On syzygies of non-complete embedding of projective varieties

Let X be a non-degenerate, not necessarily linearly normal projective variety in ℙ. Recently the generalization of property N _p to non-linearly normal projective varieties have been considered and its algebraic and geometric properties are studied extensively. One of the generalizations is the property N _d,p for the saturated ideal I _X (Eisenbud et al. in Compos Math 141: 1460-1478, 2005) and the other is the property N _d,P for the graded module of the twisted global sections of O _X(1)(Kwak and Park in J Reine Angew Math 582: 87-105, 2005). In this paper, we are interested in the algebraic and geometric meaning of properties N^S_p for every p ≥ 0 and the syzygetic behaviors of isomorphic projections and hyperplane sections of a given variety with property N^S_p.