Hyperplane Arrangements

Hyperplane arrangements are fascinating objects which do not always fit well into the “traditional” curriculum, for various definitions of traditional. Yet defining (hyper)planes of polyhedra(topes) fit in here, as do configurations of points in the complex plane, matroids, and puzzles about pieces of cake cut with straight cuts. Directly related fields include algebraic geometry, topology, and combinatorics, among others.

A standard entry point to this for graduate students has long been Orlik and Terao’s Arrangements of Hyperplanes. It is pedagogically sound, covers a wide range of topics from very basic starting points, and covers most foundational results throughout. More recently, Stanley has released a combinatorial guide, and a consortium of leading researchers has been working on a comprehensive e-text in the complex case for a number of years now. However, this means there is no truly comprehensive introductory work containing more recent developments, particularly in algebraic geometry.

The book under review aims to fill this gap, but only partly succeeds. On the one hand, it is more of an invitation to an interesting field than a comprehensive text. If you don’t know a litany of results in many fields (particularly in algebraic geometry and homological algebra) and don’t have another ready reference for combinatorial results in the field, you won’t get very far starting from scratch here.

On the other hand, it does succeed as a monograph introducing many results, including quite recent ones. Naturally, given that over half of volume is devoted to Milnor fibers and related areas of cohomological research by the author and his students, that isn’t surprising; but Huh’s very recent results on log-concavity are given a good treatment as well. Similarly, although there are many (many) occasions where proofs or even examples are only given by references to the literature, there are also plenty of places where very useful examples are given to explain the literature, or proof sketches supplied for what Dimca identifies as folklore never previously given full proofs (e.g. Theorem 8.1, Saito’s Criterion). There are also copious exercises, which is always useful in a text at this level.

This book would be a good second text for someone with a strong background in algebraic geometry who learned their arrangements from Orlik/Terao. It includes direct references to using the open source software SINGULAR to actually compute invariants, and retains the informal feel of its origins as lecture notes. Perhaps it will encourage more people to join this growing field, where the author says specialists treat outsiders “very well” — which has been this reviewer’s experience as well.

Karl-Dieter Crisman teaches mathematics at Gordon College in Massachusetts, where he also gets to work on open source software, the mathematics of voting, and examining connections between all of these and issues of belief and faith.