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Sharp bounds for higher linear syzygies and classifications of projective varieties

Title
Sharp bounds for higher linear syzygies and classifications of projective varieties
Authors
Han, Kang JinKwak, Sijong
DGIST Authors
Han, Kang Jin
Issue Date
2015-02
Citation
Mathematische Annalen, 361(1-2), 535-561
Type
Article
Article Type
Article
Keywords
GENERIC INITIAL IDEALSDEGREE-COMPLEXITYMINIMAL DEGREEGEOMETRYPOINTSCURVES
ISSN
0025-5831
Abstract
In the present paper, we consider upper bounds of higher linear syzygies i.e. graded Betti numbers in the first linear strand of the minimal free resolutions of projective varieties in arbitrary characteristic. For this purpose, we first remind ‘Partial Elimination Ideals (PEIs)’ theory and introduce a new framework in which one can study the syzygies of embedded projective varieties well using PEIs theory and the reduction method via inner projections. Next we establish fundamental inequalities which govern the relations between the graded Betti numbers in the first linear strand of an algebraic set X and those of its inner projection Xq. Using these results, we obtain some natural sharp upper bounds for higher linear syzygies of any nondegenerate projective variety in terms of the codimension with respect to its own embedding and classify what the extremal case and the next-to-extremal case are. This is a generalization of Castelnuovo and Fano’s results on the number of quadrics containing a given variety and another characterization of varieties of minimal degree and del Pezzo varieties from the viewpoint of ‘syzygies’. Note that our method could also be applied to get similar results for more general categories (e.g. connected in codimension one algebraic sets). © 2014, Springer-Verlag Berlin Heidelberg.
URI
http://hdl.handle.net/20.500.11750/5677
DOI
10.1007/s00208-014-1084-9
Publisher
SPRINGER HEIDELBERG
Files:
There are no files associated with this item.
Collection:
School of Undergraduate Studies1. Journal Articles


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