The minimal regular model of a Fermat curve of odd squarefree exponent and its dualizing sheaf

TL;DR

This paper constructs the minimal regular model of Fermat curves with odd squarefree exponents over cyclotomic integers and computes bounds for the self-intersection of its dualizing sheaf.

Contribution

It provides the first explicit construction of the minimal regular model for these Fermat curves and calculates bounds for the dualizing sheaf's self-intersection.

Findings

Constructed the minimal regular model over cyclotomic integers.

Derived bounds for the arithmetic self-intersection of the dualizing sheaf.

Enhanced understanding of the arithmetic geometry of Fermat curves.

Abstract

We construct the minimal regular model of the Fermat curve of odd squarefree composite exponent $N$ over the $N$-th cyclotomic integers. As an application, we compute upper and lower bounds for the arithmetic self-intersection of the dualizing sheaf of this model.