Tai Huy Ha

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• PhD
• Professor (Full) at Tulane University

The pure $O$-sequences of the form $(1,a,a,\ldots)$ are classified.

We develop the notions of Newton non-degenerate (NND) ideals and Newton polyhedra for regular local rings. These concepts were first defined in the context of complex analysis. We show that the characterization of NND ideals via their integral closures known in the analytical setting extends to regular local rings. We use the limiting body $\mathca...

Let $G$ be a finite graph and $\kappa(G)$ the vertex connectivity of $G$. A chordal graph $G$ is called chordal$^*$ if no vertex of $G$ is adjacent to all other vertices of $G$. Using the syzygy theory in commutative algebra, it is proved that every chordal$^*$ graph $G$ on $n$ vertices satisfies $\kappa(G) \leq (n - 1) - \lceil2\sqrt{n}-2\,\rceil$...

We show that the asymptotic regularity of a graded family $(I_n)_{n \ge 0}$ of homogeneous ideals in a standard graded algebra, i.e., the limit $\lim\limits_{n \rightarrow \infty} \text{reg } I_n/n$, exists in several cases; for example, when the family $(I_n)_{n \ge 0}$ consists of artinian ideals, or Cohen-Macaulay ideals of the same codimension,...

We identify several classes of monomial ideals that possess minimal generalized Barile-Macchia resolutions. These classes of ideals include generic monomial ideals, monomial ideals with linear quotients, and edge ideals of hypertrees. We also characterize connected unicyclic graphs whose edge ideals are bridge-friendly and, in particular, have mini...

Let $G$ be a gapfree graph and let $I(G)$ be its edge ideal. An open conjecture of Nevo and Peeva states that $I(G)^q$ has a linear resolution for $q\gg 0$. We investigate a stronger conjecture that if $I(G)^q$ has linear quotients for some integer $q$, then $I(G)^{q+1}$ also has linear quotients. We give a partial solution to this conjecture. It i...

We discuss how to understand the asymptotic resurgence number of a pair of graded families of ideals from combinatorial data of their associated convex bodies. When the families consist of monomial ideals, the convex bodies being considered are the Newton-Okounkov bodies of the families. When ideals in the second family are classical invariant idea...

Let I = { I k } k ∈ N \mathcal {I}=\{I_k\}_{k \in \mathbb {N}} be a graded family of monomial ideals. We use the Newton–Okounkov body of I \mathcal {I} to: (a) give a characterization for the Noetherian property of the Rees algebra of the family I \mathcal {I} ; and (b) present a combinatorial interpretation for the analytic spread of I \mathcal {I...

We define the resurgence and asymptotic resurgence numbers associated to a pair of graded families of ideals in a Noetherian ring. These notions generalize the well-studied resurgence and asymptotic resurgence of an ideal in a polynomial ring. We examine when these invariant are finite and rational. We investigate situations where these invariant c...

Let \({\mathbb {k}}\) be a field, let A and B be polynomial rings over \({\mathbb {k}}\), and let \(S= A \otimes _{\mathbb {k}}B\). Let \(I \subseteq A\) and \(J \subseteq B\) be monomial ideals. We establish a binomial expansion for rational powers of \(I+J \subseteq S\) in terms of those of I and J. Particularly, for \(u \in {\mathbb Q}_+\), we p...

We use initially regular sequences that consist of linear sums to explore the depth of R/I2, when I is a monomial ideal in a polynomial ring R. We give conditions under which these linear sums form regular or initially regular sequences on R/I2. We then obtain a criterion for when depthR/I2>1 and a lower bound on depthR/I2.

The purpose of this corrigendum is to correct a mistake in Section 5 of the aforementioned paper [1].

Let $k$ be a field, let $A$ and $B$ be polynomial rings over $k$, and let $S = A \otimes_k B$. Let $I$ and $J$ be monomial ideals in $A$ and $B$, respectively. We establish a binomial expansion for rational powers of $I+J \subseteq S$ in terms of those of $I$ and $J$. We give a sufficient condition for this formula to hold for the integral closures...

We investigate containment statements between symbolic and ordinary powers and bounds on the Waldschmidt constant of defining ideals of points in projective spaces. We establish the stable Harbourne conjecture for the defining ideal of a general set of points. We also prove Chudnovsky’s Conjecture and the stable version of the Harbourne–Huneke cont...

In 1999, Herzog and Hibi introduced componentwise linear ideals. A homogeneous ideal I is componentwise linear if for all nonnegative integers d, the ideal generated by the homogeneous elements of degree d in I has a linear resolution. For square-free monomial ideals, componentwise linearity is related via Alexander duality to the property of being...

We use initially regular sequences that consist of linear sums to explore the depth of $R/I^2$, when $I$ is a monomial ideal in a polynomial ring $R$. We give conditions under which these linear sums form regular or initially regular sequences on $R/I^2$. We then obtain a criterion for when $\depth R/I^2>1$ and a lower bound on $\depth R/I^2$.

There are two different notions for symbolic powers of ideals existing in the literature, one defined in terms of associated primes, the other in terms of minimal primes. Elaborating on an idea known to Eisenbud, Herzog, Hibi, and Trung, we interpret both notions of symbolic powers as suitable saturations of the ordinary powers. We prove a binomial...

In 1999 Herzog and Hibi introduced componentwise linear ideals. A homogeneous ideal $I$ is componentwise linear if for all non-negative integers $d$, the ideal generated by the homogeneous elements of degree $d$ in $I$ has a linear resolution. For square-free monomial ideals, componentwise linearity is related via Alexander duality to the property...

Let $\mathcal{I} = \{I_k\}_{k \in \mathbb{N}}$ be a graded family of monomial ideal. We use the Newton-Okounkov body of $\mathcal{I}$ to: (a) give a characterization for the Noetherian property of the Rees algebra of the family; and (b) present a combinatorial interpretation for the analytic spread of the family. We also apply these results to inve...

A Correction to this paper has been published: https://doi.org/10.1007/s40306-021-00447-w

We investigate Demailly's Conjecture for a general set of sufficiently many points. Demailly's Conjecture generalizes Chudnovsky's Conjecture in providing a lower bound for the Waldschmidt constant of a set of points in projective space. We also study a containment between symbolic and ordinary powers conjectured by Harbourne and Huneke that in par...

We survey recent studies and results on the following problem: which numerical functions can be the depth function of powers and symbolic powers of homogeneous ideals.

We show that attaching a whisker (or a pendant) at the vertices of a cycle cover of a graph results in a new graph with the following property: all symbolic powers of its cover ideal are Koszul or, equivalently, componentwise linear. This extends previous work where the whiskers were added to all vertices or to the vertices of a vertex cover of the...

We prove bounds on the saturation degrees of homogeneous ideals (and their powers) defining smooth complex projective varieties. For example, we show that a classical statement due to Macualay for zero-dimensional complete intersection ideals holds for any smooth variety. For curves, we bound the saturation degree of powers in terms of the regulari...

Let G be a finite simple graph on n vertices, that contains no isolated vertices, and let \(I(G) \subseteq S = K[x_1, \dots , x_n]\) be its edge ideal. In this paper, we study the pair of integers that measure the projective dimension and the regularity of S/I(G). We show that if \({{\,\mathrm{pd}\,}}(S/I(G))\) attains its minimum possible value \(...

We present a proof of the celebrated result due to Alexander and Hirschowitz which determines when a general set of double points in $\mathbb P^n$ has the expected Hilbert function. Our intended audience are Commutative Algebraists who may be new to interpolation problems. In particular, the main aim of our presentation is to provide a self-contain...

We present a proof of a celebrated theorem of Alexander and Hirschowitz determining when a general set of double points in \(\mathbb {P}^n\) has the expected Hilbert function. Our intended audience are Commutative Algebraists who may be new to interpolation problems. In particular, the main aim of our presentation is to provide a self-contained pro...

We survey recent studies and results on the following problem: for which function \(f: {\mathbb N} \rightarrow {\mathbb Z}_{\ge 0}\) does there exist a homogeneous ideal Q in a polynomial ring S such that (a) depthS∕Qt = f(t) for all t ≥ 1, or (b) depthS∕Q(t) = f(t) for all t ≥ 1?KeywordsDepthProjective dimensionHomogeneous idealMonomial idealPower...

We investigate the resurgence and asymptotic resurgence numbers of fiber products of projective schemes. Particularly, we show that while the asymptotic resurgence number of the k-fold fiber product of a projective scheme remains unchanged, its resurgence number could strictly increase.

A squarefree monomial ideal is called an $f$-ideal if its Stanley-Reisner and facet simplicial complexes have the same $f$-vector. We show that $f$-ideals generated in a fixed degree have asymptotic density zero when the number of variables goes to infinity. We also provide novel algorithms to construct $f$-ideals generated in small degrees.

We investigate Demailly's Conjecture for a general set of sufficiently many points. Demailly's Conjecture generalizes Chudnovsky's Conjecture in providing a lower bound for the Waldschmidt constant of a set of points in projective spaces. We also study a containment between symbolic and ordinary powers conjectured by Harbourne and Huneke that in pa...

We show that attaching a whisker (or a pendant) at the vertices of a cycle cover of a graph results in a new graph with the following property: all symbolic powers of its cover ideal are Koszul or, equivalently, componentwise linear. This extends previous work where the whiskers were added to all vertices or to the vertices of a vertex cover of the...

We derive two general bounds for the depths of powers of squarefree monomial ideals corresponding to hyperforests. These bounds generalize known bounds for the depths of squarefree monomial ideals, which were given in terms of the edgewise domination number of the corresponding hypergraphs and the lengths of initially regular sequences with respect...

In this chapter we introduce the symbolic defect of a homogeneous ideal. This concept was introduced recently by Galetto, Geramita, Shin, and Van Tuyl. There are a number of interesting questions one can ask about this invariant, and hopefully this chapter will inspire you to investigate the symbolic defect of your favourite family of homogeneous i...

The primary decomposition of ideals in Noetherian rings is a fundamental result in commutative algebra and algebraic geometry. It is a far reaching generalization of the fact that every positive integer has a unique factorization into primes. We recall one version of this result.

The most effective tool to deal with the Waring problem for forms is the so-called Apolarity Lemma (see Iarrobino and Kanev and the lecture notes of Carlini, Grieve, and Oeding). To introduce the Apolarity Lemma we need to briefly review some notion from apolarity theory, following Geramita (Inverse systems of fat points: Waring’s problem, secant v...

The study of ideals underlies both algebra and geometry. For example, the study of homogeneous ideals in polynomial rings is an aspect of both commutative algebra and of algebraic geometry. In both cases, given an ideal, one wants to understand how the ideal behaves. One way in which algebra and geometry differ is in what it means to be “given an i...

Castelnuovo-Mumford regularity (or simply regularity) is an important invariant in commutative algebra and algebraic geometry. Computing or finding bounds for the regularity is a difficult problem. In the next three chapters, we shall address the regularity of ordinary and symbolic powers of squarefree monomial ideals.

Given a fat point scheme \(Z=m_1p_1+\cdots +m_sp_s\subset {\mathbb P}^N\), the containment problem for Z is to determine for which r and m the containment (I(Z))(m) ⊆ (I(Z))r holds. In this section we present some initial results for the containment problem, and we define an asymptotic quantity, the resurgence, that measure to what extent the conta...

The notion of unexpected hypersurfaces is quite new; research on this topic is growing rapidly but an orderly unified perspective has not yet been achieved. The phenomenon itself can be defined succinctly, but the many examples of unexpectedness that are now known seem to arise in different ways, depending on specific properties available in each c...

In the previous chapter, we looked at a result of Brodmann (Theorem 1.4) concerning the associated primes of powers of ideals. This theorem inspires a number of natural questions. To state these questions, we introduce some suitable terminology.

The recent survey and the lecture notes of Grifo provide more information on symbolic powers and the containment problem for ideals.

In these short chapters we just started to explore a very large and intriguing field of mathematics. During the school, our focus was on homogeneous polynomials. However, this is just one of the many landmarks of the subject.

The papers (Cook II et al., Compos Math 154(10):2150–2194, 2018; Harbourne et al., Mich Math J (to appear). arXiv:1805.10626) are essential reading for this section. The references in these papers give additional papers that may be useful to look at. This research topic is very new but of growing interest, so there are a lot of possible unexplored...

In this chapter, we continue to explore problems related to the Waring problem introduced in the last two chapters.

As is standard at PRAGMATIC, the participants were divided into small groups to work on open research problems, based upon their ranked preferences of the problems. In this iteration of PRAGMATIC, we, as instructors, presented a number of open research problems (see the previous chapter) and some suggested approaches. After the initial assignment o...

An ubiquitous theme in mathematics is the rewriting of mathematical objects. This is usually done to reveal underlying properties, to classify, to solve problems or just for aesthetic reasons!

In this chapter, we present a number of open problems and questions for edge ideals of graphs. These problems and questions fall under the umbrella of Problem 4.8. We shall also discuss inductive techniques that have been applied in the literature.

The last two chapters introduced the Waldschmidt constant of a homogeneous ideal of set of (fat) points and some of its properties. In fact, the definition of the Waldschmidt constant makes sense for any homogeneous ideal. In this chapter we explain how to compute this invariant in the case of squarefree monomial ideals. In the case of edge ideals,...

In this chapter, we present detailed proofs of a few stated results to illustrate how the inductive techniques introduced in the last chapter can be applied to the study of the regularity of powers of edge ideals.

In this chapter we collect together the projects that were initially presented to the students of PRAGMATIC. Each project was related to the theme of the workshop, i.e., “Powers of ideals and ideals of powers”. Many of these questions are open-ended (and perhaps not well-defined). The intention, however, was to give each group of students some init...

As we have hopefully demonstrated in the last two chapters, Question 1.2 has motivated a number of interesting results, including some nice connections with combinatorics. Although we cannot cover all of the existing literature, here are some suggested references for further reading.

Banerjee’s inductive method has also been successfully applied by various authors, such as Alilooee, Beyarslan, and Selvaraja, Jayanthan, Narayanan, and Selvaraja, and Moghimian, Norouzi Seyed Fakhari, and Yassemi, pushing Theorems 5.1 and 5.2 further to the classes of unicyclic graphs (see Theorem 5.3) and very well-covered graphs (see Theorem 5.4...

In this paper, we investigate containment statements between symbolic and ordinary powers and bounds on the Waldschmidt constant of defining ideals of points in projective spaces. We establish the stable Harbourne conjecture for the defining ideal of a general set of points. We also prove Chudnovsky's Conjecture and the stable version of the Harbou...

Let G be a graph and let I=I(G) be its edge ideal. When G is unicyclic, we give a decomposition of symbolic powers of I in terms of its ordinary powers. This allows us to explicitly compute the Waldschmidt constant and the resurgence number of I. When G is an odd cycle, we explicitly compute the regularity of I(s) for all s∈N. In doing so, we also...

Let $I$ and $J$ be nonzero ideals in two Noetherian algebras $A$ and $B$ over a field $k$. We study algebraic properties and invariants of symbolic powers of the ideal $I+J$ in $A\otimes_k B$. Our main technical result is the binomial expansion $(I+J)^{(n)} = \sum_{i+j = n} I^{(i)} J^{(j)}$ for all $n > 0$. Moreover, we show that if char$(k) = 0$ o...

For an arbitrary ideal I in a polynomial ring R we define the notion of initially regular sequences on R/I. These sequences share properties with regular sequences. In particular, the length of an initially regular sequence provides a lower bound for the depth of R/I when I is homogeneous. Using combinatorial information from the initial ideal of I...

We settle a conjecture of Herzog and Hibi, which states that the function d e p t h S / Q n \mathrm {depth}\, S/Q^n , n ≥ 1 n \ge 1 , where Q Q is a homogeneous ideal in a polynomial ring S S , can be any convergent numerical function. We also give a positive answer to a longstanding open question of Ratliff on the associated primes of powers of id...

We introduce an invariant, associated to a coherent sheaf of graded modules over a projective morphism of schemes, which controls when sheaf cohomology can be passed through the given morphism. We then use this invariant to estimate the stability indexes of the regularity and a∗-invariant of powers of homogeneous ideals. Specifically, for an equige...

Let $G$ be a finite simple graph on $n$ vertices, that contains no isolated vertices, and let $I(G) \subseteq S = K[x_1, \dots, x_n]$ be its edge ideal. In this paper, we study the pair of integers that measure the projective dimension and the regularity of $S/I(G)$. We show that if the projective dimension of $S/I(G)$ attains its minimum value $2\...

This book discusses regular powers and symbolic powers of ideals from three perspectives– algebra, combinatorics and geometry – and examines the interactions between them. It invites readers to explore the evolution of the set of associated primes of higher and higher powers of an ideal and explains the evolution of ideals associated with combinato...

We present an algebraic algorithm to detect the existence of and to list all indecomposable even circuits in a given graph. We also discuss an application of our work to the study of directed cycles in digraphs.

We present an algebraic algorithm to detect the existence of and to list all indecomposable even circuits in a given graph. We also discuss an application of our work to the study of directed cycles in digraphs.

We derive two general bounds for the depths of powers of squarefree monomial ideals corresponding to hyperforests. These bounds generalize known bounds for the depths of squarefree monomial ideals, which were given in terms of the edgewise domination number of the corresponding hypergraphs and the lengths of initially regular sequences with respect...

We settle a conjecture of Herzog and Hibi, which states that the function depth $S/Q^n$, $n \ge 1$, where $Q$ is a homogeneous ideal in a polynomial ring $S$, can be any convergent numerical function. We also give a positive answer to a long-standing open question of Ratliff on the associated primes of powers of ideals.

We study the relationship between depth and regularity of a homogeneous ideal I and those of (I,f) and I:f, where f is a linear form or a monomial. Our results has several interesting consequences on depth and regularity of edge ideals of hypegraphs and of powers of ideals.

We present a close relationship between matching number, covering numbers and their fractional versions in combinatorial optimization and ordinary powers, integral closures of powers, and symbolic powers of monomial ideals. This relationship leads to several new results and problems on the containments between these powers.

We survey recent studies on the Castelnuovo–Mumford regularity of edge ideals of graphs and their powers. Our focus is on bounds and exact values of \(\text {reg}I(G)\) and the asymptotic linear function \(\text {reg}I(G)^q\), for \(q \ge 1\), in terms of combinatorial data of the given graph G.

Let G=(V,E) be a simple graph. We investigate the Cohen–Macaulayness and algebraic invariants, such as the Castelnuovo-Mumford regularity and the projective dimension, of the toric ring k[G] via those of toric rings associated to induced subgraphs of G.

We establish Chudnovsky's conjecture for a general set of points in projective spaces. This conjecture provides a lower bound for the least degree of a homogeneous polynomial that vanishes at the given set of points with a given multiplicity. We also investigate the resurgence and asymptotic resurgence numbers of fiber products of projective scheme...

We introduce an invariant, associated to a coherent sheaf over a projective morphism of schemes, which controls when sheaf cohomology can be passed through the given morphism. We then use this invariant to estimate the stability indexes of the regularity and a*-invariant of powers of homogeneous ideals.

Let 𝒟 be a weighted oriented graph and let I(𝒟) be its edge ideal. Under a natural condition that the underlying (undirected) graph of 𝒟 contains a perfect matching consisting of leaves, we provide several equivalent conditions for the Cohen–Macaulayness of I(𝒟). We also completely characterize the Cohen–Macaulayness of I(𝒟) when the underlying gra...

For an arbitrary ideal $I$ in a polynomial ring $R$ and any term order, we define a new notion of initially regular sequences on $R/I$. These sequences share properties with regular sequences, and are relatively easy to construct using information obtained from the initial ideal of $I$. For any ideal $I$ we give a combinatorial description of eleme...

We present a close relationship between matching number, covering numbers and their fractional versions in combinatorial optimization and ordinary powers, integral closures of powers, and symbolic powers of monomial ideals. This relationship leads to several new results and problems on the containments between these powers.

Let $\mathcal{D}$ be a weighted oriented graph and let $I(\mathcal{D})$ be its edge ideal. Under a natural condition that the underlying (undirected) graph of $\mathcal{D}$ contains a perfect matching consisting of leaves, we provide several equivalent conditions for the Cohen-Macaulayness of $I(\mathcal{D})$. We also completely characterize the Co...

Let $G$ be a graph and let $I = I(G)$ be its edge ideal. When $G$ is unicyclic, we give a decomposition of symbolic powers of $I$ in terms of its ordinary powers. This allows us to explicitly compute the Waldschmidt constant, the resurgence number, and the symbolic defect for $I$. When $G$ is an odd cycle, we explicitly compute the regularity of $I...

Let $I = I(G)$ be the edge ideal of a graph $G$. We give various general upper bounds for the regularity function $\text{ reg } I^s$, for $s \ge 1$, addressing a conjecture made by the authors and Alilooee. When $G$ is a gap-free graph and locally of regularity 2, we show that $\text{ reg } I^s = 2s$ for all $s \ge 2$. This is a slightly weaker ver...

We survey recent studies on the Castelnuovo-Mumford regularity of edge ideals of graphs and their powers. Our focus is on bounds and exact values of $\text{ reg } I(G)$ and the asymptotic linear function $\text{ reg } I(G)^q$, for $q \geq 1,$ in terms of combinatorial data of the given graph $G.

We study the relationship between depth and regularity of a homogeneous ideal I and those of (I,f) and I:f, where f is a linear form or a monomial. Our results has several interesting consequences on depth and regularity of edge ideals of hypegraphs and of powers of ideals.

Let $G = (V,E)$ be a simple graph. We give a necessary condition for the toric ring $k[G]$ associated to $G$ to be Cohen-Macaulay. Particularly, we investigate a "forbidden" structure in $G$ that prevents $k[G]$ from being Cohen-Macaulay. We also give a bound for the regularity and projective dimension of $k[G]$ in terms of those of its induced sub...

Let $I$ and $J$ be nonzero ideals in two Noetherian algebras $A$ and $B$ over a field $k$. Let $I+J$ denote the ideal generated by $I$ and $J$ in $A\otimes_k B$. We prove the following expansion for the symbolic powers: $$(I+J)^{(n)} = \sum_{i+j = n} I^{(i)} J^{(j)}.$$ If $A$ and $B$ are polynomial rings and if chara$(k) = 0$ or if $I$ and $J$ are...

Given arbitrary homogeneous ideals I and J in polynomial rings A and B over a field k, we investigate the depth and the Castelnuovo–Mumford regularity of powers of the sum I+J\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgree...

In attempting to understand how combinatorial modifications alter algebraic properties of monomial ideals, several authors have investigated the process of adding "whiskers" to graphs. In this paper, we study a similar construction for building a simplicial complex Delta(chi) from a coloring chi of a subset of the vertices of Delta and give necessa...

Let $G$ be a simple graph on $n$ vertices. Let $H$ be either the complete graph $K_m$ or the complete bipartite graph $K_{r,s}$ on a subset of the vertices in $G$. We show that $G$ contains $H$ as a subgraph if and only if $\beta_{i,\alpha}(H) \le \beta_{i,\alpha}(G)$ for all $i \ge 0$ and $\alpha \in \mathbb{Z}^n$. In fact, it suffices to consider...

Let G be a graph and let $$I = I(G)$$I=I(G) be its edge ideal. In this paper, when G is a forest or a cycle, we explicitly compute the regularity of $$I^s$$Is for all $$s \ge 1$$s≥1. In particular, for these classes of graphs, we provide the asymptotic linear function $${{\mathrm{reg}}}(I^s)$$reg(Is) as $$s \gg 0$$s≫0, and the initial value of s st...

We study the question of when 0-1 polytopes are normal or, equivalently, have the integer decomposition property. In particular, we shall associate to each 0-1 polytope a labeled hypergraph, and examine the equality between its Ehrhart and polytopal rings via the combinatorial structures of the labeled hypergraph.

Let $(R,\mathfrak{m})$ be a Noetherian local ring of dimension $d > 0$. Let $I_\bullet = \{I_n\}_{n \in \mathbb{N}}$ be a graded family of $\mathfrak{m}$-primary ideals in $R$. We examine how far off from a polynomial can the length function $\ell_R(R/I_n)$ be asymptotically. More specifically, we show that there exists a constant $\gamma > 0$ such...

Let G be a graph and let I = I(G) be its edge ideal. In this paper, when G is a forest or a cycle, we explicitly compute the regularity of I^s for all s > 0. In particular, for this class of graphs, we provide the asymptotic linear function reg(I^s) as s > 0, and the initial value of s starting from which reg(I^s) attains its linear form. We also g...

In a recent work, Kaiser, Stehl\'ik and \v{S}krekovski provide a family of critically 3-chromatic graphs whose expansions do not result in critically 4-chromatic graphs, and thus give counterexamples to a conjecture of Francisco, Ha and Van Tuyl. The cover ideal of the smallest member of this family also gives a counterexample to the persistence an...

We survey a number of recent studies of the Castelnuovo-Mumford regularity of squarefree monomial ideals. Our focus is on bounds and exact values for the regularity in terms of combinatorial data from associated simplicial complexes and/or hypergraphs.

We investigate symbolic and regular powers of monomial ideals. For a square-free monomial ideal I in k[x_0, ..., x_n] we show I^(t(m+e-1)-e+r) is a subset of M^{(t-1)(e-1)+r-1}(I^(m))^t for all positive integers m, t and r, where e is the big-height of I and M = (x_0, ..., x_n). This captures two conjectures (r=1 and r=e): one of Harbourne-Huneke a...

We study the question of when the Ehrhart and toric rings of 0-1 polytopes are the same. In particular, we shall associate to each 0-1 polytope a labeled hypergraph, and examine the equality between its Ehrhart and toric rings via the combinatorial structures of the labeled hypergraph.

We survey research relating algebraic properties of powers of squarefree monomial ideals to combinatorial structures. In particular, we describe how to detect important properties of (hyper)graphs by solving ideal membership problems and computing associated primes. This work leads to algebraic characterizations of perfect graphs independent of the...

We survey research relating algebraic properties of powers of squarefree monomial ideals to combinatorial structures. In particular, we describe how to detect important properties of (hyper)graphs by solving ideal membership problems and computing associated primes. This work leads to algebraic characterizations of perfect graphs independent of the...

In a 2008 paper, the first author and Van Tuyl proved that the regularity of the edge ideal of a graph G is at most one greater than the matching number of G. In this note, we provide a generalization of this result to any square-free monomial ideal. We define a 2-collage in a simple hypergraph to be a collection of edges with the property that for...