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Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



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

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


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.

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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

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

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.
Article copyright: © Copyright 2019 University Press, Inc.