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We discuss and enumerate pc-sets that are both not contained in any diatonic collection and are minimal with respect to this property, and we generalize this idea to other collections. We also consider related simplicial complexes and examine how some of their geometric properties reflect qualities of the associated pc-sets. [ABSTRACT FROM AUTHOR]

Abstract Fix a field k. When Δ is a simplicial complex on n vertices with Stanley–Reisner ideal I Δ , we define and study an invariant called the type defect of Δ. Except when Δ is a single simplex, the type defect of Δ, td (Δ) , is the difference dim ⁡ Tor c S (S / I Δ , k) − c , where c is the codimension of Δ and S = k [ x 1 , ... x n ]. We show that this invariant admits surprisingly nice properties. For example, it is well-behaved when one glues two complexes together along a face. Furthermore, Δ is Cohen–Macaulay if td (Δ) ≤ 0. On the other hand, if Δ is a simple graph (viewed as a one-dimensional complex), then td (Δ ′) ≥ 0 for every induced subgraph Δ ′ of Δ if and only if Δ is chordal. Requiring connected induced subgraphs to have type defect zero allows us to define a class of graphs that we call treeish , and which we generalize to simplicial complexes. We then extend some of our chordality results to higher dimensions, proving sharp lower bounds for most Betti numbers of ideals with linear resolution, and classifying when equalities occur. As an application, we prove sharp lower bounds for Betti numbers of graded ideals (not necessarily monomial) with linear resolution. [ABSTRACT FROM AUTHOR]

We introduce a new class of lattices, the modernistic lattices, and their duals, the comodernistic lattices. We show that every modernistic or comodernistic lattice has shellable order complex. We go on to exhibit a large number of examples of (co)modernistic lattices. We show comodernism for two main families of lattices that were not previously known to be shellable: the order congruence lattices of finite posets, and a weighted generalization of the k -equal partition lattices. We also exhibit many examples of (co)modernistic lattices that were already known to be shellable. To begin with, the definition of modernistic is a common weakening of the definitions of semimodular and supersolvable. We thus obtain a unified proof that lattices in these classes are shellable. Subgroup lattices of solvable groups form another family of comodernistic lattices that were already proven to be shellable. We show not only that subgroup lattices of solvable groups are comodernistic, but that solvability of a group is equivalent to the comodernistic property on its subgroup lattice. Indeed, the definition of comodernistic exactly requires on every interval a lattice-theoretic analogue of the composition series in a solvable group. Thus, the relation between comodernistic lattices and solvable groups resembles, in several respects, that between supersolvable lattices and supersolvable groups. [ABSTRACT FROM AUTHOR]

We study the relationship between the projective dimension of a squarefree monomial ideal and the domination parameters of the associated graph or clutter. In particular, we show that the projective dimensions of graphs with perfect dominating sets can be calculated combinatorially. We also generalize the well-known graph domination parameter τ to clutters, obtaining bounds on the projective dimension analogous to those for graphs. Through Hochster's Formula, our bounds on projective dimension also give rise to bounds on the homologies of the associated Stanley–Reisner complexes. [ABSTRACT FROM AUTHOR]

We introduce the concept of edgewise domination in clutters and use it to provide an upper bound for the projective dimension of any squarefree monomial ideal. We then compare this bound to a bound given by Faltings. Finally, we study a family of clutters associated to graphs and compute dom-ination parameters for certain classes of these clutters. [ABSTRACT FROM AUTHOR]

Abstract: We construct several pairwise-incomparable bounds on the projective dimensions of edge ideals. Our bounds use combinatorial properties of the associated graphs. In particular, we draw heavily from the topic of dominating sets. Through Hochsterʼs Formula, we recover and strengthen existing results on the homological connectivity of graph independence complexes. [Copyright &y& Elsevier]

Abstract: White has conjectured that the toric ideal of a matroid is generated by quadric binomials corresponding to symmetric basis exchanges. We prove a stronger version of this conjecture for lattice path polymatroids by constructing a monomial order under which these sets of quadrics form Gröbner bases. We then introduce a larger class of polymatroids for which an analogous theorem holds. Finally, we obtain the same result for lattice path matroids as a corollary. [Copyright &y& Elsevier]

Let L be a supersolvable lattice with nonzero Möbius function. We show that the order complex of any rank-selected subposet of L admits a convex-ear decomposition. This proves many new inequalities for the h-vectors of such complexes, and shows that their g-vectors are Mvectors. [ABSTRACT FROM AUTHOR]

We investigate properties of rim-finite subsets of the plane (those which have topological bases whose elements have finite boundaries), which are also arc-free. Recently (see [K. Bouhjar, J.J. Dijkstra, Preprint], [K. Bouhjar, J.J. Dijkstra, J. van Mill, Topology Appl., to appear], [M.N. Charatonik, W.J. Charatonik, Comment. Math. Univ. Carolin., to appear], [D.L. Fearnley, J.W. Lamoreaux, Proc. Amer. Math. Soc., to appear] and [L.D. Loveland, S.M. Loveland, Houston J. Math. 23 (1997) 485–497]) there has been considerable research regarding n-point sets (sets which intersect each line in exactly n-points). These spaces are rim-finite (since the interior of a triangle has its boundary contained in a union of three lines, each of which has n points of the space), and our investigation provides a direction to generalize them. One of our main theorems seems to generalize all known results regarding the dimension of n-point sets (see, for example, [K. Bouhjar, J.J. Dijkstra, J. van Mill, Topology Appl., to appear], [D.L. Fearnley, J.W. Lamoreaux, Proc. Amer. Math. Soc., to appear] and [J. Kulesza, Proc. Amer. Math. Soc. 116 (1992) 551–553]), and beyond that has, as corollaries, the solutions to problems of Bouhjar and Dijkstra [Preprint], and L.D. Loveland and S.M. Loveland [Houston J. Math. 23 (1997) 485–497]. In Bouhjar and Dijkstra [Preprint] it is asked if all n-point sets which are arc-free must be zero-dimensional, and our result gives a positive answer. In [L.D. Loveland, S.M. Loveland, Houston J. Math. 23 (1997) 485–497] it is asked whether a connected 2-GM set must contain an arc, and again we give a positive answer.Another main theorem states that if X is a subset of R 2 such that there is a nonnegative integer n so that every straight interval of length 1 has a local basis of open sets with boundaries which intersect X in a set of cardinality less than or equal to n, then either X is zero-dimensional or X contains an arc. We produce an example which demonstrates that, essentially, our theorem cannot be improved. The “straight interval of length 1” cannot be replaced by “point”, because our example has a base of open sets whose boundaries have cardinality less than or equal to 72 and contains no arcs, yet has dimension 1. This example seems to be the first of a positive dimensional, rim-finite and arc-free separable metric space. [Copyright &y& Elsevier]

We characterize the lcm lattices that support a monomial ideal with a pure resolution. Given such a lattice, we provide a construction that yields a monomial ideal with that lcm lattice and whose minimal free resolution is pure. [ABSTRACT FROM AUTHOR]

We study connections among structures in commutative algebra, combinatorics, and discrete geometry, introducing an array of numbers, called Borel’s triangle, that arises in counting objects in each area. By defining natural combinatorial bijections between the sets, we prove that Borel’s triangle counts the Betti numbers of certain Borel-fixed ideals, the number of binary trees on a fixed number of vertices with a fixed number of “marked” leaves or branching nodes, and the number of pointed pseudotriangulations of a certain class of planar point configurations. [ABSTRACT FROM AUTHOR]

We introduce the notion of Q-Borel ideals: ideals that are closed under the Borel moves arising from a poset Q. We study decompositions and homological properties of these ideals, and offer evidence that they interpolate between Borel ideals and arbitrary monomial ideals. [ABSTRACT FROM AUTHOR]

Abstract: We use the notion of Borel generators to give alternative methods for computing standard invariants, such as associated primes, Hilbert series, and Betti numbers, of Borel ideals. Because there are generally few Borel generators relative to ordinary generators, this enables one to do manual computations much more easily. Moreover, this perspective allows us to find new connections to combinatorics involving Catalan numbers and their generalizations. We conclude with a surprising result relating the Betti numbers of certain principal Borel ideals to the number of pointed pseudo-triangulations of particular planar point sets. [Copyright &y& Elsevier]

We give a complete classification of free and non-free multiplicities on the A 3 braid arrangement. Namely, we show that all free multiplicities on A 3 fall into two families that have been identified by Abe-Terao-Wakefield (2007) and Abe-Nuida-Numata (2009). The main tool is a new homological obstruction to freeness derived via a connection to multivariate spline theory. [ABSTRACT FROM AUTHOR]

Given an ideal in a polynomial ring, we show that the asymptotic resurgence studied by Guardo, Harbourne, and Van Tuyl can be computed using integral closures. As a consequence, the asymptotic resurgence of an ideal is the maximum of finitely many ratios involving Waldschmidt-like constants (which we call skew Waldschmidt constants) defined in terms of Rees valuations. We use this to prove that the asymptotic resurgence coincides with the resurgence if the ideal is normal (that is, all of its powers are integrally closed). For a monomial ideal, the skew Waldschmidt constants have an interpretation involving the symbolic polyhedron defined by Cooper, Embree, H'a, and Hoefel. Using this intuition, we provide several examples of squarefree monomial ideals whose resurgence and asymptotic resurgence are different. [ABSTRACT FROM AUTHOR]

Let M and N be two monomials of the same degree, and let I be the smallest Borel ideal containing M and N. We show that the toric ring of I is Koszul by constructing a quadratic Gröbner basis for the associated toric ideal. Our proofs use the construction of graphs corresponding to fibers of the toric map. As a consequence, we conclude that the Rees algebra is also Koszul. [ABSTRACT FROM AUTHOR]