Resonance varieties via blowups of P^2 and scrolls

TL;DR

This paper explores the relationship between the first resonance variety of hyperplane arrangements and the Orlik-Terao algebra, revealing geometric constraints via blowups of P^2 and scrolls.

Contribution

It establishes a novel connection between resonance varieties and algebraic geometry through blowups and scrolls, linking combinatorial and geometric properties of arrangements.

Findings

Non-local components of R^1(A) induce determinantal syzygies of C(A)

Proj(C(A)) lies on a scroll, constraining R^1(A)

C(A) corresponds to a nef divisor on a blowup of P^2

Abstract

Conjectures of Suciu relate the fundamental group of the complement M = C^n\A of a hyperplane arrangement A to the first resonance variety of H^*(M,Z). We describe a connection between the first resonance variety and the Orlik-Terao algebra C(A) of the arrangement. In particular, we show that non-local components of R^1(A) give rise to determinantal syzygies of C(A). As a result, Proj(C(A)) lies on a scroll, placing geometric constraints on R^1(A). The key observation is that C(A) is the homogeneous coordinate ring associated to a nef but not ample divisor on the blowup of P^2 at the singular points of A.