Compositio Mathematica

Restricting linear syzygies: algebra and geometry

David Eisenbud a1, Mark Green a2, Klaus Hulek a3 and Sorin Popescu a4
a1 Department of Mathematics, University of California, Berkeley, CA 94720, USA
a2 Department of Mathematics, University of California, Los Angeles, CA 90095-1555, USA
a3 Institut für Mathematik,, Universität Hannover, 30060 Hannover, Germany
a4 Department of Mathematics, SUNY at Stony Brook, Stony Brook, NY 11794-3651, USA

Article author query
eisenbud d   [Google Scholar] 
green m   [Google Scholar] 
hulek k   [Google Scholar] 
popescu s   [Google Scholar] 


Let $X\subset \mathbb{P}^r$ be a closed scheme in projective space whose homogeneous ideal is generated by quadrics. We say that X (or its ideal IX) satisfies the condition N2,p if the syzygies of IX 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 $\leq p$ 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 $I_{X\cup \Gamma}$ 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.

(Received April 29 2004)
(Accepted January 17 2005)

Key Words: linear syzygies; secants; quadratic ideals; chordal graphs.

Maths Classification

14N05; 13D02; 14M17.