The Kodaira dimension of Hilbert modular threefolds

TL;DR

This paper investigates the Kodaira dimension of Hilbert modular threefolds, demonstrating many are of general type or nonnegative Kodaira dimension through geometric and combinatorial analysis.

Contribution

It introduces a detailed study of totally positive integral elements in number fields to determine the Kodaira dimension of certain Hilbert modular threefolds.

Findings

Many Hilbert modular threefolds of genus 0 and 1 are of general type.

Some Hilbert modular threefolds have nonnegative Kodaira dimension.

A new geometric and combinatorial approach is developed for analyzing these varieties.

Abstract

Following a method introduced by Thomas-Vasquez and developed by Grundman, we prove that many Hilbert modular threefolds of arithmetic genus $0$ and $1$ are of general type, and that some are of nonnegative Kodaira dimension. The new ingredient is a detailed study of the geometry and combinatorics of totally positive integral elements $x$ of a fractional ideal $I$ in a totally real number field $K$ with the property that $\mathop{\mathrm{tr}} xy < \mathop{\mathrm{min}} I \mathop{\mathrm{tr}} y$ for some $y \gg 0 \in K$.