×

Singular fibers and Kodaira dimensions. (English) Zbl 1430.14024

A non-isotrivial family \(f:S \to \mathbb P^1\) of curves of genus \(g\) has a certain number \(s\) of singular fibers. What is the smallest possible \(s\)? A. Beauville [Astérisque 86, 97–108 (1981; Zbl 0502.14009)] proved that \(f\) admits at least 3 singular fibers, and if the fibration is semi-stable, then \(s\geq 4\). Both results are sharp. Later on S.-L. Tan et al. [Math. Z. 249, No. 2, 427–438 (2005; Zbl 1074.14009)] proved: if \(g\geq 2\) and \(S\) has non negative Kodaira dimension, then \(s\geq 6\). They also provided 2 examples: A) \(S\) is a \(K3\) surface, \(g=3\) and \(s=6\); B) \(S\) is of general type, \(g=4\) and \(s=7\).
The main result of this paper is the following.
Theorem 1. Let \(f:S \to \mathbb P^1\) be a non-isotrivial semi-stable fibration of curves of genus \(g\geq 2\). If \(S\) is of general type, then \(f\) has at least 7 singular fibers.
The proof relies on an analysis of the semi-stable fibrations coming from Teichmüller curves. In particular the author use a result of M. Möller [J. Am. Math. Soc. 19, No. 2, 327–344 (2006; Zbl 1090.32004)] to obtain the following.
Corollary 1. Let \(f:S \to \mathbb P^1\) be a semi-stable fibration of curves of genus \(g\geq 2\) and \(s=6\). If \(p_g(S)>0\), then \(f\) comes form a Teichmüller curve and \(\omega^2_{S/\mathbb{P}^1}\leq 6g-6\).
As other applications of this corollary they obtain the following theorems.
Theorem 2. Let \(f:S \to \mathbb P^1\) be a non-isotrivial semi-stable fibration of curves of genus \(g\geq 2\). If \(S\) has Kodaira dimension \(\kappa(S)=0\) and \(s=6\), then \(\omega^2_{S/\mathbb{P}^1}= 6g-6\) and the family is Teichmüller.
Theorem 3. Let \(f:S \to \mathbb P^1\) be a non-isotrivial semi-stable fibration of curves of genus \(g\geq 2\). If \(S\) has Kodaira dimension \(\kappa(S)=1\) and \(s=6\), then \(S\) is simply connected, \(p_g(S)=q(S)=0\) and the canonical elliptic fibration has exaclty 2 multiple fibers: a double fiber and one of multiplicity 3 or 5.
It is not clear if this case does occur.
In the last section the authors use the Arakelov type inequality in [J. Lu et al., Math. Ann. 368, No. 3–4, 1311–1332 (2017; Zbl 1401.14054)] to derive similar results for fibrations \(f:X\to \mathbb{P}^1\) in any dimension.
Theorem 4. Let \(f:X\to \mathbb{P}^1\) be a non-isotrivial semi-stable family of varieties of dimension \(m\) over \(\mathbb P^1\) with \(s\) singular fibers. Assume that the smooth fibers \(F\) are minimal, i.e., their canonical line bundles are semiample. Then \(\kappa(X)\leq \kappa(F)+1\). Moreover, if \(\kappa(X)\geq 0\), then \(s\geq \dfrac 4m +2\), and if \(\kappa(X)= \kappa(F)+1\), then \(s> \dfrac 4m +2\).

MSC:

14D06 Fibrations, degenerations in algebraic geometry
14H10 Families, moduli of curves (algebraic)
14J29 Surfaces of general type
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Beauville, A.: Le nombre minimum de fibres singulieres d’une courbe stable sur \[{ P}^1\] P1. Astérisque 86, 97-108 (1981). (French) · Zbl 0502.14009
[2] Beauville, A.: Les familles stables de courbes elliptiques sur \[{\bf P}^1\] P1 admettant 4 fibres singulières. C. R. Acad. Sci. Paris 294, 657-660 (1982) · Zbl 0504.14016
[3] Eskin, A., Kontsevich, M., Zorich, A.: Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow. Publ. Math. Inst. Hautes Études Sci. 120, 207-333 (2014) · Zbl 1305.32007 · doi:10.1007/s10240-013-0060-3
[4] Fujita, T.: On Kähler fiber spaces over curves. J. Math. Soc. Jpn. 30(4), 779-794 (1978) · Zbl 0393.14006 · doi:10.2969/jmsj/03040779
[5] Griffiths, P., Harris, J.: Principles of Algebraic Geometry. Wiley Classics Library. Wiley, New York (1994). Reprint of the (1978) original · Zbl 0408.14001
[6] Gong, C., Lu, X., Tan, S.-L.: Families of curves over \[{\mathbb{P}}^1\] P1 with 3 singular fibers. C. R. Math. Acad. Sci. Paris 351(9-10), 375-380 (2013) · Zbl 1282.14063 · doi:10.1016/j.crma.2013.05.002
[7] Kovács, S.J.: On the minimal number of singular fibres in a family of surfaces of general type. J. Reine Angew. Math. 487, 171-177 (1997) · Zbl 0905.14022
[8] Lu, J., Tan, S.-L., Zuo, K.: Canonical class inequality for fibred spaces. Math. Ann. (2016). doi:10.1007/s00208-016-1474-2 · Zbl 1401.14054
[9] Lu, X., Tan, S.-L., Xu, W.-Y., Zuo, K.: On the minimal number of singular fibers with non-compact Jacobians for families of curves over \[{\mathbb{P}}^1\] P1. J. Math. Pures Appl. 105(5), 724-733 (2016) · Zbl 1368.14018 · doi:10.1016/j.matpur.2015.11.011
[10] Huitrado-Mora, A., Castaneda-Salazar, M., Zamora, A. G.: Toward a conjecture of Tan and Tu on fibered general type surfaces. arXiv:1604.00050 (2016) · Zbl 1368.14018
[11] Moishezon, B.: Complex Surfaces and Connected Sums of Complex Projective Planes. With an Appendix by R. Livne. Lecture Notes in Mathematics, vol. 603. Springer, Berlin (1977) · Zbl 0392.32015
[12] Möller, M.: Variations of Hodge structures of a Teichmüller curve. J. Am. Math. Soc. 19(2), 327-344 (2006) · Zbl 1090.32004 · doi:10.1090/S0894-0347-05-00512-6
[13] Möller, M.: Teichmüller curves, mainly from the viewpoint of algebraic geometry. In: Moduli Spaces of Riemann Surfaces. IAS/Park City Mathematics Series, vol. 20. American Mathematical Society, Providence, pp. 267-318 (2013) · Zbl 1279.14031
[14] Sun, X., Tan, S.-L., Zuo, K.: Families of K3 surfaces over curves reaching the Arakelov-Yau type upper bounds and modularity. Math. Res. Lett. 10(2-3), 323-342 (2003) · Zbl 1100.14004 · doi:10.4310/MRL.2003.v10.n3.a4
[15] Tan, S.-L.: The minimal number of singular fibers of a semistable curve over \[{\bf P}^1\] P1. J. Algebraic Geom. 4(3), 591-596 (1995) · Zbl 0864.14003
[16] Tan, S.-L., Tu, Y., Yu, F.: On semistable families of curves over \[\mathbb{P}^1\] P1 with a small number of singular curves (2009, preprint) · Zbl 0502.14009
[17] Tan, S.-L., Tu, Y., Zamora, A.G.: On complex surfaces with 5 or 6 semistable singular fibers over \[{\mathbb{P}}^1\] P1. Math. Z. 249(2), 427-438 (2005) · Zbl 1074.14009 · doi:10.1007/s00209-004-0706-4
[18] Tu, Y.: Surfaces of Kodaira dimension zero with six semistable singular fibers over \[\mathbb{P}^1\] P1. Math. Z. 257(1), 1-5 (2007) · Zbl 1126.14015 · doi:10.1007/s00209-006-0090-3
[19] Viehweg, E., Zuo, K.: On the isotriviality of families of projective manifolds over curves. J. Algebraic Geom. 10(4), 781-799 (2001) · Zbl 1079.14503
[20] Viehweg, E., Zuo, K.: Families over curves with a strictly maximal Higgs field. Asian J. Math. 7(4), 575-598 (2003) · Zbl 1084.14521 · doi:10.4310/AJM.2003.v7.n4.a8
[21] Viehweg, E., Zuo, K.: A characterization of certain Shimura curves in the moduli stack of abelian varieties. J. Differ. Geom. 66(2), 233-287 (2004) · Zbl 1078.11043 · doi:10.4310/jdg/1102538611
[22] Viehweg, E., Zuo, K.: Numerical bounds for semi-stable families of curves or of certain higher-dimensional manifolds. J. Algebraic Geom. 15(4), 771-791 (2006) · Zbl 1200.14065 · doi:10.1090/S1056-3911-05-00423-6
[23] Zamora, A.G.: Semistable genus 5 general type \[\mathbb{P}^1\] P1-curves have at least 7 singular fibres. Note Mat. 32(2), 1-4 (2012) · Zbl 1281.14009
[24] Zuo, K.: Yau’s form of Schwarz lemma and Arakelov inequality on moduli spaces of projective manifolds. In: Handbook of Geometric Analysis. No. 1. Advanced Lectures in Mathematics (ALM), vol. 7, pp. 659-676. International Press, Somerville, MA (2008) · Zbl 1168.14009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.