Geometry Seminar

Date: October 20, 2017

Time: 4:00PM - 5:00PM

Location: BLOC 628

Speaker: Michael Di Pasquale, Oklahoma State University

Title: Homological Obstructions to Freeness of Multi-arrangements

Abstract:

If the module of vector fields tangent to a multi-arrangement is free over the underlying polynomial ring, we say that the multi-arrangement is free. It is of particular interest in the theory of hyperplane arrangements to investigate the relation of freeness to the combinatorics of the intersection lattice - the holy grail here is Terao's conjecture that freeness of arrangements is detectable from the intersection lattice. It is known that corresponding statements for multi-arrangements fail.

Given a multi-arrangement, we present a co-chain complex derived from work of Brandt and Terao on k-formality whose exactness encodes freeness of the multi-arrangement. The cohomology groups of this co-chain complex thus present obstructions to freeness of multi-arrangements. Using this criterion we give an example showing that the property of being totally non-free is not detectable from the intersection lattice. This builds on previous work with Francisco, Schweig, Mermin, and Wakefield.