Graded mapping cone theorem, multisecants and syzygies

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• 作者：Jeaman Ahn^a ; ¹ ; ^{jeamanahn@kongju.ac.kr} ; Sijong Kwak^b ; ² ; ^{skwak@kaist.ac.kr}
• 关键词：Linear syzygies ; Graded mapping cone ; Castelnuovo– ; Mumford regularity ; Partial elimination ideal
• 刊名：Journal of Algebra
• 出版年：2011
• 出版时间：1 April 2011
• 年：2011
• 卷：331
• 期：1
• 页码：243-262
• 全文大小：255 K

文摘

Let X be a reduced closed subscheme in . 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 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 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 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, d2, but also geometric structures for projections according to moving the center.