Families of curves over $ {\mathbb{P}}^{1}$ with 3 singular fibers

Gong, Cheng; Lu, Xin; Tan, Sheng-Li

doi:10.1016/j.crma.2013.05.002

Algebraic Geometry

Families of curves over

P^{1}

with 3 singular fibers
[Familles de courbes sur

P^{1}

avec trois fibres singulières]

Présenté par : Claire Voisin

Gong, Cheng ¹ ; Lu, Xin ¹ ; Tan, Sheng-Li ¹

¹ Department of Mathematics, East China Normal University, Dongchuan RD 500, Shanghai 200421, PR China

Comptes Rendus. Mathématique, Tome 351 (2013) no. 9-10, pp. 375-380.

Résumé
Abstract

Soit $f : S \to P^{1}$ une fibration de genre g avec trois fibres singulières, dont deux dʼentre elles sont semi-stables. Nous montrons que le groupe de Mordell–Weil de f est fini, que la surface S est rationnelle et que $2 g ⩽ - K_{S}^{2} ⩽ 4 g - 4$ . Nous construisons également des exemples montrant quʼil existe de telles fibrations pour une infinité de g.

Suppose $f : S \to P^{1}$ is a fibration of genus g with 3 singular fibers and two of them are semistable. We show that the Mordell–Weil group of f is finite, the surface S is rational and $2 g ⩽ - K_{S}^{2} ⩽ 4 g - 4$ . We construct some examples to show that such fibrations exist for infinitely many g.

Reçu le : 2012-12-04
Accepté le : 2013-05-15
Publié le : 2013-05-01

DOI : 10.1016/j.crma.2013.05.002

Affiliations des auteurs :

Gong, Cheng ¹ ; Lu, Xin ¹ ; Tan, Sheng-Li ¹

¹ Department of Mathematics, East China Normal University, Dongchuan RD 500, Shanghai 200421, PR China

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     title = {Families of curves over $ {\mathbb{P}}^{1}$ with 3 singular fibers},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {375--380},
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Gong, Cheng; Lu, Xin; Tan, Sheng-Li. Families of curves over $ {\mathbb{P}}^{1}$ with 3 singular fibers. Comptes Rendus. Mathématique, Tome 351 (2013) no. 9-10, pp. 375-380. doi : 10.1016/j.crma.2013.05.002. http://www.numdam.org/articles/10.1016/j.crma.2013.05.002/

Bibliographie
Cité par

[1] Beauville, A. Le nombre minimum de fibres singulières dʼun courbe stable sur $P^{1}$ , Séminaire sur les pinceaux de courbes de genre au moins deux (Szpiro, L., ed.) (Astérisque), Volume 86 (1981), pp. 97-108

[2] Beauville, A. Les familles stables de courbes elliptiques sur $P^{1}$ admettant quatre fibres singuliéres, C. R. Acad. Sci. Paris, Ser. I, Volume 294 (1982) no. 19, pp. 657-660

[3] Belyi, G. On Galois extensions of a maximal cyclotomic field, Math. USSR Izv., Volume 14 (1980) no. 2, pp. 247-256

[4] Kukulies, S. On Shimura curves in the Schottky locus, J. Algebraic Geom., Volume 19 (2010), pp. 371-397

[5] Liu, K. Geometric height inequalities, Math. Res. Lett., Volume 3 (1996) no. 5, pp. 693-702

[6] Liu, X.-L.; Tan, S.-L. Families of hyperelliptic curves with maximal slopes, Sci. China Math., Volume 56 (2013) | DOI

[7] Lu, J.; Tan, S.-L. Inequalities between the Chern numbers of a singular fiber in a family of algebraic curves, Trans. Amer. Math. Soc., Volume 365 (2013), pp. 3373-3396

[8] Lu, J.; Tan, S.-L.; Yu, F.; Zuo, K. A new inequality on the Hodge number $h^{1, 1}$ of algebraic surfaces | arXiv

[9] Nguyen, K.V. On Beauvilleʼs conjecture and related topics, J. Math. Kyoto Univ., Volume 35 (1995) no. 2, pp. 275-298

[10] Nguyen, K.V. On family of curves over $P^{1}$ with small number of singular fibers, C. R. Acad. Sci. Paris, Ser. I, Volume 326 (1998) no. 4, pp. 459-463

[11] Oort, F. Some questions in algebraic geometry, June 1995 http://www.math.uu.nl/people/oort/ (Manuscript)

[12] Schmickler Hirzebruch, U. Elliptische flächenüber $P^{1} (C)$ mit drei Ausnahmefasern und die hypergeometrische Differentialgleichung, Universität Münster, Mathematisches Institut, Münster, West Germany, 1985

[13] Shioda, T. Mordell–Weil lattices for higher genus fibration over a curve, New Trends in Algebraic Geometry, Cambridge Univ. Press, 1999, pp. 359-373

[14] Tan, S.-L. On the invariants of base changes of pencils of curves, I, Manuscr. Math., Volume 84 (1994), pp. 225-244

[15] Tan, S.-L. The minimal number of singular fibers of a semistable curve over $P^{1}$ , J. Algebraic Geom., Volume 4 (1995), pp. 591-596

[16] Tan, S.-L. On the invariants of base changes of pencils of curves, II, Math. Z., Volume 222 (1996), pp. 655-676

[17] Tan, S.-L. Chern numbers of a singular fiber, modular invariants and isotrivial families of curves, Acta Math. Vietnam, Volume 35 (2010) no. 1, pp. 159-172

[18] Tan, S.-L.; Tu, Y.-P.; Zamora, A.G. On complex surfaces with 5 or 6 semistable singular fibers over $P^{1}$ , Math. Z., Volume 249 (2005), pp. 427-438

[19] Viehweg, E.; Zuo, K. Arakelov inequalities and the uniformization of certain rigid Shimura varieties, J. Differ. Geom., Volume 77 (2007) no. 2, pp. 291-352

[20] Xiao, G. On the stable reduction of pencils of curves, Math. Z., Volume 203 (1990), pp. 379-389

Cité par Sources :

^☆ This work is supported by NSFC. The second author is partially supported by ECNU Reward for Excellent Doctors in Academics.