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

Titelangaben

Kühn, Ulf ; Müller, Jan Steffen:
Lower bounds on the arithmetic self-intersection number of the relative dualizing sheaf on arithmetic surfaces.
In: Transactions of the American Mathematical Society. Bd. 369 (2017) Heft 3 . - S. 1869-1894.
ISSN 1088-6850
DOI: https://doi.org/10.1090/tran/6787

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Abstract

We give an explicitly computable lower bound for the arithmetic self-intersection number ω² 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