2005/11/01 by David Eisenbud, Mark Green, Klaus Hulek +1 · 12 citations
Mathematics · Computer Science · #Commutative Algebra and Its Applications #Algebraic Geometry and Number Theory #Polynomial and algebraic computation
paper · pdf · doi:10.1112/s0010437x05001776
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 Frberg, 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 for a scheme , and apply the result to bound the number of intersection points of certain pairs of varieties such as rational normal scrolls.