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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 Grobner 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.

SummaryWe study triples of labeled dice in which the relation “is a better die than” is nontransitive. Focusing on such triples with an additional symmetry we call balance, we prove that such tripl...

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 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., Comment: 21 pages

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.

Boij-S\"oderberg theory has had a dramatic impact on commutative algebra. We determine explicit Boij-S\"oderberg coefficients for ideals with linear resolutions and illustrate how these arise from the usual Eliahou-Kervaire computations for Borel ideals. In addition, we explore a new numerical decomposition for resolutions based on a row-by-row approach; here, the coefficients of the Betti diagrams are not necessarily positive. Finally, we demonstrate how the Boij-S\"oderberg decomposition of an arbitrary homogeneous ideal with a pure resolution changes when multiplying the ideal by a homogeneous polynomial.

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 domination parameters for certain classes of these clutters.

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., 27 pages, 2 tables, 5 figures

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., Comment: 23 pages, 2 figures; very minor changes in v2. To appear in J. Algebra

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 start with, the definition of modernistic is a common weakening of the definitions of semimodular and supersolvable. We thus obtain a unified proof that lattice in these classes are shellable. Subgroup lattices of solvable groups form another family of comodernistic lattices that were already proved 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., Comment: 25 pages, 3 figures, 1 table; v2 includes minor corrections for publication

We survey the Stanley–Reisner correspondence in combinatorial commutative algebra, describing fundamental applications involving Alexander duality, associated primes, f- and h-vectors, and Betti numbers of monomial ideals.

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 Hochster's Formula, these bounds recover and strengthen existing results on the homological connectivity of graph independence complexes., Add references for Chapter 5 and 6. Add remarks 5.7 and 5.8

In this paper, we give new upper bounds on the regularity of edge ideals whose resolutions are k-step linear; surprisingly, the bounds are logarithmic in the number of variables. We also give various bounds for the projective dimension of such ideals, generalizing other recent results. By Alexander duality, our results also apply to unmixed square-free monomial ideals of codimension two. We also discuss and connect these results to more classical topics in commutative algebra.

We show that the toric ideal of a lattice path polymatroid is generated by quadrics corresponding to symmetric exchanges, and give a monomial order under which these quadrics form a Gr\"obner basis. We then obtain an analogous result for lattice path matroids., Comment: 9 pages, 4 figures

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.