The Kodaira dimension of Hilbert modular threefolds

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 $\mathrm{tr}xy<\min I\mathrm{tr}y$ for some $y\gg 0\in K$.

arXiv commentCorrected an oversight regarding the cubic field of discriminant 778. To appear, in a somewhat shorter version, in the proceedings of LUCANT 2025. With an appendix by Adam Logan and John Voight