  Based on a recent result of Voisin [2001] we describe the last nonzero syzygy
space in the linear strand of a canonical curve C of even genus g=2k lying on a
K3 surface, as the ambient space of a k-2-uple embedded P^{k+1}. Furthermore
the geometric syzygies constructed by Green and Lazarsfeld [1984] from
g^1_{k+1}'s form a non degenerate configuration of finitely many rational
normal curves on this P^{k+1}. This proves a natural generalization of Green's
conjecture [1984], namely that the geometric syzygies should span the space of
all syzygies, in this case.
