Graded Mapping Cone Theorem, Multisecants and Syzygies

Let X be a reduced closed subscheme in P. As a slight generalization of property Np due to Green-Lazarsfeld, we can say 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. [9], [10], [21] ). Recently, many algebraic and geometric results have been proved for projective varieties satisfying property N2,p(cf. [6], [9], [10] [15]). In this case, the Castelnuovo regularity and normality can be obtained by the blowingup method as reg(X) ≤ e + 1 where e is the codimension of a smooth variety X (cf. [2]). On the other hand, projection methods have been very useful and powerful in bounding Castelnuovo regularity, normality and other classical invariants in geometry(cf. [4], [14], [15], [16] [20]). In this paper, we first prove the graded mapping cone theorem on partial eliminations as a general algebraic tools and give some applications. Then, we bound the length of zero dimensional intersection of X and a linear space L in terms of graded Betti numbers and deduce a relation between X and its projections with respect to the geometry and syzygies in the case of projective schemes satisfying property N2,p scheme-theoretically. In addition, we give not only interesting information on the regularity of fibers and multiple loci for the case of Nd,p, d ≥ 2 but also geometric structures for projections according to moving the center. 2000 Mathematics Subject Classification: 14N05, 13D02, 14M17.