SOORI Atif Hasan，DAOUSSA Daniel

(1. Department of Mathematics，East China Normal University，Shanghai 200241，China；2. Department of Mathematics，Air University，Islamabad 44000，Pakistan)



Article ID：1000-5641(2016)04-0025-05

On classification of isotrivial elliptic Belyi fibrations

SOORI Atif Hasan1，2，DAOUSSA Daniel1

(1. Department of Mathematics，East China Normal University，Shanghai 200241，China；2. Department of Mathematics，Air University，Islamabad 44000，Pakistan)

In this paper we classify relatively minimal，isotrivial families of curves f：S→P1of genus 1 with three singular fibers(Belyi fibrations). Assuming that these families have a section，we find that they are exactly 12 in number up to isomorphism. Moreover，as a result of this classification，we find that except one，the dimension of all other families in M1is zero.

elliptic fibrations；isotrivial；relatively minimal；J-invariant

0　 Introduction

Belyi fibrations are the relatively minimal families of curves f：S→P1with 2 or 3 singular fibers. The term was coined by C. Gong，J. Lu and S. L. Tan[3]as these fibrations have properties similar to Belyi covers[2]. Previously，U. Schmickler Hirzebruch[4]classified all elliptic fibrations f：S→P1with two or three singular fibers. The results are classic and are available in German. S. L. Tan[9]rewrote the proof for two singular fibers. He used the local Chern invariants defined by him in his papers[7-8]and the inequalities on these invariants to prove the result for two singular fibers[5]. The result of Hirzebruch for two singular fibers is given below：

收稿日期：2015-05
第一作者：SOORI Atif Hasan，男，博士研究生，研究方向为代数几何. E-mail：ah.soori@yahoo.com.

Theorem 0.1[4]Let f：S→P1be an elliptic fibration with two singular fibers. Then f is isomorphic to one of the following families：

(I)X =(E×P1)/Zn；

(I*)y2=λ(x3+ x + c)，4 + 27c2/= 0；

(II)y2= x3+λ；

(III)y2= x3+λx；

(IV)y2= x3+λ2.

The types of singular fibers are respectively(nI0，nI0)，(I∗0，I∗0)，(II，II∗)，(III，III∗)and (IV，IV∗)(see Barth et al.[1]for the Kodaira's types of singular fibers).

In this paper，we will prove the classification for three singular fibers by Tan's method. Here we assume that the fibration f：S→P1is relatively minimal，i.e.，no fiber has a(-1)-curve as its component，isotrivial，i.e.，general fibers are isomorphic to each other and that f has a section.

1　 Main result

Theorem 1.1 Let f be as assumed above and the number of singular fibers s = 3. Then f is isomorphic to one of the following families(see Table 1)：

Tab.1 Classification

2　　 Proof

Since we assume here that f is isotrivial，so the semistable model of singular fibers F are smooth. In Kodaira's notation，these singular fibers are I∗0，II，II∗，III，III∗，IV，and IV∗(see Barth et al.[1]for the notations and the dual graphs). One can see that the minimal normal-crossing model F of F can be written as follows：

where Γi's are disjoint H-J branches(see Definition 2.4[5])，and F contains only one principal component C0which is nonsingular curve satisfying C0Γi，red= 1 for all i. Moreover，see Tan[9]for the values of local Chern invariants c2(F)of these singular fibers(see Figure 1).

Fig.1 Local Chern invariants c2(F)of the singular fibers

Suppose F0，F1，and F∞are the three singular fibers of f over 0，1 and∞respectively. By Kodaira's Formula，

12χ(OS)= c2(S)= c2(F1)+ c2(F2)+ c2(F3)≤30.(2.2)

Thus either χ(OS)= 0，1 or 2. When χ(OS)= 0，S is a product. Since we assume that f has a section，so we consider only the cases when χ(OS)= 1 or 2.

Case A χ(OS)= 1：In this case S is rational elliptic and has a Weierstrass normal form as its defining equation：

y2= x3+ a(t)x + b(t)，t∈P1，(2.3)

where a(t)and b(t)are polynomials in an affine variable t on the base P1，dega(t)<4 or degb(t)<6，and 4a(t)3+ 27b(t)2/= 0 for t /= 0，1 and∞.

Since f is isotrivial，the J-invariant，

is constant. This means，either a(t)≡0，or b(t)≡0，or b(t)2≡c·a(t)3for some nonzero constant c∈C.

a(t)≡0(i.e. J = 0)：Here b(t)has the form b(t)= b0tm(t-1)n，m+n<6，where m and n are not both zero. The standard form is

y2= x3- tm(t - 1)n.

Type(II，II∗，nI0). We use here Table IV.3.1 of Miranda[6]to determine the type of the singular fibers. If m + n = 1，i.e.，m = 1，n = 0 or m = 0，n = 1. The two cases are birational by using the linear transformation t■→t - 1. Take m = 1，n = 0. The equation is

y2= x3- t.

At t = 0；ν(△)= 2，ν(b(t))= 1，the above table gives us singular fiber of type II. At t = 1；ν(△)= 0，ν(b(t))= 0，givesnI0. At t =∞；equation becomes y2= x3- s5(replacing t by 1/s，x by x/s2，and y by y/s3)，ν(△)= 10，ν(b(t))= 5，gives reducible singular fiber with 9 components，i.e.，II∗.

Type(IV，IV∗，nI0). If m + n = 2，suppose m = 2，n = 0，then the equation is

y2= x3- t2.

The case when m = 0，n = 2 is birational to the above，because of the transformation t = t - 1. The singular fibers are(IV，IV∗，nI0).

Type(II，II，IV∗). For m = 1，n = 1，i.e.，y2= x3- t(t - 1). The type of singular fibers can be determined by using the Table IV.3.1 of Miranda[6].

Similarly，for m + n = 3，4，5，we get the families 6，7.

b(t)≡0(i.e.，J = 1)：The standard equation is

y2= x3- tm(t - 1)nx，

where m+n<4 and m and n are not both zero. In this case，we get families 3，8，with singular fibers(nI0，III，III∗)and(I∗0，III，III)respectively.

b(t)2≡c·a(t)3：Since the discriminant 4a(t)3+27b(t)2has only zeros at t = 0 and t = 1，so the Weierstrass equation is

y2= x3+ t2m(t - 1)2nx + ct3m(t - 1)3n，

where 2m + 2n<4 and 3m + 3n<6，so we have m + n = 1. The family of curves is

y2= x3+ at2x + bt3，

Case B χ(OS)= 2：In this case，S is by definition an elliptic K3 surface. To prove that the families 9，10，11 and 12 have configurations of singular fibers as stated in the theorem，we first note the following：

Lemma 2.1 With the assumptions in case B，S does not havenI0，II，and III in any of the configuration of its singular fibers. Also there is no configuration of these types：(IV，IV，F)，(I∗0，I∗0，F)，(IV，IV∗，F)and(IV，I∗0，F)，for any singular fiber F.

Proof Since χ(OS)= 2，so c2(S)= 24，by Noether's formula. As c2(S)= c2(F1)+ c2(F2)+ c2(F3)，also c2(F)≤10，for any F satisfying(2.1). As c2(nI0)= 0，so c2(S)<24，a contradiction. So，nI0is not one of F0，F1，F∞，Similarly it can be shown for other configurations.

Back to the Proof of Theorem The Weierstrass equation for an elliptic K3 surface is given by(2.3)with deg(a(t))≤8 and deg(b(t))≤12. Again，as above，f is isotrivial，so J is constant. So，either，a(t)≡0，b(t)≡0 or b(t)2≡c·a(t)3，for some nonzero c∈C.

a(t)≡0：The equation for f will be of the form

y2= x3- tm(t - 1)n，

where m + n≤12，m≤5，n≤5. By the above Lemma 2.1nI0and II are not included in any configuration，thus by using the Table IV. 3.1 of Miranda[6]，m，n /= 0 and m，n /= 1. Also the lemma rules out the existence of the following values of(m，n)：(2，2)，(2，3)，(2，4)，(3，3)，(4，2).

Now，we find the equations for the families of curves for the remaining possible values of (m，n).

Type(II∗，II∗，IV). Suppose m = 2，n = 5，the equation is：

y2= x3- t2(t - 1)5.

At t = 0，ν(b(t))= 2，ν(△)= 4，so from the Miranda's Table IV.3.1，the singular fiber is IV. At t = 1，the singular fiber is II∗. At t =∞，the equation(after replacing y by y/s6，x by x/s4and t by 1/s)is y2= x3- s5(1 - s)5and the singular fiber at s = 0 is II∗. Similarly，for m = 5，n = 5，we get the singular fibers are(II∗，II∗，IV). We can similarly prove the equations for the families 11 and 12.

b(t)≡0(i.e.，J = 1)The equation is：

y2= x3- tm(t - 1)nx，

where m + n≤8，m≤3，n≤3. Since，by Lemma 2.1，nI0，III are not included in the singular fibers of f，so m，n /= 0 and m，n /= 1.

Type(I∗0，III∗，III∗). As(m，n)/=(2，2)，since there is no configuration of singular fibers that contain two I∗0(by Lemma 2.1). Thus the remaining possibilities are(2，3)and(3，3)，the equation of the family for m = 2，n = 3 is

y2= x3- t2(t - 1)3x

at t = 0 and y2= x3- s3(1 - s)3x at t =∞. The singular fibers are(I∗0，III∗，III∗).

b(t)2= c·a(t)3：The equation can be written in the form：

y2= x3- t2m(t - 1)2nx - ct3m(t - 1)3n，

where m + n≤4. By Lemma 2.1，there is no such family.

So，we have all the families of curves over P1with 3 singular fibers，classified.

Corollary 2.2 The elliptic Belyi family with three singular fibers(I∗0，I∗0，nI0)is dense in the compact moduli space M1. The dimension of all other families in M1with three singular fibers is zero.

Proof The proof is clear and follows easily from the fact that the image of P1under J：P1→Mgis a manifold of dimension equal to the number of parameters in the equation of branch locus，i.e.，y2= h(x，t).

Remark 2.3 Note here that the families 1 to 4 are the equations of families of curves over P1with 2 singular fibers，simply because the third fiber here is rational smooth，namely nI0.

Conjecture 2.4 The case when S is a product of curves has still to be determined. We conjecture S =(E×P1)/Zn，for an elliptic curve E and for a suitable group action.

Acknowledgements The authors would like to thank Prof. Lu Jun for his helpful comments and suggestions.

[References]

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[3]GONG C，LU J，TAN S L. On the classification and Mordell-Weil groups of families of curves with two singular fibers[J]. preprint.

[4]SCHMICKLER-HIRZEBRUCH U. Elliptische fl¨achen¨uber P1C mit drei Ausnahmefasern und die hypergeometrische Differentialgleichung[M]. M¨unster：Universit¨at M¨unster，1985.

[5]LU J，TAN S L. Inequalities between the Chern numbers of a singular fiber in a family of algebraic curves[J]. Trans Amer Math Soc，2013，365：3373-3396.

[6]MIRANDA R. The Basic Theory of Elliptic Surfaces[R]. Fort Collins，Colorado：Colorado State Univ，1989.

[7]TAN S L. On the base changes of pencils of curves，I[J]. Manusc Math，1994，84：225-244.

[8]TAN S L. On the base changes of pencils of curves，II[J]. Math Z，1996，222：655-676.

[9]TAN S L. Chern numbers of a singular fiber，modular invariants and isotrivial families of curves[J]. Acta Math Vietnam，2010，35：159-172.

(责任编辑：林磊)

10.3969/j.issn.1000-5641.2016.04.003

论常模Belyi纤维化的分类

SOORI Atif Hasan1，2，DAOUSSA Daniel1

(1.华东师范大学数学系，上海200241；2.爱尔大学数学系，伊斯兰堡44000，巴基斯坦)

本文对P1上带有三条奇异纤维的常模椭圆纤维化(简称Belyi纤维化)进行了分类，给出了精确的12类带有截面的Belyi纤维化.作为这一分类的推论，还发现，除了一种情形外，其余情形对应的M1中的轨迹都是零维的.

椭圆纤维化；常模；相对极小；J-不变量

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