Generic syzygy schemes

For a finite dimensional vector space GG we define the kk-th generic syzygy scheme Gensyzk(G) by giving explicit equations. If X⊂PnX⊂Pn is cut out by quadrics and ff is a pp-th syzygy of rank p+k+1p+k+1 we show that the syzygy scheme Syz(f) of ff is a cone over a linear section of Gensyzk(G). We also give a geometric description of Gensyzk(G) for k=0,1,2k=0,1,2; in particular Gensyz2(G) is the union of a Plücker embedded Grassmannian and a linear space. From this we deduce that every smooth, non-degenerate projective curve C⊂PnC⊂Pn which is cut out by quadrics and has a pp-th linear syzygy of rank p+3p+3 admits a rank 2 vector bundle EE with detE=OC(1)detE=OC(1) and h0(E)≥p+4h0(E)≥p+4.