  We discuss various applications of a uniform vanishing result for the graded
components of the finite length Koszul module associated to a subspace in the
second wedge product of a vector space. Previously Koszul modules of finite
length have been used to give a proof of Green's Conjecture on syzygies of
generic canonical curves. We now give applications to effective stabilization
of cohomology of thickenings of algebraic varieties, divisors on moduli spaces
of curves, enumerative geometry of curves on K3 surfaces and to skew-symmetric
degeneracy loci. We also show that the stability of sufficiently positive rank
2 vector bundles on curves is governed by resonance.
