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

TL;DR

This paper provides an explicit lower bound for the arithmetic self-intersection number of the dualizing sheaf on certain arithmetic surfaces, with conditions ensuring positivity, especially for minimal surfaces with specific fiber properties.

Contribution

It introduces a computable lower bound for the self-intersection number applicable to a broad class of arithmetic surfaces, including those with non-reduced fibers.

Findings

Lower bounds are positive under certain technical conditions.

Bounds are explicitly computable for minimal arithmetic surfaces with simple multiplicities.

Applicable to some surfaces with non-reduced fibers.

Abstract

We give an explicitly computable lower bound for the arithmetic self-intersection number 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 have also used our techniques to obtain lower bounds for some arithmetic surfaces with non-reduced fibers.