Restricting linear syzygies: algebra and geometry

Compositio Mathematica, 2004

165 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/89fb53e4db85875d84da7f4cded10cefffb9412e

Citations in the corpus

• The regularity of Tor and graded Betti Numbers (2004, unknown): “Corol-lary 2.5 we give a bound for the saturation and regularity of a plane section, generalizing Theorem 1.2 of Eisenbud-Green-Hulek-Popescu [2005]: C OROLLARY .1 .”
• A Beginner’s Guide to Edge and Cover Ideals (2013, unknown): “al [17] describe how long the resolution will stay linear (sometimes called the N2,p property) in terms of the length of the smallest induced cycle in the complement of G (also see Tutorial Exercise 5.”
• Asymptotic syzygies of algebraic varieties (2011, unknown): “In the case of curves, the fact that highly secant planes carry syzygies goes back to [15] and [16], and in [11] special secants of the sort appearing in our arguments were used to study the failure of Property (Np) on surfaces.”
• Regularity 3 in edge ideals associated to bipartite graphs (2012, unknown): “Another proof of Fröberg’s characterization can be found in [6] where, moreover, the least i such that ui > i + 2 is characterized in a combinatorial way when the edge ideal does not have a linear resolution.”
• The regularity of powers of edge ideals (2014, unknown): “Bounds on the regularity of edge ideals have been studied by a number of researchers (see [1], [2], [3], [4], [5], [6], [7],[8] [9], [11],[12],[13]).”
• Open Problems on Syzygies and Hilbert Functions (2009, unknown): “For example, [46] provides a criterion for monomial ideals to satisfy Np.”
• Koszul homology and syzygies of Veronese subalgebras (2009, unknown): “In their interesting paper [13] Eisenbud, Green , Hulek and Popescu reproved some of these statements using different methods.”
• Regularity of powers of forests and cycles (2014, unknown): “[18, 13, 21]) imply that if the complement graph of G contains no induced cycles of length ≥ 4 then b = 0 and s0 = 1.”
• Regularity of Squarefree Monomial Ideals (2013, unknown): “The following result was originally stated and proved by Wegner [50] using topological language, and re-stated in terms of monomial ideals by Fröberg [22] (see also [17, 26]).”