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

Kühn, Ulf; Müller, J.

doi:10.1090/tran/6787

Lower bounds on the arithmetic self-intersection number of the relative dualizing sheaf on arithmetic surfaces
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by Ulf Kühn and J. Steffen Müller PDF

Trans. Amer. Math. Soc. 369 (2017), 1869-1894 Request permission

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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