Generic syzygy schemes

Abstract

For a finite dimensional vector space G we define the k -th generic syzygy scheme Gensyz k ( G ) by giving explicit equations. If X ⊂ P n is cut out by quadrics and f is a p -th syzygy of rank p + k + 1 we show that the syzygy scheme Syz ( f ) of f is a cone over a linear section of Gensyz k ( G ) . 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 Plücker embedded Grassmannian and a linear space. From this we deduce that every smooth, non-degenerate projective curve C ⊂ P n 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 ) ≥ p + 4 .