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TitleOn syzygies, degree, and geometric properties of projective schemes with property
Author(s) Jeaman Ahn, Sijong Kwak
TypeArticle in Journal
AbstractAbstract 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 &#8805; 2 ) if the i-th syzygies of the homogeneous coordinate ring are generated by elements of degree < d + i for 0 &#8804; i &#8804; 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 &#8805; 3 . Assuming property N d , e , we give a sharp upper bound deg &#8289; ( X ) &#8804; ( e + d &#8722; 1, d &#8722; 1 ) . It is natural to ask whether deg &#8289; ( X ) = ( e + d &#8722; 1, d &#8722; 1 ) implies that X is arithmetically Cohen–Macaulay (ACM) with a d-linear resolution. In case of d = 3 , by using the elimination mapping cone sequence and the generic initial ideal theory, we show that deg &#8289; ( X ) = ( e + 2, 2 ) if and only if X is ACM with a 3-linear 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.
ISSN0022-4049
URL http://www.sciencedirect.com/science/article/pii/S002240491400262X
LanguageEnglish
JournalJournal of Pure and Applied Algebra
Volume219
Number7
Pages2724 - 2739
Year2015
Edition0
Translation No
Refereed No
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