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Lower bounds on the arithmetic self-intersection number of the relative dualizing sheaf on arithmetic surfaces


Authors: Ulf Kühn and J. Steffen Müller
Journal: Trans. Amer. Math. Soc. 369 (2017), 1869-1894
MSC (2010): Primary 14G40; Secondary 11G50, 11G30, 14H25
DOI: https://doi.org/10.1090/tran/6787
Published electronically: June 2, 2016
MathSciNet review: 3581222
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Abstract: We give an explicitly computable lower bound for the arithmetic self-intersection number $ \overline {\omega }^2$ of the dualizing sheaf on a large class of arithmetic surfaces. If some technical conditions are satisfied, then this lower bound is positive. In particular, these technical conditions are always satisfied for minimal arithmetic surfaces with simple multiplicities and at least one reducible fiber, but we also use our techniques to obtain lower bounds for some arithmetic surfaces with non-reduced fibers.


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Additional Information

Ulf Kühn
Affiliation: Fachbereich Mathematik, Universität Hamburg, Bundesstrasse 55, 20146 Hamburg, Germany
Email: kuehn@math.uni-hamburg.de

J. Steffen Müller
Affiliation: Fachbereich Mathematik, Universität Hamburg, Bundesstrasse 55, 20146 Hamburg, Germany
Address at time of publication: Institut für Mathematik, Carl von Ossietzky Universität Oldenburg, 26111 Oldenburg, Germany
Email: jan.steffen.mueller@uni-oldenburg.de

DOI: https://doi.org/10.1090/tran/6787
Received by editor(s): October 15, 2013
Received by editor(s) in revised form: March 11, 2015
Published electronically: June 2, 2016
Additional Notes: The second author was supported by DFG grant KU 2359/2-1
Article copyright: © Copyright 2016 American Mathematical Society