The effective Chen ranks conjecture

Koszul modules and their associated resonance schemes are objects appearing in a variety of contexts in algebraic geometry, topology, and combinatorics. We present a proof of an effective version of the Chen ranks conjecture describing the Hilbert function of any Koszul module verifying natural conditions inspired by geometry. We give applications to hyperplane arrangements, describing in a uniform effective manner the Chen ranks of the fundamental group of the complement of every arrangement whose projective resonance is reduced. Finally, we formulate a sharp generic vanishing conjecture for Koszul modules and present a parallel between this statement and the Prym--Green Conjecture on syzygies of general Prym canonical curves.