Restricting linear syzygies: Algebra and geometry

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Eisenbud, D.; Green, M.; Hulek, K.; Popescu, S.: Restricting linear syzygies: Algebra and geometry. In: Compositio Mathematica 141 (2005), Nr. 6, S. 1460-1478. DOI: https://doi.org/10.1112/S0010437X05001776

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Let X ⊂ ℙr be a closed scheme in projective space whose homogeneous ideal is generated by quadrics. We say that X (or its ideal I X) satisfies the condition N2,p if the syzygies of I X are linear for p steps. We show that if X satisfies N2,p then a zero-dimensional or one-dimensional intersection of X with a plane of dimension ≤ p is 2-regular. This extends a result of Green and Lazarsfeld. We give conditions when the syzygies of X restrict to the syzygies of the intersection. Many of our results also work for ideals generated by forms of higher degree. As applications, we bound the p for which some well-known projective varieties satisfy N2,p. Another application, carried out by us in a different paper, is a step in the classification of 2-regular reduced projective schemes. Extending a result of Fröberg, we determine which monomial ideals satisfy N2,p. We also apply Green's 'linear syzygy theorem' to deduce a relation between the resolutions of IX and IX∪Γur for a scheme Γ, and apply the result to bound the number of intersection points of certain pairs of varieties such as rational normal scrolls. © Foundation Compositio Mathematica 2005.
License of this version: Es gilt deutsches Urheberrecht. Das Dokument darf zum eigenen Gebrauch kostenfrei genutzt, aber nicht im Internet bereitgestellt oder an Außenstehende weitergegeben werden. Dieser Beitrag ist aufgrund einer (DFG-geförderten) Allianz- bzw. Nationallizenz frei zugänglich.
Document Type: article
Publishing status: publishedVersion
Issue Date: 2005
Appears in Collections:Fakultät für Mathematik und Physik

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1 image of flag of Germany Germany 16 80.00%
2 image of flag of United States United States 1 5.00%
3 image of flag of Netherlands Netherlands 1 5.00%
4 image of flag of United Kingdom United Kingdom 1 5.00%
5 image of flag of Algeria Algeria 1 5.00%

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