Restricting linear syzygies: algebra and geometry
Sorin Popescu2005, Compositio Mathematica
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Abstract
Let X ⊂ P r 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 N 2,p if the syzygies of I X are linear for p steps. We show that if X satisfies N 2,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 N 2,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 N 2,p. We also apply Green's 'linear syzygy theorem' to deduce a relation between the resolutions of I X and I X∪Γ 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.
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