Graded mapping cone theorem, multisecants and syzygies

Abstract

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) [8], Vermeire (2001) [20]). Recently, many algebraic and geometric results have been proved for projective varieties satisfying property N(2,p) (cf. Choi, Kwak, and Park (2008) [6], Eisenbud et al. (2005) [8], Kwak and Park (2005) [15]). 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) [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 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.