Let X be a reduced closed subscheme in P(n). As a slight generalization of property N(p) due to Green-Lazarsfeld, one says that X satisfies property N(2,p) scheme-theoretically if there is an ideal I generating the ideal sheaf J(X)/P(n) such that I is generated by quadrics and there are only linear syzygies up to p-th step (cf. Eisenbud et al. (2005) , Vermeire (2001) ). Recently, many algebraic and geometric results have been proved for projective varieties satisfying property N(2,p) (cf. Choi, Kwak, and Park (2008) , Eisenbud et al. (2005) , Kwak and Park (2005) ). In this case, the Castel-nuovo regularity and normality can be obtained by the blowing-up method as reg(X) <= e + 1 where e is the codimension of a smooth variety X (cf. Bertram, Ein, and Lazarsfeld (2003) ). On the other hand, projection methods have been very useful and powerful in bounding Castelnuovo regularity, normality and other classical invariants in geometry (cf. Beheshti and Eisenbud (2010)  Kwak (1998) , Kwak and Park (2005), Lazarsfeld (1987) . We first prove the graded mapping cone theorem on partial eliminations as a general algebraic tool to study syzygies of the non-complete embedding of X. For applications, we give an optimal bound on the length of zero-dimensional intersections of X and a linear space L in terms of graded Betti numbers. We also deduce several theorems about the relationship between X and its projections with respect to the geometry and syzygies for a projective scheme X satisfying property N(2,p) scheme-theoretically. In addition, we give not only interesting information on the regularity of fibers of the projection for the case of N(d,p), d >= 2, but also geometric structures for projections according to moving the center. (C) 2010 Elsevier Inc. All rights reserved.