Fibered varieties over curves with low slope and sharp bounds in dimension three

Hu, Yong; Zhang, Tong

doi:10.1090/jag/739

Authors: Yong Hu and Tong Zhang
Journal: J. Algebraic Geom. 30 (2021), 57-95
DOI: https://doi.org/10.1090/jag/739
Published electronically: December 4, 2019
MathSciNet review: 4233178
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Abstract | References | Additional Information

Abstract:

In this paper, we first construct varieties of any dimension $n>2$ fibered over curves with low slopes. These examples violate the conjectural slope inequality of Barja and Stoppino [Springer Proc. Math. Stat. 71 (2014), pp. 1–40].

Led by their conjecture, we focus on finding the lowest possible slope when $n=3$. Based on a characteristic $p > 0$ method, we prove that the sharp lower bound of the slope of fibered $3$-folds over curves is $4/3$, and it occurs only when the general fiber is a $(1, 2)$-surface. Otherwise, the sharp lower bound is $2$. We also obtain a Cornalba-Harris-Xiao-type slope inequality for families of surfaces of general type over curves, and it is sharper than all known results with no extra assumptions.

As an application of the slope bound, we deduce a sharp Noether-Severi-type inequality that $K_X^3 \ge 2\chi (X, \omega _X)$ for an irregular minimal $3$-fold $X$ of general type not having a $(1,2)$-surface Albanese fibration. It answers a question in [Canad. J. Math. 67 (2015), pp. 696–720] and thus completes the full Severi-type inequality for irregular $3$-folds of general type.

References

Miguel A. Barja, Lower bounds of the slope of fibred threefolds, Internat. J. Math. 11 (2000), no. 4, 461–491. MR 1768169, DOI https://doi.org/10.1142/S0129167X00000234
Miguel A. Barja, Generalized Clifford-Severi inequality and the volume of irregular varieties, Duke Math. J. 164 (2015), no. 3, 541–568. MR 3314480, DOI https://doi.org/10.1215/00127094-2871306

Miguel A. Barja, Rita Pardini, and Lidia Stoppino, Linear systems on irregular varieties, arXiv:1606.03290, 2016.

Miguel A. Barja and Lidia Stoppino, Positivity properties of relative complete intersections, arXiv:1410.3009, 2014.

M. A. Barja and L. Stoppino, Stability conditions and positivity of invariants of fibrations, Algebraic and complex geometry, Springer Proc. Math. Stat., vol. 71, Springer, Cham, 2014, pp. 1–40. MR 3278568, DOI https://doi.org/10.1007/978-3-319-05404-9_1

Miguel Ángel Barja and Lidia Stoppino, Stability and singularities of relative hypersurfaces, Int. Math. Res. Not. IMRN 4 (2016), 1026–1053. MR 3493441, DOI https://doi.org/10.1093/imrn/rnv158

E. Bombieri, Canonical models of surfaces of general type, Inst. Hautes Études Sci. Publ. Math. 42 (1973), 171–219. MR 318163

Jean-Benoît Bost, Semi-stability and heights of cycles, Invent. Math. 118 (1994), no. 2, 223–253. MR 1292112, DOI https://doi.org/10.1007/BF01231533

F. Catanese, The moduli and the global period mapping of surfaces with $K^{2}=p_{g}=1$: a counterexample to the global Torelli problem, Compositio Math. 41 (1980), no. 3, 401–414. MR 589089

Jungkai A. Chen and Meng Chen, The Noether inequality for Gorenstein minimal 3-folds, Comm. Anal. Geom. 23 (2015), no. 1, 1–9. MR 3291363, DOI https://doi.org/10.4310/CAG.2015.v23.n1.a1

Meng Chen, Inequalities of Noether type for 3-folds of general type, J. Math. Soc. Japan 56 (2004), no. 4, 1131–1155. MR 2092941, DOI https://doi.org/10.2969/jmsj/1190905452

Meng Chen and Christopher D. Hacon, On the geography of Gorenstein minimal 3-folds of general type, Asian J. Math. 10 (2006), no. 4, 757–763. MR 2282363, DOI https://doi.org/10.4310/AJM.2006.v10.n4.a8

Maurizio Cornalba and Joe Harris, Divisor classes associated to families of stable varieties, with applications to the moduli space of curves, Ann. Sci. École Norm. Sup. (4) 21 (1988), no. 3, 455–475. MR 974412

Steven Dale Cutkosky, Resolution of singularities for 3-folds in positive characteristic, Amer. J. Math. 131 (2009), no. 1, 59–127. MR 2488485, DOI https://doi.org/10.1353/ajm.0.0036

O. Debarre, Inégalités numériques pour les surfaces de type général, Bull. Soc. Math. France 110 (1982), no. 3, 319–346 (French, with English summary). With an appendix by A. Beauville. MR 688038

Yong Hu, Inequality for Gorenstein minimal 3-folds of general type, Comm. Anal. Geom. 26 (2018), no. 2, 347–359. MR 3805162, DOI https://doi.org/10.4310/CAG.2018.v26.n2.a3

Masanori Kobayashi, On Noether’s inequality for threefolds, J. Math. Soc. Japan 44 (1992), no. 1, 145–156. MR 1139663, DOI https://doi.org/10.2969/jmsj/04410145

János Kollár, Subadditivity of the Kodaira dimension: fibers of general type, Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, North-Holland, Amsterdam, 1987, pp. 361–398. MR 946244, DOI https://doi.org/10.2969/aspm/01010361

János Kollár and Shigefumi Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998. With the collaboration of C. H. Clemens and A. Corti; Translated from the 1998 Japanese original. MR 1658959

Kazuhiro Konno, A lower bound of the slope of trigonal fibrations, Internat. J. Math. 7 (1996), no. 1, 19–27. MR 1369903, DOI https://doi.org/10.1142/S0129167X96000037

Adrian Langer, Generic positivity and foliations in positive characteristic, Adv. Math. 277 (2015), 1–23. MR 3336081, DOI https://doi.org/10.1016/j.aim.2015.02.015

Jun Lu, Mao Sheng, and Kang Zuo, An Arakelov inequality in characteristic $p$ and upper bound of $p$-rank zero locus, J. Number Theory 129 (2009), no. 12, 3029–3045. MR 2560851, DOI https://doi.org/10.1016/j.jnt.2009.05.015

Jun Lu, Sheng-Li Tan, and Kang Zuo, Canonical class inequality for fibred spaces, Math. Ann. 368 (2017), no. 3-4, 1311–1332. MR 3673655, DOI https://doi.org/10.1007/s00208-016-1474-2

Margarida Mendes Lopes and Rita Pardini, The geography of irregular surfaces, Current developments in algebraic geometry, Math. Sci. Res. Inst. Publ., vol. 59, Cambridge Univ. Press, Cambridge, 2012, pp. 349–378. MR 2931875

Atsushi Moriwaki, Bogomolov conjecture over function fields for stable curves with only irreducible fibers, Compositio Math. 105 (1997), no. 2, 125–140. MR 1440719, DOI https://doi.org/10.1023/A%3A1000139117766

Koji Ohno, Some inequalities for minimal fibrations of surfaces of general type over curves, J. Math. Soc. Japan 44 (1992), no. 4, 643–666. MR 1180441, DOI https://doi.org/10.2969/jmsj/04440643

Zsolt Patakfalvi, Semi-positivity in positive characteristics, Ann. Sci. Éc. Norm. Supér. (4) 47 (2014), no. 5, 991–1025 (English, with English and French summaries). MR 3294622, DOI https://doi.org/10.24033/asens.2232

Miles Reid, Young person’s guide to canonical singularities, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 345–414. MR 927963

Lidia Stoppino, Slope inequalities for fibred surfaces via GIT, Osaka J. Math. 45 (2008), no. 4, 1027–1041. MR 2493968

Eckart Viehweg, Positivity of direct image sheaves and applications to families of higher dimensional manifolds, School on Vanishing Theorems and Effective Results in Algebraic Geometry (Trieste, 2000) ICTP Lect. Notes, vol. 6, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2001, pp. 249–284. MR 1919460

Eckart Viehweg and Kang Zuo, On the isotriviality of families of projective manifolds over curves, J. Algebraic Geom. 10 (2001), no. 4, 781–799. MR 1838979

Gang Xiao, Fibered algebraic surfaces with low slope, Math. Ann. 276 (1987), no. 3, 449–466. MR 875340, DOI https://doi.org/10.1007/BF01450841

Xinyi Yuan and Tong Zhang, Relative Noether inequality on fibered surfaces, Adv. Math. 259 (2014), 89–115. MR 3197653, DOI https://doi.org/10.1016/j.aim.2014.03.018

Oscar Zariski, Introduction to the problem of minimal models in the theory of algebraic surfaces, Publications of the Mathematical Society of Japan, no. 4, The Mathematical Society of Japan, Tokyo, 1958. MR 0097403

Tong Zhang, Severi inequality for varieties of maximal Albanese dimension, Math. Ann. 359 (2014), no. 3-4, 1097–1114. MR 3231026, DOI https://doi.org/10.1007/s00208-014-1025-7

Tong Zhang, Geography of irregular Gorenstein 3-folds, Canad. J. Math. 67 (2015), no. 3, 696–720. MR 3339537, DOI https://doi.org/10.4153/CJM-2014-033-0

Tong Zhang, Irregular varieties of Albanese fiber dimension one with small volume, Math. Res. Lett. 23 (2016), no. 6, 1833–1856. MR 3621109, DOI https://doi.org/10.4310/MRL.2016.v23.n6.a12

Additional Information

Yong Hu
Affiliation: School of Mathematics, Korea Institute for Advanced Study, 85 Hoegiro, Dongdaemun-gu, Seoul 02455, South Korea
MR Author ID: 1227227
Email: yonghu11@kias.re.kr

Tong Zhang
Affiliation: School of Mathematical Sciences, Shanghai Key Laboratory of PMMP, East China Normal University, 500 Dongchuan Road, Shanghai 200241, People’s Republic of China
MR Author ID: 1017747
Email: tzhang@math.ecnu.edu.cn, mathtzhang@gmail.com

Received by editor(s): April 18, 2018
Received by editor(s) in revised form: January 7, 2019
Published electronically: December 4, 2019
Additional Notes: The first author was supported by the National Researcher Program of the National Research Foundation of Korea (Grant No. 2010-0020413). The second author was supported by the Science and Technology Commission of Shanghai Municipality (STCSM) Grant No. 18dz2271000 and the Leverhulme Trust Research Project Grant ECF-2016-269 when he started working on this problem at Durham University.
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