Abstract
Modular invariants of families of curves are Arakelov invariants in arithmetic algebraic geometry. All the known uniform lower bounds of these invariants are not sharp. In this paper, we aim to give explicit lower bounds of modular invariants of families of curves, which is sharp for genus 2. According to the relation between fractional Dehn twists and modular invariants, we give the sharp lower bounds of fractional Dehn twist coefficients and classify pseudo-periodic maps with minimal coefficients for genus 2 and 3 firstly. We also obtain a rigidity property for families with minimal modular invariants, and other applications.
Funding source: National Key Research and Development Program of China
Award Identifier / Grant number: 2018AAA0101001
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 11601504
Award Identifier / Grant number: 11731004
Award Identifier / Grant number: 11761141005
Funding source: Fundamental Research Funds of the Central Universities
Award Identifier / Grant number: DUT18RC(4)065
Funding statement: This work was funded by the National Key Research and Development Program of China (Grant No. 2018AAA0101001), the National Natural Science Foundation of China (Grants No. 11601504, No. 11731004, and No. 11761141005), the Shanghai Science and Technology Commission Foundation (Grants No. 18dz2271000 and No. 20511100200) and Fundamental Research Funds of the Central Universities (No. DUT18RC(4)065).
Acknowledgements
The authors would like to thank Professor Shouwu Zhang for his helpful comments. They are very grateful to Professor Jun Lu for his discussion for a long time. They also thank Professor Tadashi Ashikaga, Professor Kazuhiro Konno, and Professor Yukio Matsumoto for useful comments on pseudo-periodic maps. They would like to thank the referees sincerely for pointing out mistakes and useful detailed suggestions.
References
[1] S. J. Arakelov, Families of algebraic curves with fixed degeneracies, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 1269–1293. 10.1070/IM1971v005n06ABEH001235Search in Google Scholar
[2] T. Ashikaga and M. Ishizaka, Classification of degenerations of curves of genus three via Matsumoto–Montesinos’ theorem, Tohoku Math. J. (2) 54 (2002), no. 2, 195–226. 10.2748/tmj/1113247563Search in Google Scholar
[3] W. Barth, C. Peters and A. Van de Ven, Compact complex surfaces, Ergeb. Math. Grenzgeb. (3) 4, Springer, Berlin 1984. 10.1007/978-3-642-96754-2Search in Google Scholar
[4]
A. Beauville,
Le nombre minimum de fibres singulières d’une courbe stable sur
[5] Z. Cinkir, Zhang’s conjecture and the effective Bogomolov conjecture over function fields, Invent. Math. 183 (2011), no. 3, 517–562. 10.1007/s00222-010-0282-7Search in Google Scholar
[6] M. Cornalba and J. Harris, Divisor classes associated to families of stable varieties, with applications to the moduli space of curves, Ann. Sci. Éc. Norm. Supér. (4) 21 (1988), no. 3, 455–475. 10.24033/asens.1564Search in Google Scholar
[7] G. Faltings, Calculus on arithmetic surfaces, Ann. of Math. (2) 119 (1984), no. 2, 387–424. 10.2307/2007043Search in Google Scholar
[8] D. Gabai and U. Oertel, Essential laminations in 3-manifolds, Ann. of Math. (2) 130 (1989), no. 1, 41–73. 10.2307/1971476Search in Google Scholar
[9]
C. Gong, J. Lu and S.-L. Tan,
On families of complex curves over
[10] M. Hedden and T. E. Mark, Floer homology and fractional Dehn twists, Adv. Math. 324 (2018), 1–39. 10.1016/j.aim.2017.11.008Search in Google Scholar
[11] K. Honda, W. H. Kazez and G. Matić, Right-veering diffeomorphisms of compact surfaces with boundary, Invent. Math. 169 (2007), no. 2, 427–449. 10.1007/s00222-007-0051-4Search in Google Scholar
[12] Y. Imayoshi, A construction of holomorphic families of Riemann surfaces over the punctured disk with given monodromy, Handbook of Teichmüller theory. Vol. II, IRMA Lect. Math. Theor. Phys. 13, European Mathematical Society, Zürich (2009), 93–130. 10.4171/055-1/4Search in Google Scholar
[13] M. Ishizaka, Monodromies of hyperelliptic families of genus three curves, Tohoku Math. J. (2) 56 (2004), no. 1, 1–26. 10.2748/tmj/1113246379Search in Google Scholar
[14] T. Ito and K. Kawamuro, Essential open book foliations and fractional Dehn twist coefficient, Geom. Dedicata 187 (2017), 17–67. 10.1007/s10711-016-0188-7Search in Google Scholar
[15] A. Javanpeykar, Polynomial bounds for Arakelov invariants of Belyi curves, Algebra Number Theory 8 (2014), no. 1, 89–140. 10.2140/ant.2014.8.89Search in Google Scholar
[16] W. H. Kazez and R. Roberts, Fractional Dehn twists in knot theory and contact topology, Algebr. Geom. Topol. 13 (2013), no. 6, 3603–3637. 10.2140/agt.2013.13.3603Search in Google Scholar
[17] X. Liu, Modular invariants and singularity indices of hyperelliptic fibrations, Chinese Ann. Math. Ser. B 37 (2016), no. 6, 875–890. 10.1007/s11401-016-1045-6Search in Google Scholar
[18] X. Liu and S. Tan, Families of hyperelliptic curves with maximal slopes, Sci. China Math. 56 (2013), no. 9, 1743–1750. 10.1007/s11425-013-4634-9Search in Google Scholar
[19] X.-L. Liu, Fractional Dehn twists and modular invariants, Sci. China Math. 64 (2021), no. 8, 1735–1744. 10.1007/s11425-019-1716-4Search in Google Scholar
[20] X.-L. Liu and S.-L. Tan, Uniform bound for the effective Bogomolov conjecture, C. R. Math. Acad. Sci. Paris 355 (2017), no. 2, 205–210. 10.1016/j.crma.2017.01.003Search in Google Scholar
[21] Y. Liu, A characterization of virtually embedded subsurfaces in 3-manifolds, Trans. Amer. Math. Soc. 369 (2017), no. 2, 1237–1264. 10.1090/tran/6707Search in Google Scholar
[22] Y. Matsumoto and J. M. Montesinos-Amilibia, Pseudo-periodic maps and degeneration of Riemann surfaces, Lecture Notes in Math. 2030, Springer, Heidelberg 2011. 10.1007/978-3-642-22534-5Search in Google Scholar
[23] A. Moriwaki, Relative Bogomolov’s inequality and the cone of positive divisors on the moduli space of stable curves, J. Amer. Math. Soc. 11 (1998), no. 3, 569–600. 10.1090/S0894-0347-98-00261-6Search in Google Scholar
[24] D. Mumford, Towards an enumerative geometry of the moduli space of curves, Arithmetic and geometry, Vol. II, Progr. Math. 36, Birkhäuser, Boston (1983), 271–328. 10.1007/978-1-4757-9286-7_12Search in Google Scholar
[25] Y. Namikawa and K. Ueno, The complete classification of fibres in pencils of curves of genus two, Manuscripta Math. 9 (1973), 143–186. 10.1007/BF01297652Search in Google Scholar
[26] J. Nielsen, Surface transformation classes of algebraically finite type, Danske Vid. Selsk. Mat.-Fys. Medd. 21 (1944), no. 2, 89. Search in Google Scholar
[27] A. N. Paršin, Algebraic curves over function fields. I, Math. SSSR Izv. 2 (1968), 1145–1170. 10.1070/IM1968v002n05ABEH000723Search in Google Scholar
[28] S. Takamura, Towards the classification of atoms of degenerations. II: Cyclic quotient construction of degenerations of complex curves, Preprint Ser. No. 1334, Research Institute for Mathematical Sciences Kyoto University, (2001). Search in Google Scholar
[29] S. L. Tan, On the invariants of base changes of pencils of curves. I, Manuscripta Math. 84 (1994), no. 3–4, 225–244. 10.1007/BF02567455Search in Google Scholar
[30] S.-L. Tan, On the invariants of base changes of pencils of curves. II, Math. Z. 222 (1996), no. 4, 655–676. 10.1007/BF02621887Search in Google Scholar
[31] S.-L. Tan, Chern numbers of a singular fiber, modular invariants and isotrivial families of curves, Acta Math. Vietnam. 35 (2010), no. 1, 159–172. Search in Google Scholar
[32] S.-L. Tan, Poincaré–Painlevé problem and Chern numbers of a holomorphic foliation, preprint. Search in Google Scholar
[33]
S.-L. Tan, Y. Tu and A. G. Zamora,
On complex surfaces with 5 or 6 semistable singular fibers over
[34] R. Wilms, New explicit formulas for Faltings’ delta-invariant, Invent. Math. 209 (2017), no. 2, 481–539. 10.1007/s00222-016-0713-1Search in Google Scholar
[35] G. Xiao, Fibered algebraic surfaces with low slope, Math. Ann. 276 (1987), no. 3, 449–466. 10.1007/BF01450841Search in Google Scholar
[36] G. Xiao, On the stable reduction of pencils of curves, Math. Z. 203 (1990), no. 3, 379–389. 10.1007/BF02570745Search in Google Scholar
[37] G. Xiao, The fibrations of algbraic surfaces (in Chinese), Shanghai Scientific & Technical, Shanghai 1992. Search in Google Scholar
[38] S. Zhang, Heights and reductions of semi-stable varieties, Compos. Math. 104 (1996), no. 1, 77–105. Search in Google Scholar
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Articles in the same Issue
- Frontmatter
- Artin–Mazur heights and Yobuko heights of proper log smooth schemes of Cartier type, and Hodge–Witt decompositions and Chow groups of quasi-F-split threefolds
- Kazhdan–Lusztig conjecture via zastava spaces
- Monodromic model for Khovanov–Rozansky homology
- Hyperbolic secant varieties of M-curves
- On the sharp lower bounds of modular invariants and fractional Dehn twist coefficients
- Degeneration of curves on some polarized toric surfaces
- Noether–Severi inequality and equality for irregular threefolds of general type
- Erratum to Table of Contents (J. reine angew. Math. 785 (2022), i–iv)
Articles in the same Issue
- Frontmatter
- Artin–Mazur heights and Yobuko heights of proper log smooth schemes of Cartier type, and Hodge–Witt decompositions and Chow groups of quasi-F-split threefolds
- Kazhdan–Lusztig conjecture via zastava spaces
- Monodromic model for Khovanov–Rozansky homology
- Hyperbolic secant varieties of M-curves
- On the sharp lower bounds of modular invariants and fractional Dehn twist coefficients
- Degeneration of curves on some polarized toric surfaces
- Noether–Severi inequality and equality for irregular threefolds of general type
- Erratum to Table of Contents (J. reine angew. Math. 785 (2022), i–iv)