Fermat-type arrangements

TL;DR

This paper consolidates known Fermat-type arrangements, explores their configurations and higher-dimensional generalizations, and presents new results related to symbolic powers of ideals and unexpected hypersurfaces.

Contribution

It provides a comprehensive overview of Fermat arrangements, introduces new findings on their configurations, and connects these to modern algebraic geometry problems.

Findings

New results on arrangements and their configurations

Connections to symbolic powers and unexpected hypersurfaces

Higher-dimensional generalizations explored

Abstract

The purpose of this work is to collect in one place available information on line arrangements known in the literature as braid, monomial, Ceva or Fermat arrangement. They have been studied for a long time and appeared recently in connection with highly interesting problems, namely: the containment problem between symbolic and ordinary powers of ideals and the existence of unexpected hypersurfaces. We also study also derived configurations of points (or more general: linear flats) which arise by intersecting hyperplanes in Fermat arrangements or by taking duals of these hyperplanes. Furthermore we discuss briefly higher dimensional generalizations and present results arising by applying this approach to problems mentioned above. Some of our results are original and appear for the first time in print.