Generic Syzygy Schemes

Details Generic Syzygy Schemes Generic Syzygy Schemes

2004-03-25

arxiv.org/abs/math/0403432v1

Journal of Pure and Applied Algebra, Volume 208, Issue 3 , March 2007, Pages 867-876

Mathematics

Algebraic Geometry

12 Pages. This paper is a completely rewritten version of the first part of math.AG/0108078. It also contains several new resu

Scientific paper

10.1016/j.jpaa.2006.03.021

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.

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