Details:
Title  On syzygies, degree, and geometric properties of projective schemes with property  Author(s)  Jeaman Ahn, Sijong Kwak  Type  Article in Journal  Abstract  Abstract Let X be a reduced, but not necessarily irreducible closed subscheme of codimension e in a projective space. One says that X satisfies property N d , p ( d ≥ 2 ) if the ith syzygies of the homogeneous coordinate ring are generated by elements of degree < d + i for 0 ≤ i ≤ p (see [10] for details). Much attention has been paid to linear syzygies of quadratic schemes ( d = 2 ) and their geometric interpretations (cf. [1,9,15–17]). However, not very much is actually known about algebraic sets satisfying property N d , p , d ≥ 3 . Assuming property N d , e , we give a sharp upper bound deg ⁡ ( X ) ≤ ( e + d − 1, d − 1 ) . It is natural to ask whether deg ⁡ ( X ) = ( e + d − 1, d − 1 ) implies that X is arithmetically Cohen–Macaulay (ACM) with a dlinear resolution. In case of d = 3 , by using the elimination mapping cone sequence and the generic initial ideal theory, we show that deg ⁡ ( X ) = ( e + 2, 2 ) if and only if X is ACM with a 3linear resolution. This is a generalization of the results of Eisenbud et al. ( d = 2 ) [9,10]. We also give more general inequality concerning the length of the finite intersection of X with a linear space of not necessary complementary dimension in terms of graded Betti numbers. Concrete examples are given to explain our results.  ISSN  00224049 
URL 
http://www.sciencedirect.com/science/article/pii/S002240491400262X 
Language  English  Journal  Journal of Pure and Applied Algebra  Volume  219  Number  7  Pages  2724  2739  Year  2015  Edition  0  Translation 
No  Refereed 
No 
