  Generalizing the well-known Green Conjecture on syzygies of canonical curves,
Green and Lazarsfeld formulated in 1986 the Secant Conjecture predicting that a
line bundle L of sufficiently high degree on a curve has a non-linear p-syzygy
if and only if L fails to be (p+1)-very ample. Via lattice theory for special
K3 surfaces, Voisin's solution of the classical Green Conjecture and
calculations on moduli stacks of pointed curves, we prove: (1) The
Green-Lazarsfeld Secant Conjecture in various degree of generality, including
its strongest possible form in the divisorial case in the universal Jacobian.
(2) The Prym-Green Conjecture on the naturality of the resolution of a general
Prym-canonical curve of odd genus.
