De Concini-Procesi Wonderful Arrangement Models: A Discrete Geometer's Point of View

Summary

This article outlines the construction of De Concini-Procesi arrangement models and describes recent progress in understanding their significance from the algebraic, geometric, and combinatorial point of view. Throughout the exposition, strong emphasis is given to the combinatorial and discrete geometric data that lie at the core of the construction.

1. An Invitation to Arrangement Models The complements of coordinate hyperplanes in a real or complex vector space are easy to understand: The coordinate hyperplanes in ℝⁿ dissect the space into 2ⁿ open orthants; removing the coordinate hyperplanes from ℂⁿ leaves the complex torus (ℂ*)ⁿ. Arbitrary subspace arrangements, i.e., finite families of linear subspaces, have complements with far more intricate combinatorics in the real case, and far more intricate topology in the complex case. Arrangement models improve this complicated situation locally — constructing an arrangement model means to alter the ambient space so as to preserve the complement and to replace the arrangement by a divisor with normal crossings, i.e., a collection of smooth hypersurfaces which locally intersect like coordinate hyperplanes. Almost a decade ago, De Concini and Procesi provided a canonical construction of arrangement models — wonderful arrangement models — that had significant impact in various fields of mathematics.

Why should a discrete geometer be interested in this model construction?

Because there is a wealth of wonderful combinatorial and discrete geometric structure lying at the heart of the matter. Our aim here is to bring these discrete pearls to light.

First, combinatorial data plays a descriptive role in various places: The combinatorics of the arrangement fully prescribes the model construction and a natural stratification of the resulting space. We will see details and examples in Section 2.