Syzygy Bound on the Cubic Strand of a Projective Variety and 3-linear Resolutions

Let X be any projective variety in P^N over an algebraically closed fieldK. Suppose that X is nondegenerate, i.e. not contained in any hyperplane of P^N. Few years ago, K. Han and S. Kwak developed a technique to compare syzygies under projections, as applications they proved sharp upper bounds on the ranks of higher linear syzygies, and characterized the extremal and next-to-extremal cases.

In this talk, we report generalizations of these results, which are on-going with S. Kwak and J. Ahn. First, let us consider any variety X such that the defining ideal I_X has no generators of degree less than 3. Since I_X has no generators of degree ≤ 2, so the first non-vanishing strand of the resolution comes from linear syzygies of minimal generators of degree 3. We consider a basic degree bound and sharp bounds for generators and syzygies in this cubic strand. Further, the extremal cases will be discussed in the end.