  We prove a strong vanishing result for finite length Koszul modules, and use
it to derive Green's conjecture for every g-cuspidal rational curve over an
algebraically closed field k with char(k) = 0 or char(k) >= (g+2)/2. As a
consequence, we deduce that the general canonical curve of genus g satisfies
Green's conjecture in this range. Our results are new in positive
characteristic, whereas in characteristic zero they provide a different proof
for theorems first obtained in two landmark papers by Voisin. Our strategy
involves establishing two key results of independent interest: (1) we describe
an explicit, characteristic-independent version of Hermite reciprocity for
sl_2-representations; (2) we completely characterize, in arbitrary
characteristics, the (non-)vanishing behavior of the syzygies of the tangential
variety to a rational normal curve.
