Sharp bounds for higher linear syzygies and classifications of projective varieties

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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 and those of its inner projection . 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).
Publisher
SPRINGER HEIDELBERG
Issue Date
2015-02
Language
English
Article Type
Article
Keywords

GENERIC INITIAL IDEALS; DEGREE-COMPLEXITY; MINIMAL DEGREE; GEOMETRY; POINTS; CURVES

Citation

MATHEMATISCHE ANNALEN, v.361, no.1-2, pp.535 - 561

ISSN
0025-5831
DOI
10.1007/s00208-014-1084-9
URI
http://hdl.handle.net/10203/195288
Appears in Collection
MA-Journal Papers(저널논문)
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