Every finite-type graded algebra defines a complex of finitely generated, graded modules over a symmetric algebra, whose homology modules are called the Koszul modules of the given algebra. Particularly interesting in a variety of contexts is the geometry of the support loci of these modules, known as the resonance schemes of the algebra. In this talk, I will describe several conditions that ensure the reducedness of the associated projective resonance schemes and yield asymptotic formulas for the Hilbert series of the corresponding Koszul modules. For the exterior Stanley-Reisner algebra associated to a finite simplicial complex, we show that the resonance schemes are reduced, and give bounds on the regularity and projective dimension of the Koszul modules. This leads to a relationship between resonance and Hilbert series that generalizes a known formula for the Chen ranks of a right-angled Artin group. Based on joint work with Marian Aprodu, Gavril Farkas, Claudiu Raicu, and Alessio Sammartano.