Degeneration in discriminantal arrangements

TL;DR

This paper investigates the properties of discriminantal arrangements, introduces the concept of non-very generic varieties to classify them, and demonstrates how degeneration can be used to construct these varieties, with applications to small line arrangements.

Contribution

It introduces non-very generic varieties in hyperplane arrangements and shows how degeneration helps classify discriminantal arrangements, expanding understanding of their combinatorial properties.

Findings

Zariski open set of arrangements characterized by non-very generic varieties

Degeneration as a construction tool for non-very generic varieties

Explicit examples for small line arrangements

Abstract

Discriminantal arrangements are hyperplane arrangements, which are generalized braid ones. They are constructed from given hyperplane arrangements, but their combinatorics are not invariant under combinatorial equivalence. However, it is known that the combinatorics of the discriminantal arrangement are constant on a Zariski open set of the space of hyperplane arrangements. In the present paper, we introduce non-very generic varieties in the space of hyperplane arrangements to classify discriminantal arrangements and show that the Zariski open set is the complement of non-very generic varieties. We study their basic properties and construction and provide examples, including infinite families of non-very generic varieties. In particular, the construction we call degeneration is a powerful tool for constructing non-very generic varieties. As an application, we provide lists of non-very generic varieties for spaces of small line arrangements.