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We give an explicitly computable lower bound for the arithmeticself-intersection number of the dualizing sheaf on a large class of arithmeticsurfaces. If some technical conditions are satisfied, then this lower bound ispositive. In particular, these technical conditions are always satisfied forminimal arithmetic surfaces with simple multiplicities and at least onereducible fiber, but we have also used our techniques to obtain lower boundsfor some arithmetic surfaces with non-reduced fibers.

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