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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 combinatorial objects like graphs or hypergraphs in terms of the original combinatorial objects. It also addresses similar questions concerning our understanding of the Castelnuovo-Mumford regularity of powers of combinatorially defined ideals in terms of the associated combinatorial data. From a more geometric point of view, the book considers how the relations between symbolic and regular powers can be interpreted in geometrical terms. Other topics covered include aspects of Waring type problems, symbolic powers of an ideal and their invariants (e.g., the Waldschmidt constant, the resurgence), and the persistence of associated primes.

### Chapter 1. Associated Primes of Powers of Ideals

Abstract
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.
Enrico Carlini, Huy Tài Hà, Brian Harbourne, Adam Van Tuyl

### Chapter 2. Associated Primes of Powers of Squarefree Monomial Ideals

Abstract
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.
Enrico Carlini, Huy Tài Hà, Brian Harbourne, Adam Van Tuyl

Abstract
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.
Enrico Carlini, Huy Tài Hà, Brian Harbourne, Adam Van Tuyl

### Chapter 4. Regularity of Powers of Ideals and the Combinatorial Framework

Abstract
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.
Enrico Carlini, Huy Tài Hà, Brian Harbourne, Adam Van Tuyl

### Chapter 5. Problems, Questions, and Inductive Techniques

Abstract
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.
Enrico Carlini, Huy Tài Hà, Brian Harbourne, Adam Van Tuyl

### Chapter 6. Examples of the Inductive Techniques

Abstract
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.
Enrico Carlini, Huy Tài Hà, Brian Harbourne, Adam Van Tuyl

Abstract
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). The core of given arguments in these works is an understanding of ideals of the form I q+1 : 〈M〉, where I = I(G) is the edge ideal of a simple graph G and M is a minimal generator of I q.
Enrico Carlini, Huy Tài Hà, Brian Harbourne, Adam Van Tuyl

### Chapter 8. The Containment Problem: Background

Abstract
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 ideal”. For an algebraist it typically means being given generators of the ideal. For a geometer it often means being given a locus of points (or a scheme) in projective space, the ideal then being all elements of the polynomial ring which vanish on the given locus or scheme. Determining generators for the ideal defining a scheme sometimes requires significant effort, and if given generators a geometer will usually want to know what vanishing locus they cut out. Thus while both algebraists and geometers study ideals, their starting points are different.
Enrico Carlini, Huy Tài Hà, Brian Harbourne, Adam Van Tuyl

### Chapter 9. The Containment Problem

Abstract
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 containment hold for a given Z.
Enrico Carlini, Huy Tài Hà, Brian Harbourne, Adam Van Tuyl

### Chapter 10. The Waldschmidt Constant of Squarefree Monomial Ideals

Abstract
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, we will also give a combinatorial interpretation of this invariant. Throughout this chapter, $$R = \mathbb {K}[x_1,\ldots ,x_n]$$ is a polynomial ring over a field $$\mathbb {K}$$, where $$\mathbb {K}$$ has characteristic zero and is algebraically closed. All ideals I ⊆ R will be assumed to be homogeneous, and in most cases, I will be a squarefree monomial ideal.
Enrico Carlini, Huy Tài Hà, Brian Harbourne, Adam Van Tuyl

### Chapter 11. Symbolic Defect

Abstract
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 ideals. Throughout this lecture, we will assume that $$R = \mathbb {K}[x_1,\ldots ,x_n]$$ is a polynomial ring over an algebraically closed field of characteristic zero, and I will be a homogeneous ideal of R.
Enrico Carlini, Huy Tài Hà, Brian Harbourne, Adam Van Tuyl

Abstract
The recent survey and the lecture notes of Grifo provide more information on symbolic powers and the containment problem for ideals.
Enrico Carlini, Huy Tài Hà, Brian Harbourne, Adam Van Tuyl

### Chapter 13. Unexpected Hypersurfaces

Abstract
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 context (such as special properties of line arrangements, or of cones, or of characteristic p > 0). This currently makes presenting an exposition of reasonable length futile. Thus here we content ourselves with mostly just describing some of the ways unexpectedness arises, with pointers to the literature.
Enrico Carlini, Huy Tài Hà, Brian Harbourne, Adam Van Tuyl

Abstract
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 directions to take.
Enrico Carlini, Huy Tài Hà, Brian Harbourne, Adam Van Tuyl

### Chapter 15. An Introduction to the Waring Problem

Abstract
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!
Enrico Carlini, Huy Tài Hà, Brian Harbourne, Adam Van Tuyl

### Chapter 16. Algebra of the Waring Problem for Forms

Abstract
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 varieties of veronese varieties and parameter spaces for Gorenstein ideals. In The curves seminar at Queen’s, vol 10, pp 2–114, 1996).
Enrico Carlini, Huy Tài Hà, Brian Harbourne, Adam Van Tuyl

### Chapter 17. More on the Waring Problem

Abstract
In this chapter, we continue to explore problems related to the Waring problem introduced in the last two chapters.
Enrico Carlini, Huy Tài Hà, Brian Harbourne, Adam Van Tuyl

Abstract
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.
Enrico Carlini, Huy Tài Hà, Brian Harbourne, Adam Van Tuyl

### Chapter 19. Proposed Research Problems

Abstract
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 initial suggestions to guide their own research.
Enrico Carlini, Huy Tài Hà, Brian Harbourne, Adam Van Tuyl

### Chapter 20. The Art of Research

Abstract
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 of projects, we shifted our focus from lecturing to a focus on mentoring the groups. Not only did we suggest how to make progress on their specific projects, but we also gave more general advice on how to do research and how to present the results.
Enrico Carlini, Huy Tài Hà, Brian Harbourne, Adam Van Tuyl