  The resonance varieties are cohomological invariants that are studied in a
variety of topological, combinatorial, and geometric contexts. We discuss their
scheme structure in a general algebraic setting and introduce various
properties that ensure the reducedness of the associated projective resonance
scheme. We prove an asymptotic formula for the Hilbert series of the associated
Koszul module, then discuss applications to vector bundles on algebraic curves
and to Chen ranks formulas for finitely generated groups, with special emphasis
on K\"ahler and right-angled Artin groups.
