  For a finite dimensional vector space G we define the k-th generic syzygy
scheme Gensyz_k(G) by explicit equations. We show that the syzygy scheme Syz(f)
of any syzygy in the linear strand of a projective variety X which is cut out
by quadrics is a cone over a linear section of a corresponding generic syzygy
scheme. We also give a geometric description of Gensyz_k(G) for k=0,1,2. In
particular Gensyz_2(G) is the union of a Pl"ucker embedded Grassmannian and a
linear space. From this we deduce that every smooth, non-degenerate projective
curve C which is cut out by quadrics and has a p-th linear syzygy of rank p+3
admits a rank 2 vector bundle E with det E = O_C(1) and h^0(E) at least p+4.
