Graded mapping cone theorem, multisecants and syzygies

Let X be a reduced closed subscheme in Pn. As a slight generalization of property Np due to Green–Lazarsfeld, one says that X satisfies property N2,p scheme-theoretically if there is an ideal I generating the ideal sheaf IX/Pn such that I is generated by quadrics and there are only linear syzygies up to p-th step (cf. Eisenbud et al. (2005) [8], , Vermeire (2001) [20], ). Recently, many algebraic and geometric results have been proved for projective varieties satisfying property N2,p (cf. Choi, Kwak, and Park (2008) [6], , Eisenbud et al. (2005) [8], , Kwak and Park (2005) [15], ). In this case, the Castelnuovo 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) [3], ). 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) [2], , Kwak (1998) [14], , Kwak and Park (2005) [15], , Lazarsfeld (1987) [16].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 N2,p scheme-theoretically. In addition, we give not only interesting information on the regularity of fibers of the projection for the case of Nd,p, d⩾2, but also geometric structures for projections according to moving the center.