. Each of these has three reducible fibers of types A , A and D . Forexample the components of the image of the reducible fibers for σ are { P , Θ (1)1 , Θ (1)2 , P } , { P , Θ (2)1 , Θ (2)0 , P } , { P , Θ (1)4 , Θ (2)3 } and the image of the reducible fibers for σ are LLIPTIC FIBRATIONS ON COVERS OF THE ELLIPTIC MODULAR SURFACE 7 { P , Θ (1)0 , Θ (1)4 , P } , { P , Θ (2)1 , Θ (2)2 P } , { P , Θ (1)2 , Θ (2)4 } . (2) | B | = | P + Θ (1)0 + Θ (1)1 + Θ (1)2 + P | . The components of one reducible fiberare P , Θ (1)0 , Θ (1)1 , Θ (1)2 , P . This fiber is of type A . The curves Θ (1)3 , Θ (1)4 , Θ (2)0 ,Θ (2)4 and P are sections of the bundle. There is another reducible fiber of type A which is made up by the curves P , Θ (2)1 , Θ (2)2 , Θ (2)3 , P .Similarly here we can write: B = F + m + E + m + E = 2 h − E − F − E − F − E − F . Hence B corresponds to the pencil of conics through Q , Q , Q ′ , and Q . Moreexplicitly this is given by conics passing through Q and Q , and tangent to ℓ at Q and in P this pencil is given by the equation(3.2) x x = τ ( x x − x ) . The two reducible fibers correspond to the reducible conics m ∪ m and ℓ ∪ ℓ .The conic bundle | B | is sent to other conic bundles by the i -th powers of σ ( i = 1 , , ,
4) and σ .(3) | B | = | P + Θ (1)0 + Θ (1)1 + Θ (1)2 + Θ (1)3 + P | . The components of one reduciblefiber are P , Θ (1)0 , Θ (1)1 , Θ (1)2 , Θ (1)3 , P . This fiber is of type A . The curves Θ (2)0 ,Θ (2)1 , P and P are sections of the bundle. The curve Θ (1)4 is a multisection ofdegree 2. There is another reducible fiber of type D which is made up by thecurves P , Θ (2)3 , Θ (2)4 , Θ (2)2 .We can also describe this using: B = F + m + E + m + E + F = 2 h − E − F − E − F − E . Therefore B comes from the pencil of conics in P through Q , Q , Q ′ and Q ;that is conics through Q and Q , tangent to ℓ at Q , and in P this pencil isgiven by the equation(3.3) x x = ( τ + 1) x x − τ x . We can again understand the reducible fibers as coming from reducible conics ℓ ∪ ℓ and m ∪ m .The conic bundle | B | is sent to other conic bundles by the i -th powers of σ ( i = 1 , , ,
4) and σ .(4) | B | = | P + Θ (2)0 + Θ (2)1 + Θ (2)2 + Θ (2)3 + P | . The components of one reduciblefiber are P , Θ (2)0 , Θ (2)1 , Θ (2)2 , Θ (2)3 , P . This fiber is of type A . The curves Θ (1)0 ,Θ (1)4 , P and P are sections of the bundle. The curve Θ (2)4 is a multisection ofdegree 2. There is another reducible fiber of type D which is made up by thecurves P , Θ (1)2 , Θ (1)1 , Θ (1)3 .We can also describe this using: B = F + E + l + l + E + F = 2 h − E − F − E − F . Therefore B comes from the pencil of conics in P through Q , Q ′ , Q and Q ′ ;that is conics through Q and Q , tangent to ℓ at Q and to ℓ at Q , and in P this pencil is given by the equation(3.4) x x = τ ( x − x ) . The reducible fibers correspond to the singular conics ℓ ∪ ℓ and 2 m .The conic bundle | B | is sent to other conic bundles by the i -th powers of σ ( i = 1 , , ,
4) and σ .Since the reducible fibers of | B | are (2 A , D ), the reducible fibers of | B | are(2 A ) and the reducible fibers of | B | are ( A , D ), they cannot be equivalentup to automorphisms. So there are at least three different conic bundles up toautomorphisms on R , . It remains to discuss the possibilities that B and B areequivalent up to automorphisms (they are not equivalent up to σ and σ since theseautomorphism preserves each fiber). If there was an automorphism sending B to B , then there would be an automorphism which switches the two fibers of type I .Moreover this automorphism sends sections to sections. Since the intersections ofthe sections with one fiber of type I are not equivalent to the intersections of thesections with the other fiber of type I , one obtains that it is impossible to define anautomorphism which switches these two fibers. So the conic bundles B and B arenot equivalent. Hence there are four conic bundles on R , up to automorphisms.4. K3 surfaces obtained by R , Now we consider K3 surfaces obtained from R , by a base change of order 2branched on two fibers. Of course the K3 surface obtained depends on the branchfibers. Let us explicitly give the description of the K3 surfaces that we can obtainin this way. They will be both described as elliptic fibrations (induced by the oneof R , ) and as double covers of P .4.1. The branch fibers are I : the K3 surface S , . Let us consider theK3 surface S , obtained by a base change of order 2 of R , whose branch locuscorresponds to the two fibers of type I . This means that all the components ofthe fibers of type I are in the branch locus of the double cover S , R , .4.1.1. The surface e R . The double cover of R , branched over the two fibers oftype I has ordinary double point singularities at the 10 points over the nodes inthe branch fibers. In order to obtain a K3 surface one can blow-up these 10 pointson the double cover, introducing 10 exceptional divisors. Equivalently one mayfirst blow-up the 10 nodes of the branch fibers to obtain a non-minimal rationalelliptic surface e R and then normalize the double cover of this surface branched overthe preimage of the branch fibers. Note that in the preimage the 10 exceptionalcurves all occur with multiplicity 2, and so they are not in the branch locus afternormalization.We will make use of this non-minimal rational elliptic surface e R ; it is simplythe blow-up of P in 9 + 10 points, some of which are infinitely near to each other.The 10 additional points are T , T and the 2 points on each of the exceptionaldivisors E i for i = 1 , , , Q i specifiedby respectively ℓ and ℓ , ℓ and ℓ , m and m , and finally m and m .We will denote by E Q , E T and E T the exceptional divisors over Q , T and T respectively. These three divisors are ( − E i , F i , G i and H i the four exceptional divisors over Q i for i = 1 , , ,
4. For each i , the divisor E i is a ( − F i , G i , and H i , which are orthogonal ( − F i is the LLIPTIC FIBRATIONS ON COVERS OF THE ELLIPTIC MODULAR SURFACE 9 tangent direction corresponding to the basepoint of the pencil of cubics, and H i and G i correspond to the tangent directions specified in the following table: i G i ℓ ℓ m m H i ℓ ℓ m m .Note that the E i and F j are the strict transforms of the curves of the same nameon R , .The strict transforms of the lines ℓ j and m j , j = 1 , , e R are the following: ℓ : = h − E − F − G − H − E − F − G − H − E Q ; ℓ : = h − E − F − G − H − E − F − G − H − E T ; ℓ : = h − E − F − G − H − E − F − G − H − E T ; m : = h − E − F − G − H − E − F − G − H − E T ; m : = h − E − F − G − H − E − F − G − H − E T ; m : = h − E − F − G − H − E − F − G − H − E Q . The sections of the non–relatively minimal fibration on e R are F j , j = 1 , , , E Q (i.e. the strict transform of the sections of the fibration on R , ).4.1.2. Geometric description of S , and its N´eron–Severi group. The surface S , admits a non-symplectic involution ι which is the cover involution of the doublecover S , → e R . This involution fixes 10 rational curves (the curves m i , ℓ i , i = 1 , , E j with j = 1 , , ,
4) and it acts trivially on the N´eron-Severi group.The elliptic fibration E S , : S , → P induced by E R , : R , → P has twofibers of type I (induced by the fibers of type I on R , ) and four other singularfibers, all of type I . Since both the rank of the N´eron-Severi group and the rank ofthe trivial lattice are equal to 20, we conclude that there are no sections of infiniteorder for the fibration E S , : S , → P . The 5-torsion sections of the fibration on R , induce 5-torsion sections of E S , . Hence, MW( E S , ) ⊇ Z / Z . The possibletorsion parts of the Mordell–Weil group of an elliptic fibration on a K3 surface are Z /n Z for 2 ≤ n ≤
8, and Z / Z × Z /m Z , for m = 2 , ,
6, ( Z /k Z ) for k = 3 , E S , ) = Z / Z .The curves Θ ( j ) i are in the branch locus and we denote by Ω ( j ) i the rational curveon S , which maps 1 : 1 to Θ ( j ) i . Moreover, we have 10 other rational curves on S , : the curves Ω ( j ) i ,i − , for i , i − ∈ Z / Z , j = 1 ,
2, which are the curvesresolving the singularities of the intersection point between Θ ( j ) i and Θ ( j ) i and arethe double cover of the 10 exceptional curves of the blow up e R → R , . The curves P i are not in the branch locus and we denote by Q i their 2 : 1 cover in S , . Thedual graph of this configuration is given in Figure 4. We observe that this givesexactly the diagram given in [11, Figure 1] as dual graph of certain rational curveson the K3 surface whose transcendental lattice is h i , which is a different way todescribe the surface S , .Let π : S , → e R denote the double cover. As can be deduced from (2.4) and theabove identifications, pushing forward curve classes has the following effect: • •••• • •••• • •••• • Ω (2)3 , • Ω (2)4 , • Ω (2)2 , • Ω (2)4 , • Ω (2)1 , • Ω (1)3 , • Ω (1)4 , • Ω (1)4 , • Ω (1)1 , • Ω (1)2 , Ω (1)0 Q Ω (2)0 Ω (1)1 Q Ω (2)2 Ω (1)2 Q Ω (2)4 Ω (1)3 Q Ω (2)1 Ω (1)4 Q Ω (2)3 Figure 4.
Dual graph of relevant negative curves on S , . π ∗ Ω (1)0 = m π ∗ Ω (1)3 , = 2 G π ∗ Ω (2)0 = E π ∗ Ω (2)3 , = 2 G π ∗ Q = 2 F π ∗ Ω (1)1 , = 2 G π ∗ Ω (1)3 = E π ∗ Ω (2)1 , = 2 H π ∗ Ω (2)3 = E π ∗ Q = 2 F π ∗ Ω (1)1 = E π ∗ Ω (1)4 , = 2 H π ∗ Ω (2)1 = ℓ π ∗ Ω (2)4 , = 2 H π ∗ Q = 2 F π ∗ Ω (1)2 , = 2 H π ∗ Ω (1)4 = m π ∗ Ω (2)1 , = 2 E T π ∗ Ω (2)4 = ℓ π ∗ Q = 2 F π ∗ Ω (1)2 = m π ∗ Ω (1)4 , = 2 E T π ∗ Ω (2)2 = ℓ π ∗ Ω (2)4 , = 2 G π ∗ Q = 2 E . (4.1)4.1.3. Weierstrass equation of S , . By the Weierstrass equation (2.3) the fibers of E R , of type I are the fibers over µ = 0 and µ = ∞ . So the base change branchedon these fibers is given by µ → µ and the elliptic fibration on S , induced by theone on R , is(4.2) y = x + A ( µ ) x + B ( µ ) , where A ( µ ) := − µ − µ − µ + 14 µ − , and B ( µ ) := − µ − µ − µ − µ + 148 µ − . The discriminant is µ ( − µ + µ ), so there are, as expected, two fibersof type I over µ = 0 and µ = ∞ . Moreover there are four fibers of type I over µ = ± q − ± √ Double cover of P . One the other hand, e R and R , are blow-ups of P andthe branch fibers of π : S , R , corresponds to the cubics f := x x ( x − x ) =0 and g := x ( x − x − x )( x − x ) = 0. This exhibits S , as a double cover of P branched along the union of these two cubics. So we obtain a different equationfor S , , as a double cover of P , i.e.(4.3) w = x x ( x − x ) x ( x − x − x )( x − x ) . LLIPTIC FIBRATIONS ON COVERS OF THE ELLIPTIC MODULAR SURFACE 11
We observe that S , is rigid, since R , is rigid and the choice of the branch fiberis rigid.4.2. The branch fibers are I : the K3 surface X , . Let us consider theK3 surface X , obtained by a base change of order 2 of R , whose branch locuscorresponds to two fibers of type I . This K3 surface lies in a 2-dimensional familyof K3 surfaces, whose parameters depend on the choice of the two branch fibers.4.2.1. Geometric description of X , and its N´eron–Severi group. The surface X , admits a non-symplectic involution ι which is the cover involution of the doublecover X , → R , and which fixes two elliptic curves.The elliptic fibration E X , : X , → P induced by E R , : R , → P has fourfibers of type I and four fibers of type I . Moreover it has a 5-torsion section,induced by the one of sections of the elliptic fibration on E R , . The N´eron–Severigroup and the transcendental lattice of this K3 surface are computed in [2] and aset of generators of the N´eron–Severi group is given by the class of the fiber of thefibration, the zero section, one section of order 5 and the irreducible componentsof the reducible fibers of the fibration.The curves Θ ( j ) i are not in the branch locus and we denote by Ω ( j,k ) i for k = 1 , ( j ) i by the quotient map X , → R , . The curves P i are not in the branch locus and we denote by Q i their2 : 1 cover in X , . The dual graph of this configuration is shown in Figure 5. ••••• •• •• •• Q • Q • Q • Q • Q •••••• •• •• Ω (1 , Ω (2 , Ω (1 , Ω (2 , Ω (1 , Ω (2 , Ω (1 , Ω (2 , Ω (1 , Ω (2 , Ω (1 , Ω (2 , Ω (1 , Ω (2 , Ω (1 , Ω (2 , Ω (1 , Ω (2 , Ω (1 , Ω (2 , Figure 5.
Dual graph of relevant negative curves on X , .4.2.2. Weierstrass equation of X , . Let us denote by µ and µ two arbitrarypoints of P µ such that the fibers of (2.3) over µ and µ are smooth. Let X , bethe surface obtained from R , by a base change of order 2 branched in µ and µ . The surface X , depends on the two parameters µ and µ , i.e., it lives in a2-dimensional family of K3 surfaces. Let us consider the base change P α : β ) → P µ : λ ) branched over ( µ : λ ) = ( µ : 1)and ( µ : 1), i.e. the base change given by(4.4) µ = µ α + β , λ = α + β /µ . It induces on X , the elliptic fibration(4.5) y = x + A ( α : β ) x + B ( α : β )whose discriminant is (cid:16)(cid:0) α µ + β (cid:1) (cid:0) µ α + β (cid:1) (cid:0) α µ µ + 11 α µ µ − α µ + 11 α µ µ β ++2 µ α µ β + 11 α µ β − α µ β + µ β + 11 β µ − β (cid:1)(cid:1) / (cid:0) µ (cid:1) (4.6)For generic values of µ and µ the elliptic fibration has 4 I + 4 I as singularfibers.4.2.3. Double cover of P . One the other hand, X , is the double cover of P branched on the union of the two cubics x x ( x − x )+ µ x ( x − x − x )( x − x ) =0 and x x ( x − x ) + µ x ( x − x − x )( x − x ) = 0. So X , can be describedby the equation(4.7) w = (cid:0) x x ( x − x ) + µ x ( x − x − x )( x − x ) (cid:1)(cid:0) x x ( x − x ) + µ x ( x − x − x )( x − x ) (cid:1) . Branch fibers I and I . If one uses as branch fibers a fiber of type I and oneof type I one obtains a rigid K3 surface, whose singular fibers are I +2 I + I +2 I and theoretically one has four different admissible choices to do that. Indeed onecan decide that the fiber of type I which is the branch fiber is the fiber eitherover µ = 0 or over µ = ∞ and similarly one can decide that the fiber of type I which is the branch fiber is the fiber either over µ = ( −
11 + 5 √ / µ = ( − − √ /
2. In order to obtain the Weierstrass equation of these ellipticfibrations it suffices to apply the base change (4.4) with the chosen values for µ and µ .In particular we assume that µ = ( −
11 + 5 √ / µ = 0 the base change (4.4) applied to µ = 0 and µ = ( −
11 + 5 √ / y = x + A ( α : β ) x + B ( α : β )whose discriminant is − (cid:16) − α + 5 β √ (cid:17) (cid:16) − α − β + 5 β √ (cid:17) α β . In order to choose µ = ∞ , one has to slightly change the equation of the basechange (4.4), which now is µ = α and λ = α /µ + β and one obtains y = x + A ( α : β ) x + B ( α : β )whose discriminant is15120 (cid:16) − √ (cid:17) (cid:16) − α + β √ (cid:17) (cid:16) − α + 11 β + 5 β √ (cid:17) β α . LLIPTIC FIBRATIONS ON COVERS OF THE ELLIPTIC MODULAR SURFACE 13
We observe that in the first case the K3 surface obtained is described as a doublecover of P by the equation w = x x ( x − x ) (cid:16) x x x (13 + 5 √ − x x (11 + 5 √
5) + 2 x − x x − x x + 2 x x (cid:17) , in the second by the equation w = x ( x − x − x )( x − x ) (cid:16) x x x (13 + 5 √ − x x (11 + 5 √
5) + 2 x − x x − x x + 2 x x (cid:17) . Branch fibers I and I . If one chooses as branch fibers one fiber of type I and one of type I one obtains a 1-dimensional family of K3 surfaces, whosesingular fibers are I + 2 I + 4 I and theoretically one has two different admissibleways to do that. Indeed one can chose that the fiber of type I which is the branchfiber is the fiber either over µ = 0 or over µ = ∞ , while µ is the parameter ofthe family. In order to obtain the equations of these elliptic fibrations one has toapply the base changes ( µ = β , λ = α + β /µ ) or ( µ = α , λ = α /µ + β ) tothe equation (2.3), exactly as in the previous sections.Similarly one can describe these K3 surfaces as a double cover of P substitutingin (4.7) the appropriate values of µ and µ .4.5. Branch fibers I and I . If one chooses as branch fibers one fiber of type I and one of type I one obtains a 1-dimensional family of K3 surfaces, whosesingular fibers are 4 I + I + 2 I and theoretically one has two different admissibleways to do that. Indeed one can choose that the fiber of type I which is the branchfiber is the fiber either over µ = ( − − √ / µ = ( −
11 + 5 √ /
2, while µ is the parameter of the family. In order to obtain the equations of these ellipticfibrations one has to apply the base change (4.4) with the chosen µ to the equation(2.3) and to obtain an equation of this surface as a double cover of P one has tosubstitute the chosen µ in (4.7).4.6. Branch fibers I . If one chooses as branch fibers the two fibers of type I one obtains a rigid K3 surface, whose reducible fibers are 4 I + 2 I . In order toobtain the equation of this elliptic fibration one has to apply the base change (4.4)with the chosen µ = ( − − √ / µ = ( −
11 + 5 √ / P onehas to substitute µ = ( − − √ / µ = ( −
11 + 5 √ / Elliptic fibrations on K3 surfaces induced by the conic bundles
The aim of this section is to describe both geometrically and by the Weierstrassequations the elliptic fibrations induced by the conic bundles B i (described in Sec-tion 3) on the K3 surfaces described in Section 4. We also provided a generalmethod to find these Weiersatrss equations, under some assumption on the conicbundles, see Section 5.2.5.1. An example.
Let us consider the K3 surface S , , whose equation as a doublecover of P is given by (4.7). Let us consider also the conic bundle | B | from Section3. By [3, Theomrem 5.3], the conic bundle | B | induces an elliptic fibration on S , with three reducible fibers of type I ∗ : one whose components are Q , Ω (1)4 , , Ω (1)0 ,Ω (1)1 , , Ω (1)1 , Q , and Ω (1)2 , , one whose components are Q , Ω (2)4 , , Ω (2)4 , Ω (2)4 , , Ω (2)3 , Q ,and Ω (2)3 , , and one whose components are Ω (1)4 , , Ω (1)3 , , Ω (1)3 , Q , Ω (2)1 , Ω (2)2 , , and Ω (2)1 , . Equation of the elliptic fibration on S , induced by | B | . Let us consider theconic bundle B on R , associated to the pencil of lines x = τ x ⊂ P . It inducesan elliptic fibration on S , . To find the equation of this elliptic fibration we usethe equation of S , as double cover of P , i.e. the equation (4.3) and we substitutein x = τ x in (4.3).This gives: w = ( τ x ) x ( x − τ x ) x ( x − τ x − x )( x − x )We put x = 1 and we obtain w = τ (1 − τ ) x ( x − τ x − x − . Let us consider the change of coordinates w wx and divide both the membersof x . We obtain w = τ (1 − τ ) x ( x − τ x − x − . This is the equation of an elliptic fibration over P τ .To be more precise one canexplicitly compute the Weierstrass form: first one uses the changes of coordinates w τ (1 − τ ) w and x τ x obtaining w τ (1 − τ ) = τ (1 − τ ) x ( τ x (1 − τ ) − τ x − x )and so w = x (cid:18) x − τ (1 − τ ) (cid:19) (cid:18) x − τ (cid:19) Second, one considers the changes of coordinates w w/τ (1 − τ ) and x x /τ (1 − τ ) and multiplies all the equation by τ (1 − τ ) . So one obtains w = x ( x − τ (1 − τ ))( x − τ (1 − τ ) ) . (5.1)The discriminant is τ (1 − τ ) and so there are three fibers of type I ∗ over τ = 0, τ = 1, τ = ∞ .5.2. A Method.
The aim of this Section is to formalize systematically what wedid above.
Setup.
Let V be a K3 surface obtained by a base change of order 2 from a rationalelliptic surface R . Therefore, V can be described as double cover of P branchedon the union of two (possibly reducible) plane cubics from the pencil determining R . It has an equation of the form(5.2) w = f ( x : x : x ) g ( x : x : x ) . Let B be a conic bundle on R , e.g. a basepoint-free linear system of rationalcurves giving R → P τ . Pushing forward to P , B is given by a pencil of planerational curves with equation h ( x : x : x , τ ). The polynomial h ( x : x : x , τ )is homogeneous in x , x , x , say of degree e ≥ τ .As the anticanonical series on R coincides with the elliptic fibration, the ad-junction formula implies that every curve with equation h ( x : x : x , τ ) meetsboth of the branch curves (the proper transforms on R of) f = 0 and g = 0in two additional points. It therefore meets (the proper transform of) their union f g = 0 in four points. (Note that there may be additional points of intersectionon P which are separated in the blow-up R ). Therefore the preimage in V is thedouble cover of a rational curve branched over 4 points, e.g. the standard presen-tation of an elliptic curve. For general τ , we must find an isomorphism of the curve LLIPTIC FIBRATIONS ON COVERS OF THE ELLIPTIC MODULAR SURFACE 15 h ( x : x : x , τ ) = 0 with P , and extract the images of the four intersection pointswith f g = 0.When e ≤
3, an isomorphism with P is provided by projection from a point oforder e − P if e = 1, a point on the conic if e = 2,and a double point of the cubic if e = 3). Such a point necessarily exists (in thecase e = 3 the singularity must be a basepoint of the pencil) and is also necessarilya basepoint of the original pencil of cubics giving E R . Up to acting by PGL ( C ),we may assume that this point is (0 : 1 : 0). Algorithm when e ≤ . (1) Compute the resultant of the polynomials f ( x : x : x ) g ( x : x : x )and h ( x : x : x , τ ) with respect to the variable x . The result is apolynomial r ( x : x , τ ) which is homogeneous in x and x , correspondingto the images of all of the intersection points { f g = 0 } ∩ { h τ = 0 } afterprojection from (0 : 1 : 0).(2) Since B is a conic bundle, r ( x : x , τ ) will be of the form a ( x : x , τ ) b ( x : x , τ ) c ( τ ), where a and b are homogeneous in x and x , the degree of a depends upon e and the degree of b in x and x is 4.(3) The equation of V is now given by w = r ( x : x , τ ), which is birationallyequivalent to(5.3) w = c ( τ ) b ( x : x , τ ) , by the change of coordinates w wa ( x : x , τ ). Since for almost every τ ,the equation (5.3) is the equation of a 2 : 1 cover of P x : x ) branched in 4points, (5.3) is the equation of the genus 1 fibration on the K3 surface V induced by the conic bundle B .(4) If there is a section of fibration (5.3), then it is possible to obtain theWeierstrass form by standard transformations. Remark 5.1.
The algorithm can be applied exactly in the same way to the gen-eralized conic bundles, and not only to the conic bundles.When e ≥
4, projection from a point may suffice, for example if all curves havea basepoint of degree e −
1. However there are several conic bundles whose generalmember can not be parametrized by lines.We now consider one parametrization which can be done by conics. Let P be apencil of rational curves of degree e passing through the (possibly infinitely near)base points T , . . . T r with certain multiplicities. Let C be a pencil of conics whosebase points are among T , . . . T r and such that 2 e − P and a generic curve in C are in { T , . . . T r } . So there is exactly oneextra intersection between a generic curve in P and a generic curve in C . This allowsto parametrize the curves in P by the curves in C . If moreover the base points of C are three distinct points and one infinitely near point we will say that the condition( ⋆ ) is satisfied. So ( ⋆ ) is satisfied if the curves in P can be parametrized by pencilof conics passing through 4 points, exactly two of which are infinitely near.If the condition ( ⋆ ) is satisfied, up to changing coordinates by some matrix M ∈ PGL ( C ), we may assume that the base points of C are p = (0 : 0 : 1) , p =(0 : 1 : 0) , p = (1 : 0 : 0) and the infinitely near point p ′ corresponds to the line x = x . The pencil of degree 2 maps P z → P sending1 p , ∞ 7→ p , p with derivative at z = 1 in the direction of the line x = x is given by( x : x : x ) = ( z − z ( z −
1) : p · z ) , p ∈ P . In the following we are interested in pencil of quartics which satisfy the condition( ⋆ ), so we specilize this condition to certain quartics. We say that a pencil ofquartics satisfies ( † ) if, up to a change of coordinates, it is of one of the followingtype: (1) each quartic is double at p and has a tacnode at p with principal tangentspecified by p ′ ; (2) each quartic is double at p and p and has a cusp at p withprincipal tangent specified by p ′ .We recall that, by the construction of the conic bundles, all the base points of P (and thus also of C ) are also singular points for the sextic f · g = 0. Algorithm assuming ( † ) . (1) Factor h ( z − z ( z −
1) : pz, τ ) = c ( τ ) z a ( z − b r ( z, p, τ ) where r ( z, p, τ )is linear in z and a + b is 5 or 6 depending on the multiplicity of h at p . The solution z of r ( z, p, τ ) = 0 gives the rational parameterization h ( z − z ( z −
1) : pz , τ ) = 0 with parameter p ∈ P .(2) A birational equation of the K3 surface is given by w = ( f · g )( z − z ( z −
1) : pz ) . (3) If there is a section of fibration (5.3), then it is possible to obtain theWeierstrass form by standard transformations.This algorithm can be generalized to pencil of curves satisfying ( ⋆ ).5.3. The elliptic fibrations induced by conic bundles.
Here we can describeand compute the equations of all the elliptic fibrations induced by the conic bundleson the different K3 surfaces introduced. Since we have an algorithm it sufficesto apply it to the equations of the conic bundles given in Section 3 and to theWeierstrass equations given in Section 4.5.3.1.
The K3 surface S , . The conic bundle | B | induces an elliptic fibration on S , with two reducible fibers of type I ∗ : one whose components are Q , Ω (1)0 , , Ω (1)0 ,Ω (1)1 , , Ω (1)1 , Ω (1)2 , , Ω (1)2 , Q , and Ω (1)3 , , and one whose components are Q , Ω (2)1 , , Ω (2)1 ,Ω (2)2 , , Ω (2)2 , Ω (2)3 , , Ω (2)3 , Q , and Ω (2)4 , .The Weierstrass equation is computed applying the algorithm to f g = x x x ( x − x )( x − x )( x − x − x ), by (4.3), and h ( x : x : x , τ ) = x x − τ ( x x − x ).One obtains the Weierstrass equation(5.4) y = x − τ ( − τ + τ + 1)3 x − (1 / τ ( − τ )( − τ )( τ + 1)27 . So the discriminant ∆( τ ) is − τ ( − τ ) ( τ + 1) . Hence this fibration has twofibers of type I ∗ over τ = 0 , ∞ and two fibers of type I over τ = ± | B | induces an elliptic fibration on S , with two reduciblefibers: one of type I ∗ whose components are Q , Ω (1)0 , , Ω (1)0 , Ω (1)1 , , Ω (1)1 , Ω (1)2 , , Ω (1)2 ,Ω (1)3 , , Ω (1)3 , Q , and Ω (1)4 , , one of type III ∗ whose components are Q , Ω (2)3 , Ω (2)3 , ,Ω (2)4 , , Ω (2)2 , Ω (2)4 , Ω (2)2 , , and Ω (2)0 , .The Weierstrass equation is(5.5) y = x − τ ( τ + 6 τ + 9 τ + 3)3 x − τ ( τ + 3)(2 τ + 12 τ + 18 τ + 9)27 . LLIPTIC FIBRATIONS ON COVERS OF THE ELLIPTIC MODULAR SURFACE 17
So the discriminant ∆( τ ) is − τ ( τ + 4)( τ + 1) . Hence this fibration has one fiberof type I ∗ over τ = ∞ , one fiber of type III ∗ over τ = 0, one fiber of type I over τ = − I over τ = − | B | gives a fibration which is analogous to the fibration givenby the conic bundle | B | , since the construction of S , is totally symmetric withrespect the the fibers I of R , , i.e. both of them are branch fibers.5.3.2. The K3 surface X , . Here we consider the elliptic fibrations induced by B i on X , . We recall that in this case one has to apply the algorithm to(5.6) f g = (cid:0) x x ( x − x ) + µ x ( x − x − x )( x − x ) (cid:1)(cid:0) x x ( x − x ) + µ x ( x − x − x )( x − x ) (cid:1) , by equation (4.2.3). We summarize the results in the following table, where theWeierstrass equation is y = x + A ( τ ) x + B ( τ ), ∆( τ ) = 4 A ( τ ) + 27 B ( τ ) and r = rank( M W ):(5.7) ∆ singularf ibers rB − τ ( τ − ( µ − µ ) ( τ − τ − τ µ + 6 τ µ + τ + τ µ − µ )(6 τ µ + τ − τ − µ + τ − τ µ + τ µ ) I ∗ τ = 02 I τ = ∞ , I B − τ ( µ τ + 2 µ τ + µ − τ − τ + 2 τ )( − τ + µ )( µ τ + 2 µ τ + µ − τ − τ + 2 τ )( − τ + µ )( µ − µ ) I τ = 0 , ∞ I B − τ ( τ + 4 τ + 2 µ τ + 6 µ τ + µ + 2 µ τ + µ τ )( µ τ + µ + τ + 4 τ + 2 µ τ + 6 µ τ + 2 µ τ )( µ − µ ) I τ = 0 I ∗ τ = ∞ I B gives a fibration analogous to the one of the conic bundle B , since the construction of X is symmetric with respect to the two fibers I of R , , i.e. both of them are not branch fibers. In the table we used B since thecomputation in this case are easier with the equation of B than with the equationof B .5.3.3. Other K3 surfaces.
As we saw when computing the Weierstrass equationsand the models as double covers of P of K3 surfaces which are obtained as doublecovers of R , branched over some special fibers, all the equations for these K3surfaces can be obtained by the general equation for X , substituting particularvalues of µ and µ . In particular in order to find the Weierstrass equations ofthe elliptic fibrations induced by the conic bundles B i on a specific K3 surface itsuffices to substitute in (5.7) the appropriate values of µ and µ . As an examplehere we construct a table analogous to (5.7) if the K3 surface is obtained by R , branching along one fiber of type I and one generic fiber. We already noticed thatone has two different choices for the branch fibers. Once one chooses the branchfiber I , the construction is not symmetric in I , thus the conic bundles B and B do not give necessarily similar elliptic fibration. In the following we will choosethe fiber I to be the fiber over µ = 0. So that we will chose µ to be 0 and µ is the parameter of the 1-dimensional family of K3 surfaces we are considering. Sowe obtain the following (where r = rank( M W )): (5.8) ∆ singularf ibers rB − τ µ ( − τ ) ( τ − τ − τ µ + τ + 6 τ µ + τ µ − µ ) I ∗ τ = 0 , I ∗ τ = 1 I τ = ∞ , I B − τ µ ( − τ ) ( µ + τ µ + 2 τ µ − τ − τ + 2 τ )( µ − τ ) I ∗ τ = 0 , I τ = 1 I , τ = ∞ , I B − τ µ ( τ + 1) ( τ + 2 τ − τ µ + τ + τ µ − τ µ + 4 µ ) III ∗ τ = 0 , I τ = − I τ = ∞ , I B − τ µ (1 + 4 τ )(4 τ + τ µ + τ + 6 τ µ + 2 τ µ + 2 τ µ + µ ) I ∗ τ = 0 I ∗ τ = ∞ , I The K3 surface S , and its elliptic fibrations The aim of this section is to prove the following results, computing the equationsof all the elliptic fibrations on S , . Proposition 6.1.
The K3 surface S , admits 13 different elliptic fibrations. Oneof them is induced by E R , 3 are induced by conic bundles, 3 by splitting genus onepencils and 6 by generalized conic bundles. The equations of these elliptic fibrationsare given in 4.2 (the one induced by E R ), in 5.1, 5.4, 5.5 (the ones induced by theconic bunldes), in 6.7 (the ones induced by splitting genus 1 pencil) and in 6.8, 6.9(the ones induced by generalized conic bundles). Since S , has fibrations induced by splitting genus 1 pencils and generalizedconic bundles, in this section we analyze deeper these two types of elliptic fibra-tions. In particular we describe a method to find Weierstarss equations for ellipticfibrations induced by splitting genus 1 pencils.The K3 surface S , can be also described as the (unique!) K3 surface whichadmits a non-symplectic involution fixing 10 rational curves. Indeed, by our con-struction it is clear that ι fixes 10 rational curves (the inverse image of the com-ponents of the two I fibers). The fact that this K3 surface is unique is classicallyknown, and due to Nikulin, see [6]. The transcendental lattice of this K3 surface is T S ≃ h i⊕h i . The elliptic fibrations on this K3 surface are classified by Nishiyama,see [7, Table 1.2], who used a lattice theoretic method that we will apply later to adifferent K3 surface, in Section 7. The complete list of the elliptic fibrations is thefollowing:(6.1) n o singular fibers M W II ∗ + 2 I { } II ∗ + I ∗ + 2 I { } I ∗ + 2 I + 2 I Z / Z III ∗ + I ∗ Z / Z III ∗ + I ∗ + I + I Z / Z I + I + 4 I Z / Z n o singular fibers M W I ∗ + 4 I { } I ∗ + I ∗ + 2 I Z / Z I ∗ + 2 I ( Z / Z ) I + I + 4 I Z / Z IV ∗ + I + 4 I Z × Z / Z
12 3 I ∗ ( Z / Z )
13 2 I + 4 I Z / Z Since the involution ι acts as the identity on the N´eron–Severi group of S , ,there are no fibrations of type 3 on this K3 surface. LLIPTIC FIBRATIONS ON COVERS OF THE ELLIPTIC MODULAR SURFACE 19
The fibration 13 in Table (6.1) is the one induced by the fibration E R , and ithas equation (4.2). By Section 5 the fibrations 5, 9, and 12 are induced by conicbundles and their equations are (5.5), (5.4), and (5.1), respectively.The other fibrations are induced either by generalized conic bundles or by split-ting genus 1 pencils.An elliptic fibration induced by a splitting genus 1 pencil corresponds to a fibra-tion of genus 1 curves on a non-relatively minimal rational elliptic surface (that is,this fibration admits ( − R , can be recovered from this non-minimalsurface by blowing down some divisors. A different choice of divisors to blow-down,namely the ( − − curves which are components of the fibers of the splitting genus 1pencil, gives us another rational elliptic surface on which the splitting genus 1 pencilabove corresponds to a relatively minimal elliptic fibration. Hence each fibrationgiven by a splitting genus 1 pencil is indeed induced by a rational elliptic surface(different from R , ) by a base change of order 2 whose branch locus consists of10 curves. Considering the list of elliptic fibrations on S , given in Table 6.1 oneobserves immediately that the fibrations 6, 10 and 11 are of this type, i.e., they areinduced by splitting genus 1 pencils, see also [3]. The others are not. We alreadyobserved that the fibration 13 is induced by the elliptic fibration on R , and thatthe fibrations 5,9 and 12 are induced by conic bundles, so the fibrations 1,2,3,4,7,8are induced by generalized conic bundles.6.1. Splitting genus 1 fibrations.
An example, the fibration 6.
We give an example of splitting genus 1 pencilof curves: we are looking for a fiber of type I and it is given by W X := Q + Ω (2)0 + Ω (2)4 , + Ω (2)4 + Ω (2)4 , + Ω (2)3 + Ω (2)3 , + Ω (2)2 + Ω (2)2 , + Ω (2)1 ++ Q + Ω (1)3 + Ω (1)3 , + Ω (1)2 + Ω (1)2 , + Ω (1)1 + Ω (1)1 , + Ω (1)0 . Indeed this is the class of a fiber of a fibration with a reducible fiber of type I .Using (4.1), the class W R := π ∗ W X is(6.2)2 F + E +2 G + ℓ +2 H + E +2 G + ℓ +2 E T + ℓ +2 F + E +2 G + m +2 H + E +2 G + m =5 h − E − F − G − H − E − F − G − H − E − F − G − H − E − F − G − H − E Q − E T . Since we expect that the curves in the linear system described split in the doublecover, we can assume that the class W R is both the push down and the geometricimage of a fiber of our fibration (i.e. π ∗ ( W X ) = π ( W X ) = W R ).This class is the strict transform on e R of quintics in P with the following prop-erties: they have a tacnode in Q with principal tangent ℓ ; they have a tacnodein Q with principal tangent ℓ ; they have a node in Q ; the tangent in Q is m ;the tangent in Q is m ; they pass through T .This gives the following families of quintics(6.3) − bx + bx x + bx x − ex x x + ex x x + ex x x +( − b − e ) x x x + bx x + bx x = 0 . We observe that this equation is obtained by requiring all the conditions on Q , Q , Q , Q , and T above to be satisfied, but only requiring that Q is a cusp (notethat Q is automatically a tacnode). So (6.3) is the equation of the splitting genus b : e ).We now have to intersect the branch sextic given in equation (4.2) with (6.3).The resultant of the polynomials (4.3) and (6.3) with respect to the variable x is − x x b ( x − x ) ( x + x ) ( e + 2 b ) . We observe that all the solutions in ( x , x ) have even multiplicities, as is necessarilyif the double cover splits in two curves, isomorphic to the base curve, see [3, Section3]. More precisely, the curve splits after the base change of order two branched in( b : e ) = (0 : 1) and ( b : e ) = (1 : − X , one first finds therational elliptic fibration given by the splitting pencil of genus 1 curves (6.3), andthen one performs the base change of order two. This can be computed by Mapleand it is y = x + A ( b : e ) x + B ( b : e ) , where A ( b : e ) := 2348 b − b e − b e + 112 be − e and B ( b : e ) := − b − b e + 31288 b e − b e − b e + 1144 be − e . The discriminant of this rational elliptic fibration is116 b ( e + 2 b )( e − be + 13 b )and indeed the fibration has one fiber of type I and three other singular fibers, allof type I .Now we consider the base change of order 2 branched in ( b : e ) = (0 : 1) and( b : e ) = (1 : − P β : ǫ ) → P b : e ) , where b = β and e = − β + 2 βǫ + ǫ .So we obtain a new elliptic fibration on S , whose equation is(6.4) y = x + A ′ ( β : ǫ ) x + B ′ ( β : ǫ ) ,A ′ ( β : ǫ ) := 2348 β − β ( − β + 2 βǫ + ǫ ) − β ( − β + 2 βǫ + ǫ ) ++ 112 β ( − β + 2 βǫ + ǫ ) − (1 / − β + 2 βǫ + ǫ ) and B ′ ( β : ǫ ) := − (1 / β − (5 / β ǫ − (7 / β ǫ +(28 / β ǫ +(7 / β ǫ − (11 / β ǫ + − (35 / β ǫ +(1 / β ǫ +(5 / β ǫ − (5 / β ǫ − (1 / β ǫ − (1 / ǫ β − (1 / ǫ . The discriminant is(1 / β (19 β − β ǫ − β ǫ + 4 βǫ + ǫ )( ǫ + β ) . This fibration has a fiber of type I , as expected, one fiber of type I (in ( β : ǫ ) =(1 : − I . By construction this elliptic fibration exhibits S , as the double cover of an extremal rational elliptic surface with one fiber oftype I and three fibers of type I . LLIPTIC FIBRATIONS ON COVERS OF THE ELLIPTIC MODULAR SURFACE 21
Splitting genus 1 fibration: an algorithm.
The aim of this section is to gen-eralize the previous construction to other splitting genus 1 pencils.
Setup.
Let π : V → P be a K3 surface which is a double cover of a (not necessarilyminimal) rational elliptic surface ˜ R branched over two fibers. If H : ˜ R → P b : e ) is asplitting genus 1 pencil, then the induced elliptic fibration on V comes via pullbackfrom a double cover P β : ǫ ) → P b : e ) . Hence given a Weierstrass form for H , it sufficesto find the branch points of the map P β : ǫ ) → P b : e ) in order to find the Weierstrassequation for the elliptic fibration on V . We now explain how to do this in general,when the branch curves and equations for H are given in P .Assume that the equation of V as double cover of P is given by w = f ( x : x : x ) g ( x : x : x ). Let h (( x : x : x ) , ( b : e )) be the equation of the pushforwardof a splitting genus 1 pencil from ˜ R to P . The equation h is homogeneous ofsome degree in the coordinate ( x : x : x ) on P and linear in the coordinate( b : e ) of the base P of the pencil. For every ( b : e ), the curve with equation h (( x : x : x ) , ( b : e )) is of arithmetic genus 1.Write ∪ k ∈ K C k for the irreducible components of the branch curves f g = 0.For D ⊂ P , write mult C k ( D ) for the multiplicity of the component C k in D . Thenwe have the following. Lemma 6.2.
A plane curve D ( b : e ) ⊂ P that is member of the pencil H is a branchcurve for the double cover V ˜ R P β : ǫ ) P b : e ) H induced by the splitting genus 1 pencil if and only if there exists k ∈ K such thatmult C k ( D ( b : e ) ) = 0 .Proof. If D meets the branch curves transversely, it does so only in points of evenmultiplicity, and so splits in the double cover as two disjoint elliptic curves. Itsuffices, therefore, to show that if D contains at least one component C k , thesupport of the preimage of D under the 2 : 1 map π : V → P is connected. Thisfollows from the fact that the preimage of C k is a double curve, the support ofwhich maps isomorphically onto C k , and so every component of π ∗ D must meetthis curve. (cid:3) We will also make use of the following elementary fact. Let D ( b : e ) ⊂ P be aplane curve in the pencil H with equation h (( x : x : x ) , ( b : e )). Denote by r (( x : x )( b : e )) the resultant of f ( x : x : x ) g ( x : x : x ) and h (( x : x : x ) , ( b : e )) with respect to x . Lemma 6.3.
The resultant r (( x : x )( b : e )) vanishes to order = X k mult C k ( D ( b : e ) ) · ( deg C k : [1 : 0 : 0] C k deg C k − ∈ C k at ( b : e ) = ( b : e ) .Proof. This follows from the geometric description of the zeros of the resultant interms of projecting the scheme-theoretic intersection of h = 0 and f g = 0 fromthe point (1 : 0 : 0). (cid:3) Writing ord ( b : e ) for the order of vanishing at ( b : e ), we may combine thesetwo Lemmas to show the following: Corollary 6.4.
The curve D ( b : e ) is a branch curve for the double cover inducedby the splitting genus 1 pencil if ord ( b : e ) (cid:0) r (( x : x )( b : e )) (cid:1) > . Furthermore, this is an if and only if when (1 : 0 : 0)
6∈ ∪ k C k . We can therefore determine the relevant branch points from the resultant. Thisleads to the following algorithm.
Algorithm. (1) Compute the resultant r (( x : x )( b : e )) of the two polynomials f ( x : x : x ) g ( x : x : x ) and h (( x : x : x ) , ( b : e )) in one variable, say x .(2) Observe that r (( x : x )( b : e )) = c ( b : e ) c ( b : e ) s (( x : x ) , ( b : e )) ,where c i ( b : e ) are homogeneous polynomials each with a unique root,denoted by ( b i : e i ). (If (1 : 0 : 0) is in the branch curves, it may benecessary to change coordinates first.)(3) Write the Weierstrass form of h (( x : x : x ) , ( b : e )), by applying thestandard transformations. This is the equation of a rational elliptic surface,and the base of the fibration is P b : e ) .(4) Consider the base change P β : ǫ ) → P b : e ) given by ( b = β ( b /e ) + ǫ , e = β + ( e /b ) ǫ ) (cf. (4.4)). Substituting this base change in the previ-ous Weierstrass equation, one finds the Weierstrass equation of the ellipticfibration on the K3 surface V whose base is P β : ǫ ) .6.2. The elliptic fibrations on S , . In this section we want to describe all theelliptic fibrations on S , giving equations for each of them.We observe that in [1] a model of S , as a double cover of P was given andthe elliptic fibrations induced by (generalized) conic bundles are already studiedgeometrically in that context. Here we explicitly describe in our context both thefibrations induced by generalized conic bundles and the ones induced by splittinggenus 1 curves, giving also a Weierstrass equation for each of them.6.2.1. Elliptic fibrations induced by splitting genus 1 pencils.
The fibration 11 ofTable (6.1) is induced by the class of the fiber M := Q + Ω (1)0 + Ω (1)1 , + Ω (1)1 + Q + Ω (2)2 + Ω (2)3 , + Ω (2)3 + Ω (2)4 , + Ω (2)4 + Ω (2)4 , + Ω (2)0 which is the class of a fiber of type I . The curves Ω (1)3 , Ω (1)3 , , Ω (1)2 , Ω (1)3 , , Ω (1)4 , Q , and Ω (2)1 are orthogonal to the components of the I -fiber and form a fiberof type IV ∗ . The classes Q , Q , Ω (1)1 , , Ω (1)4 , , Ω (2)1 , , Ω (1)2 , are sections of the ellipticfibration induced by | M | . We observe that there are 6 curves fixed by ι among thecomponents of the fiber of type I (the curves Ω ( j ) i for ( i, j ) = (0 , , , , , , IV ∗ -fiber (the curves Ω ( j ) j for ( i, j ) = (3 , , , , M is3 h − E − F − G − H − E − F − G − H − E − F − G − H − E T − E T − E which corresponds to a pencil of cubics passing through Q with tangent line ℓ ,through Q with tangent line m , through Q , E T , E T , and Q . The equation ofthis pencil is(6.5) b ( x x − x x + x x + x x − x x ) + e ( x x x − x x ) = 0 . The Weierstrass form of the (rational) elliptic fibration associated to the pencil(6.5) is y = x − b + 4 eb + e )(2 b + e ) x − (376 b + 176 eb + 60 e b + 8 e b + e )(2 b + e ) , whose discriminant is b (31 b + 4 eb + e )(2 b + e ) /
16. This elliptic fibration has afiber of type I on b = 0 and one of type IV on ( b : e ) = (1 : − M := Q +Ω (1)0 +Ω (1)1 , +Ω (1)1 +Ω (1)2 , +Ω (1)2 +Ω (1)3 , +Ω (1)3 +Ω (1)4 , +Ω (1)4 + Q +Ω (2)3 +Ω (2)4 , +Ω (2)4 +Ω (2)4 , +Ω (2)0 , which is the class of a fiber of type I .The push-down of the class M is4 h − E − F − G − H − E − F − G − H − E − F − G − H − E − F − G − H − E T − E which corresponds to a pencil of quartics with bitangent lines l (in Q and Q )and l (in Q and Q ) and having two nodes in E T and E . The equation of thispencil is(6.6) bx − bx x + bx x − bx x + (3 b + 4 e ) x x x + ( − b − e ) x x x + ex x + bx x + ( − b − e ) x x x + 2 ex x + ex x = 0 . Using the algorithm, we find the following Weierstrass equations for elliptic fi-brations on S , induced by splitting genus 1 pencils.(6.7) i elliptic fibrations6 A = β − β ( − β + 2 βǫ + ǫ ) − β ( − β + 2 βǫ + ǫ ) ++ β ( − β + 2 βǫ + ǫ ) − (1 / − β + 2 βǫ + ǫ ) B = − (1 / β − (5 / β ǫ − (7 / β ǫ + (28 / β ǫ + (7 / β ǫ − (11 / β ǫ + − (35 / β ǫ + (1 / β ǫ + (5 / β ǫ − (5 / β ǫ − (1 / β ǫ − (1 / ǫ β − (1 / ǫ ∆ = (1 / β (19 β − β ǫ − β ǫ + 4 βǫ + ǫ )( ǫ + β ) A = − (24 ǫ + β ) β / , B = − (216 ǫ + 36 β ǫ + β ) β / ǫ (27 ǫ + β ) β / A = ( − β + 16 ǫ β − ǫ ) / , B = − ( β − ǫ )( − ǫ − ǫ β + β ) / ǫ β (2 ǫ − β )(2 ǫ + β )(4 ǫ + β ) / Elliptic fibrations induced by generalized conic bundles.
Let us consider thedivisors: N : = 2Ω (1)0 + 4 Q + 6Ω (2)0 + 5Ω (2)1 , + 4Ω (2)1 + 3Ω (2)2 , + 2Ω (2)2 + Ω (2)3 , + 3Ω (2)4 , ,N : = 2Ω (2)4 + 4Ω (2)4 , + 6Ω (2)0 + 5Ω (2)1 , + 4Ω (2)1 + 3Ω (2)2 , + 2Ω (2)2 + Ω (2)3 , + 3 Q ,N : = Q + Ω (1)4 , + Q + Ω , + 2 (cid:16) Ω (1)0 + Ω (1)1 , + Ω (1)1 + Ω (1)2 , ++ Ω (1)2 + Ω (1)3 , + Ω (1)3 + Q + Ω (2)1 + Ω (2)2 , + Ω (2)2 + Ω (2)3 , + Ω (2)3 (cid:17) ,N : = Ω (2)1 , + 2Ω (2)0 + 3 Q + 4Ω (1)0 + 3Ω (1)1 , + 2Ω (1)1 + Ω (1)2 , + 2Ω (1)4 , ,N : = Ω (1)1 , + Q + Ω (2)4 , + Ω (2)4 , + 2 (cid:16) Ω (1)0 + Ω (1)4 , + Ω (1)4 + Ω (1)4 , + Ω (1)3 + Ω (1)3 , + Ω (1)2 + Q + Ω (2)4 (cid:17) . The linear system | N i | induces the elliptic fibration number i of the Table (6.1),indeed the divisor N i corresponds to an elliptic fibration with a fibre of type II ∗ , II ∗ , I ∗ , III ∗ , I ∗ if i = 1 , , , , N are Ω (1) i,j , for i, j ∈ { , , , } , i < j , Ω (1) k for k = 1 , , , Q and Q andthey span the lattice f E , so N corresponds to the fibration 1 in Table (6.1); thecurves orthogonal to N are Ω (1) i,j , for i, j ∈ { , , , , } , i < j , Ω (1) k for k = 1 , , , Q and they span the lattice D , so N corresponds to the fibration 2 in Table(6.1).We now analyze briefly the fibration induced by | N | . The fiber associated to N is a fiber of type III ∗ . The curves Ω (1)2 and Ω (1)1 are sections of the fibration.The curves Ω (2)2 , , Ω , Ω (2)3 , , Ω (2)3 , Ω (2)4 , , Ω (2)4 , Q , Q are orthogonal to | N | and arethe components of another fiber of type III ∗ . The curves Q , Ω (1)3 , Ω (1)4 , and Ω (1)3 , are orthogonal to N and span a lattice isometric to D . So there is a fiber of type I ∗ in the fibration | N | . Four curves of this fiber are Q , Ω (1)3 , Ω (1)4 , and Ω (1)3 , , thefifth component is another curve, say V . So that we have a special fiber of thefibration | N | , which is 2Ω (1)3 + Ω (1)4 , + Ω (1)3 , + Q + V . Hence we can express theclass of the curve V as N − (2Ω (1)3 + Ω (1)4 , + Ω (1)3 , + Q ). From this expression oneidentifies all the intersection properties of V with all the curves Ω ki,j , Ω nm and Q k .One can also observe that since V is a component of a fiber of the fibration | N | ,it is orthogonal to all the components of the two other reducible fibers of the samefibration, i.e. to the components of the III ∗ -fibers.Let us now consider the class: N := Ω (1)1 , + Ω (1)4 , + V + Ω (1)4 , + 2(Ω (1)0 + Q + Ω (2)0 + Ω (2)1 , + Ω (2)1 + Ω (2)2 , ++ Ω (2)2 + Ω (2)3 , + Ω (2)3 + Ω (2)4 , + Ω (2)4 + Q + Ω (1)2 + Ω (1)3 , + Ω (1)3 ) . It is the class of the elliptic fibration 7 in the Table (6.1) since it is the class of anelliptic fibration on S , with a fiber of type I ∗ .All the classes N i , for i = 1 , , , , , e R : LLIPTIC FIBRATIONS ON COVERS OF THE ELLIPTIC MODULAR SURFACE 25 i ( π ∗ ( N i )) /
21 4 h − ( E + F + G + H ) − (2 E + 3 F + 3 G + 2 H ) − E + 2 F + G + H ) − E T h − (2 E + 2 F + 2 G + 3 H ) − E + 2 F + G + H ) − ( E + 2 F + G + H ) − E h − (2 E + 2 F + 2 G + 3 H ) − ( E + F + G + 2 H ) − E + 2 F + G + H ) − E h − ( E + F + 2 G + H ) − ( E + 2 F + G + H )7 7 h − E + F + 2 G + H ) − ( E + 2 F + G + H )+ − E + 2 F + G + H ) − (4 E + 6 F + 4 G + 5 H )8 4 h − E + − F + G + H ) − (2 E + 3 F + 2 G + 2 H )+ − ( E + 2 F + G + H ) − (2 E + 2 F + 2 G + 3 H )This allows to compute explicitly the equations of pencil of singular rationalcurves in P corresponding to our elliptic fibrations on S , . i pencil in P a ( x − x x + 3 x x x − x x + 6 x x − x x x − x x + x x + x ) + gx x x n ( x − x x + x x + 7 x x x − x x x − x x + 6 x x − x x x )++ n (2 x x − x x + 3 x x + x ) + m (8 x x x − x x )3 f ( − x + x x x + x x − x x ) + n ( x + x x − x x − x x x )4 τ x + σ ( x x − x x )7 τ x x x ( x − x )( x − x )( x − x x + x x ) + σ ( − x x + 2 x x x − x x x ++7 x x x − x x x + 6 x x x − x x x + x − x x − x x x − x x x ++10 x x x + 6 x x − x x + 3 x x x + x x − x x )8 s ( x − x x + x x + 3 x x x − x x x − x x + x x − x x x )++ t ( − x x x + x x x + 2 x x x − x x )Now it remains to find the Weierstrass equations of the elliptic fibrations on S , corresponding to the linear systems N j on e R . In case 4, the curves whichare fibers of the conic bundle have degree 2, so we can directly apply the firstalgorithm in Section 5.2. In cases 1 , , , and 8 the curves are quartics which satisfycondition ( † ) and so we may apply the second algorithm in Section 5.2. The rationalparameterizations and induced Weierstrass equations are given by(6.8) i rational parameterization elliptic fibration1 x = − agp − agp − agpx = − g p − agp − a x = g p − agp + g p − agp A = − a g , B = a g + a g ∆ = − g a ( a − g ) ( a + g ) x = − m p + 8 nmp − nmpx = − m p + 16 nmp − n p − nmp + 2 n p − n x = − m p + 8 nmp + 64 m p − nmp A = 108 m n ( − n + 384 m ) B = − m n ( n − n m + 55296 m )∆ = 570630428688384 n m ( n − m )3 x = p ( − np + fp + n )( − np + fp + fp − n + f ) x = ( p + 1)( − n + f ) p ( − np + fp + n ) x = ( − np + fp + fp − n + f ) A = (3 f − f n + 36 f n − f n − f n + 6 fn − n ) · f − n ) B = (9 f − f n + 117 f n − f n + 39 f n − fn + n ) · n ( f − n ) (3 f − fn + 2 n )∆ = − n − f )(3 n − f ) f (2 n − f ) ( n − f ) A = τ (1 + τ ) , B = 0 , ∆ = 4 τ (1 + τ ) x = ( p − sp + t )( tp − sp − t ) x = sp ( tp − sp − t ) x = ( p − − sp + tp − t )( sp + t ) A = − t s ( s + 4 s t + t ) B = − t s ( s + 2 t )(2 s + 8 s t − t )∆ = 8503056 s t ( s + 4 t ) The fibration induced by | N | is the unique elliptic fibration on a K3 surface witha fiber of type I ∗ (which is a maximal fiber, since if a K3 surface admits an ellipticfibration with this reducible fiber, then this is the unique reducible fiber). So it suffices to know the equation of this elliptic fibration, which is classically known[10, Theorem 1.2]. Hence for | N | we only re-write here the known equation, a partfrom the fact that we chose the parameters over P τ : σ ) in such a way that 4 fibersof type I are over the points ( τ : σ ) = (( ± p − ± i √ / | N | which are tangent to m in a smooth point. Wetherefore have equation:(6.9) A = τ ( − σ − (3 / σ τ − (15 / σ τ − (9 / τ ) ,B = τ σ ( − σ − (3 / τ σ − (21 / σ τ − (35 / τ σ − (63 / τ ) , ∆ = − (729 / τ (2048 σ + 52 τ σ + τ ) . The K3 surface X , and its elliptic fibrations The K3 surface X , is very well-known and studied, in particular since it is themost general K3 surface which admits an elliptic fibration with a 5-torsion sectionand thus its N´eron–Severi group allows to describe the moduli spaces of K3 surfaceswith an elliptic fibration with a 5-torsion section, see [4].7.1. The list of all the elliptic fibrations.
The transcendental lattice of X , is known to be T X ≃ U (5) ⊕ U (5), see e.g. [2]. This allows to apply the Nishiyamamethod in order to classify (at least lattice theoretically) the elliptic fibrations on X , . This method is studied and applied in several papers, and we do not intendto describe it in details. Here we just observe that we can apply the method using A ⊕ A as the lattice T , so that T is a negative-definite lattice with the samediscriminant form as the one of the transcendental lattice T X and whose rank isrank( T ) = rank( T X ) + 4. Then one has to find the primitive embeddings of A ⊕ A in the Niemeier lattice up to isometries, and this can be done by embedding T intothe root lattice of the Niemeier lattices. The list of the Niemeier lattices and thepossible embeddings of root lattices in Niemeier lattices up to the Weil group canbe found in [7]. In particular one has that A embeds primitively in a unique way(up to the action of the Weil group) in A n for n >
4, in D m for m >
4, in E h for h = 6 , ,
8, see [7, Lemmas 4.2 and 4.3]. On the other hand A ⊕ A has a primitiveembedding in A n for n ≥
9, in D m for m ≥
10, and has no primitive embeddings in E h for h = 6 , ,
8, see [7, Lemma 4.5]. All these primitive embeddings are uniquewith the exception of A ⊕ A ֒ → D , for which there are two possible primitiveembeddings, see [7, Page 325, just before Step 3]. The orthogonal of the embeddedcopy of A ⊕ A in the root lattice of each Niemeier lattices is a lattice L which canbe computed by [7, Corollary 4.4] and which encodes information about both thereducible fibers and the rank of the Mordell–Weil group of the elliptic fibrations. Inparticular the root lattice of L is the lattice spanned by the irreducible componentsof the reducible fibers. More precise information on the sections can be obtainedby a deeper analysis of these embeddings, but this is out of our interest.So in the following list we give the root lattice of the Niemeier lattice that weare considering, the embeddings of A ⊕ A in this root lattice, the root lattices ofthe orthogonal, and in the last two columns the properties of the associated ellipticfibration. This gives the complete list of the types of elliptic fibrations on X , :(7.1) LLIPTIC FIBRATIONS ON COVERS OF THE ELLIPTIC MODULAR SURFACE 27 n o Niemeier embedding(s) roots orthogonal singular fibers rk(
M W )1 E A ⊂ E A ⊂ E A ⊕ A ⊕ E I + II ∗ + 2 I E ⊕ D A ⊂ E A ⊂ D A ⊕ D I + I ∗ + 6 I E ⊕ D A ⊕ A ⊂ D E ⊕ D II ∗ + I ∗ + 6 I E ⊕ D A ⊂ E A ⊂ E A ⊕ D I + I ∗ + 6 I E ⊕ D A ⊂ E A ⊂ D E ⊕ A ⊕ D III ∗ + I + I ∗ + 5 I E ⊕ D A ⊕ A ⊂ D E III ∗ + 6 I E ⊕ D A ⊕ A ⊂ D E III ∗ + 6 I E ⊕ A A ⊂ E A ⊂ A A ⊕ A I + I + 8 I E ⊕ A A ⊕ A ⊂ A E ⊕ A III ∗ + I + 7 I D A ⊕ A ⊂ D D I ∗ + 8 I D ⊕ D A ⊂ D A ⊂ D D ⊕ D I ∗ + 6 I D ⊕ D A ⊕ A ⊂ D D ⊕ A I ∗ + 2 I + 6 I D A ⊂ D A ⊂ D D ⊕ A ⊕ A I ∗ + 2 I + 6 I D ⊕ A A ⊂ D A ⊂ A D ⊕ A I ∗ + I + 7 I D ⊕ A A ⊕ A ⊂ A D ⊕ A I ∗ + I + 7 I E A ⊂ E A ⊂ E A ⊕ E I + 2 IV ∗ + 4 I E ⊕ D ⊕ A A ⊂ E A ⊂ D A ⊕ A I + I + 6 I E ⊕ D ⊕ A A ⊂ E A ⊂ A A ⊕ D ⊕ A I + I ∗ + I + 6 I E ⊕ D ⊕ A A ⊂ D A ⊂ A E ⊕ A ⊕ A IV ∗ + 2 I + I + 5 I E ⊕ D ⊕ A A ⊕ A ⊂ A E ⊕ D ⊕ A IV ∗ + I ∗ + I + 5 I D A ⊂ D A ⊂ D D I ∗ + 8 I D ⊕ A A ⊂ D A ⊂ A A ⊕ A I + I + 9 I D ⊕ A A ⊂ A A ⊂ A D ⊕ A ⊕ A I ∗ + 2 I + 6 I D ⊕ A A ⊕ A ⊂ A D ⊕ A I ∗ + I + 6 I D ⊕ A A ⊂ D A ⊂ D A I + 8 I D ⊕ A A ⊂ D A ⊂ A A ⊕ D ⊕ A I + I ∗ + I + 6 I D ⊕ A A ⊂ A A ⊂ A D ⊕ A I ∗ + 2 I + 4 I A A ⊂ A A ⊂ A A ⊕ A I + I + 7 I A A ⊕ A ⊂ A A I + 9 I A A ⊂ A A ⊂ A A I + 8 I A A ⊕ A ⊂ A A ⊕ A I + I + 8 I D ⊕ A A ⊂ A A ⊂ A D ⊕ A I ∗ + 2 I + 6 I A A ⊂ A A ⊂ A A ⊕ A I + 2 I + 6 I A A ⊂ A A ⊂ A A I + 4 I A ⊕ A in D . The K3 surface X , is obtained as double cover of a rational surface R , branched on two smooth fibers, so there are no elliptic fibrations induced bygeneralized conic bundles or by splitting genus 1 pencils, see [3]. So an ellipticfibration on X , is either induced by a conic bundle on R , or it is of type 3.Putting together these considerations with the results of Sections 4 and 5, weproved the following proposition. Proposition 7.1.
The elliptic fibrations on X , are of 34 types, listed in Table7.1. The fibration in line 34 of Table 7.1 is induced by E R and its equation is givenin Section 4.2.2; the fibrations of lines 22, 30 and 32 are induced by conic bundleson R , and their equations are given in Section 5.3.2. The other fibrations on X , are of type 3. Fibration of type 3: an example, the fibration 26.
The aim of thissection is to construct explicitly an example of an elliptic fibration of type 3 and todiscuss the geometry of the non complete linear system on P which induces thisfibration.The class D := Ω (2 , + Ω (1 , + 2 Q + 2Ω (2 , + Ω (2 , + Ω (2 , corresponds to the class of the fiber of a fibration which has one reducible fiber oftype I ∗ . The curves Ω (2 , , Ω (2 , , Q , Q , Ω (2 , , Ω (2 , , Ω (1 , , Ω (1 , are sectionsof the fibration | D | . The other reducible fibers are I + I (assuming that Ω (2 , is the zero section, the non trivial components of the reducible fibers are Ω (2 , ,Ω (2 , , Q , Ω (1 , , Ω (1 , , Ω (1 , , Ω (1 , for the fiber I and Ω (1 , , Ω (1 , for the fiber I ).We have π ( Q i ) = P i and π (Ω ( j, i ) = π (Ω ( j, i ) = Θ ( j ) i , for j = 1 , ι (Ω ( j, i ) = Ω ( j, i , thus, ι ( D ) = D and indeed D := ι ( D ) is D := Ω (2 , + Ω (1 , + 2 Q + 2Ω (2 , + Ω (2 , + Ω (2 , . We observe that Ω (2 , D = 0, Ω (1 , D = 2, Q D = 0, Ω (2 , D = 0, Ω (2 , D =1, Ω (2 , D = 1, and thus D D = 0 + 2 + 0 + 0 + 1 + 1 = 4.So D D = 4 and ( D + D ) = 8. In particular a smooth member of the linearsystem | D + D | is a curve of genus 5.We are interested in the class of the curve π ( D + D ). By the projectionformula π ∗ ( D + D ) = 2 π ( D + D ), so we are looking for π ∗ ( D + D ). Since π ( D ) = π ( D ), π ∗ ( D + D ) = π ∗ ( D ) = π ∗ ( D ).We recall that the map π restricted to Ω ( j, i is a 1 : 1 map to Θ ( j ) i , and similarlythe map π restricted to Ω ( j, i is a 1 : 1 map to Θ ( j ) i . On the other hand the map π restricted to the sections Q i is a 2 : 1 map to the section P i . So π ∗ ( D ) = Θ (2)0 + Θ (1)0 + 4 P + 2Θ (2)0 + Θ (2)1 + Θ (2)4 = 3Θ (2)0 + Θ (2)1 + Θ (2)4 + Θ (1)0 + 4 P . Hence by (2.4) π ∗ ( D ) = 3 E + ℓ + ℓ + m + 4 F = 3 h − E − F − E Q − E − F − E − F , which is the class of the strict transforms of cubics in P passing through Q , Q , Q with tangent ℓ , and Q .The equation of the cubics satisfying these properties is(7.2) ax + bx x + ( − a − b ) x x + dx x + ex x x + f x x + ( − a − d ) x x + ( a − e − f ) x x + ( d + 2 a ) x = 0 . This equation depends on 5 parameters (4 projective parameters) and specializesto equations of cubics which split on the double cover and induces elliptic fibrationson X , .The generic cubic c as in equation (7.2) is a cubic in P . We recall that X , is the double cover of P branched along the reducible sextic c whose equation is f g = 0 for two cubics f and g . So c and c meet in 18 points in P countedwith multiplicity. Recall that c is singular at Q , Q , Q , and Q and in Q , c is the union of two cubics, both with tangent direction ℓ . Hence c intersects c in Q , Q , Q with multiplicity 2, and in Q with multiplicity 4. Outside these LLIPTIC FIBRATIONS ON COVERS OF THE ELLIPTIC MODULAR SURFACE 29 points, c and c intersect in 18 − − − − c on X , is a double cover of c branched in 8 points. Since generically c is asmooth cubic in P , it has genus 1 and then its inverse image on X , has genus g such that 2 g − . So g = 5. On the other hand the inverse image of c on X , is a curve in the linear system | D + D | and indeed has genus 5.We already observed that c is the union of two smooth cubic in P , which are theimage of the branch curves of the double cover X , → R , . Denoted by b : f = 0and b := g = 0 these two cubics, c intersects b in nine points, five of which are Q , Q , Q with tangent ℓ , and Q . So c and b generically intersect in four otherpoints. In the case c splits in the double cover (which is the case in which c is theimage both of D and D ), these four points are either one point with multiplicity4 or two points with multiplicity 2. The strict transform of b (resp. b ) on R , isa smooth fiber of the fibration on R , and its pullback to X , is a smooth fiber,denoted by B (resp. B ) , of the fibration induced on X , by the one on R , .Since the section of this fibration is Q , B Q = B Q = 1, B Ω ( k,j ) i = 0 and thus D B = D B = D B = D B = 2. In particular D and D intersect in fourpoints, two on B and two on B . Considering the image of these curves in P , onerealizes that if c is the image of D , then it intersects b (resp. b ) in two pointseach with multiplicity 2.So, theoretically, in order to find the specializations of a curve c which is theimage of D , one has to require that the intersection c ∩ b consists of the points Q , Q , Q with tangent ℓ , and Q , and of two other points each with multiplicity2. We recall that the equation on b is given by a generic smooth element in thepencil defining R , .As in the previous context one can compute the resultant between the equationof the cubics c and the branch locus of the double cover X , → P , given in(4.7). Since not all the curves in the linear system | c | split in the double cover,the resultant of the equation of c and the equation of the branch locus of X → P is not the square of a polynomial. Nevertheless one recognizes some factors witheven multiplicity (which correspond to the conditions that c passes with a certainmultiplicity through a certain base point of the pencil of cubics from which R , arises) and one can also observe that for certain choices of the values ( a, d, e, f ) theresultant becomes a square. For examples one observes that for f = − a − d − e theresultant with respect to x is x ( x − x ) ( x x ( em + am − a + dm ) + x ( am + bm ) − x ( d − a )) ( x x ( el + al − a + dl ) + x ( al + bl ) − x ( d − a )) so in this case we know that the intersection between the generic member of | c | and the branch curve is always with even degree, which is the necessary conditionto have a splitting in any point. References [1] P. Comparin, A. Garbagnati
Van Geemen–Sarti involutions and elliptic fibrations on K3surfaces double cover of P , Journal of Mathematical Society of Japan (2014) 479-522.[2] A. Garbagnati, A. Sarti, Symplectic automorphisms of prime order on K3 surfaces , J. Algebra (2007), 323–350.[3] A. Garbagnati, C. Salgado,
Linear systems on rational elliptic surfaces and elliptic fibrationson K3 surfaces , arXiv:1703.02783. Not to be confused with the conic bundles B and B in Section 3 [4] A. Garbagnati, Elliptic K3 surfaces with abelian and dihedral groups of symplectic automor-phisms , Communications in Algebra (2013) 583–616.[5] Mi R. Miranda The basic theory of elliptic surfaces ∼ miranda/BTES-Miranda.pdf. R. Miranda[6] V. V. Nikulin, Factor groups of groups of automorphisms of hyperbolic forms with respect tosubgroups generated by 2-reflections , J. Soviet Math., (1983), 1401–1475.[7] K. Nishiyama, The Jacobian fibrations on some K3 surfaces and their Mordell-Weil groups .Japan. J. Math. (N.S.) (1996), 293–347.[8] M. Schuett, T. Shioda, Elliptic surfaces , Algebraic geometry in East Asia - Seoul 2008,Advanced Studies in Pure Mathematics (2010), 51–160.[9] I. Shimada, On elliptic K3 surfaces , Michigan Math. J. (2000), 423–446, arXiv versionwith the complete Table arXiv:math/0505140.[10] T. Shioda The elliptic K3 surfaces with a maximal singular fibre , C. R. Acad. Sci. Paris, Ser.I, 337 (2003), pp. 461–466.[11] E.B. Vinberg,
The two most algebraic K3 surfaces , Math. Ann. 265 (1983), 1–21.[12] D. Q. Zhang,
Quotients of K3 surfaces modulo involutions , Japan J. Math. (N.S.) (1998),335–366.(Francesca Balestrieri)- University of Oxford, Mathematical Institute, Oxford, OX2 6HD, United King-dom - Partially supported by the EPSRC Scholarship EP /L / . E-mail address , Francesca Balestrieri: [email protected] (Julie Desjardins)-
Universit´e Grenoble Alpes, Institut Fourier, CNRS UMR 5582, 100 rue desMaths, BP 74, 38402 St Martin d’H`eres, France
E-mail address , Julie Desjardins: [email protected] (Alice Garbagnati)-
Universit`a Statale degli Studi di Milano Dipartimento di Matematica, via Sal-dini, 50 I20133 Milano - Partially supported by FIRB 2012 “Moduli Spaces and their Applications”
E-mail address , Alice Garbagnati: [email protected] (C´eline Maistret)-
University of Bristol, School of Mathematics, Bristol, UK. ... (Cec´ılia Salgado)-
Universidade Federal do Rio de Janeiro (UFRJ) Instituto de Matem´atica,Cidade Universit´aria, Ilha do Fund˜ao, Rio de Janeiro. - Partially supported by Cnpq grant / − and by Faperj Jovem cien-tista do Nosso estado grant E / / . E-mail address , Cec´ılia Salgado: [email protected] (Isabel Vogt)-
Department of Mathematics, Massachusetts Institute of Technology, 77 Mas-sachusetts Ave, Cambridge, MA 02139, USA - Partially supported by National Science Foundation Graduate Research Fel-lowship Program and grant DMS-1601946.
E-mail address , Isabel Vogt:, Isabel Vogt: