A Class of Hypergraph Arrangements with Shellable Intersection Lattice

AbstractFor every hypergraph onnvertices there is an associated subspace arrangement in Rncalled a hypergraph arrangement. We prove shellability for the intersection lattices of a large class of hypergraph arrangements. This class incorporates all the hypergraph arrangements which were previously shown to have shellable intersection lattices

Pure O-sequences: known results, applications, and open problems

This note presents a discussion of the algebraic and combinatorial aspects of the theory of pure O-sequences. Various instances where pure O-sequences appear are described. Several open problems that deserve further investigation are also presented.Comment: Some minor revisions with respect to the previous version. 20 pages. To appear in a Springer volume edited by Irena Peeva and dedicated to David Eisenbud on the occasion of his 65th birthda

Hilbert functions of d-regular ideals

In the present paper, we characterize all possible Hilbert functions of graded ideals in a polynomial ring whose regularity is smaller than or equal to $d$, where $d$ is a positive integer. In addition, we prove the following result which is a generalization of Bigatti, Hulett and Pardue's result: Let $p \geq 0$ and $d>0$ be integers. If the base field is a field of characteristic 0 and there is a graded ideal $I$ whose projective dimension $\mathrm{proj\ dim}(I)$is smaller than or equal to$p$and whose regularity$\mathrm{reg}(I)$is smaller than or equal to$d$, then there exists a monomial ideal$L$having the maximal graded Betti numbers among graded ideals$J$which have the same Hilbert function as$I$and which satisfy$\mathrm{proj dim}(J) \leq p$and$\mathrm{reg}(J) \leq d$. We also prove the same fact for squarefree monomial ideals. The main methods for proofs are generic initial ideals and combinatorics on strongly stable ideals.Comment: 33 pages, minor changes, to appear in J. Algebr

On the shape of a pure O-sequence

An order ideal is a finite poset X of (monic) monomials such that, whenever M is in X and N divides M, then N is in X. If all, say t, maximal monomials of X have the same degree, then X is pure (of type t). A pure O-sequence is the vector, h=(1,h_1,...,h_e), counting the monomials of X in each degree. Equivalently, in the language of commutative algebra, pure O-sequences are the h-vectors of monomial Artinian level algebras. Pure O-sequences had their origin in one of Richard Stanley's early works in this area, and have since played a significant role in at least three disciplines: the study of simplicial complexes and their f-vectors, level algebras, and matroids. This monograph is intended to be the first systematic study of the theory of pure O-sequences. Our work, making an extensive use of algebraic and combinatorial techniques, includes: (i) A characterization of the first half of a pure O-sequence, which gives the exact converse to an algebraic g-theorem of Hausel; (ii) A study of (the failing of) the unimodality property; (iii) The problem of enumerating pure O-sequences, including a proof that almost all O-sequences are pure, and the asymptotic enumeration of socle degree 3 pure O-sequences of type t; (iv) The Interval Conjecture for Pure O-sequences (ICP), which represents perhaps the strongest possible structural result short of an (impossible?) characterization; (v) A pithy connection of the ICP with Stanley's matroid h-vector conjecture; (vi) A specific study of pure O-sequences of type 2, including a proof of the Weak Lefschetz Property in codimension 3 in characteristic zero. As a corollary, pure O-sequences of codimension 3 and type 2 are unimodal (over any field); (vii) An analysis of the extent to which the Weak and Strong Lefschetz Properties can fail for monomial algebras; (viii) Some observations about pure f-vectors, an important special case of pure O-sequences.Comment: iii + 77 pages monograph, to appear as an AMS Memoir. Several, mostly minor revisions with respect to last year's versio