2008/04/23 by Jeaman Ahn, Ahn, Jeaman, Sijong Kwak +1
Computer Science · Mathematics · #13D02 #14M17 #14N05 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.0804.3757
openalex publication_date 2008/04/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let X be a reduced closed subscheme in \mathbb Pn. 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 \mathcal IX/¶n such that I is generated by quadrics and there are only linear syzygies up to p-th step (cf. \citeEGHP1, \citeEGHP2, \citeV). Recently, many algebraic and geometric results have been proved for projective varieties satisfying property N2,p(cf. \citeCKP, \citeEGHP1, \citeEGHP2 \cite KP). 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. \citeBEL). On the other hand, projection methods have been very useful and powerful in bounding Castelnuovo regularity, normality and other classical invariants in geometry(cf. \citeBE, \citeK, \citeKP, \citeL \cite R). 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.