Resonance Varieties Via Blowups of ℙ2 and Scrolls

Conjectures of Suciu [“Fundamental groups of line arrangements: Enumerative aspects,” 43–79. Contemporary Mathematics 276. Providence, RI: American Mathematical Society, 2001.] relate the fundamental group of an arrangement complement
$M= \mathbb{C}^{n} {\backslash} \, {\cal A}$
to the first resonance variety of
$H^*(M, \mathbb{Z})$
. We describe a connection between the first resonance variety and the Orlik–Terao algebra
$C (\cal A)$
of the arrangement. In particular, we show that nonlocal components of
$R^1 (\cal A)$
give rise to determinantal syzygies of
$C (\cal A)$
. As a result,
$C (\cal A)$
lies on a scroll, placing geometric constraints on
$R^1 (\cal A)$
. The key observation is that
$C (\cal A)$
is the homogeneous coordinate ring associated to a nef but not an ample divisor on the blowup of
$\mathbb{P}^2$
at the singular points of
$\cal {A}$
.