# Ordinary and Symbolic Powers of Ideals (17w5027)

Arriving in Oaxaca, Mexico Sunday, May 14 and departing Friday May 19, 2017

## Organizers

(Tulane University)

(McMaster University)

(Oklahoma State University)

## Objectives

The overall objective of this workshop is to bring together experts in commutative algebra, combinatorics, and algebraic geometry to discuss ordinary and symbolic powers from their different perspectives. Despite the diverse contexts in which they arise, the problems discussed in the previous section are closely related to each other, and already communication among mathematicians in different fields has led to significant progress. More specifically, our goals include:

1. Develop a better dialogue between combinatorialists and algebraists to exploit the interplay between the two subjects in studying ordinary and symbolic powers. For example, we would like to understand the properties and implications of a 2013 counterexample to the conjecture that persistence of associated primes holds for all squarefree monomial ideals. Commutative algebraists translated the algebraic conjecture to a stronger conjecture in graph theory in 2008. In 2013, graph theorists Kaiser, Skrekovski, and Stehlik found a counterexample to the graph-theoretic statement. Algebraists then verified that this work led to a counterexample (and subsequently, an infinite family of counterexamples) to the algebraic conjecture, which was previously widely believed. However, we understand very little algebraically about why the graph that Kaiser, Skrekovski, and Stehlik constructed should lead to the failure of persistence of associated primes. There are many interesting combinatorial and topological properties of the graph; what should they imply algebraically?
2. Moreover, it is especially surprising that the polarization operation in algebra often takes a monomial ideal without the persistence property and produces a squarefree monomial ideal for which the persistence property does hold, despite the fact that these two ideals share many other characteristics. We would like to understand why this occurs, and combinatorial ideas will be very important.

3. Understand the interplay among algebraic geometry, combinatorics, and commutative algebra in the conjectures of Harbourne and Huneke on containment of symbolic powers in ordinary powers. For example, one of Harbourne and Huneke's conjectures proposed that if $I$ is a homogeneous ideal in $K[\mathbb{P}^N]$, then $I^{(rN-(N-1))} \subseteq I^r$ for all $r$. Three algebraic geometers, Dumnicki, Szemberg, and Tutaj-Gasinska, found a counterexample that proved that $I^{(3)}$ is not always contained in $I^2$. The ideal arises from a special configuration of 12 points in $\mathbb{P}^2$ over the complex numbers. Bocci, Cooper, and Harbourne found a related counterexample in characteristic three. They note that the conjecture holds when $I$ is an ideal of points over a ground field of characteristic two and whenever $I$ is a monomial ideal. What, then, are the important geometric properties of the configuration that lead to failure of the conjecture in the counterexample, and when should the characteristic of the field play a role? What combinatorial properties do monomial ideals have that salvage the statement in that case?
4. Discuss algebraic approaches to the Conforti-Cornu\'ejols conjecture. This is a tantalizing conjecture from an algebraic perspective because monomial ideals with all their ordinary and symbolic powers equal are so special. When $I$ is generated in degree two, this occurs if and only if $I$ is the edge ideal of a bipartite graph, yet the situation is much less clear for more general monomial ideals.
5. There are a number of problems in the literature that would shed considerable light on the question. For example, suppose $I$ is a squarefree monomial ideal. If $I^{(q)} = I^q$ for all $q \gg 0$, what algebraic and combinatorial properties does $I$ have? For what $\ell$ would $I^{(q)} = I^q$ for all $0 \le q \le \ell$ imply that $I^{(q)}=I^q$? Any squarefree monomial ideal is the edge ideal of some hypergraph. If $I^{(q)} \neq I^q$ for some $q$, what combinatorial properties of the hypergraph give us information about the least such $q$ where equality fails?

Our list of proposed participants includes a diverse group of researchers, providing a good representation of researchers from algebraic geometry, combinatorics, and commutative algebra, and ensuring that we have a good balance of mathematicians from different backgrounds and at a variety of career stages. Our hope is to structure our days to maximize the potential for participants both to learn about problems and to work with each other on conjectures. Prior to the meeting, we would encourage participants to suggest relevant problems to work on during the meeting. At BIRS, we would likely have morning talks that include identification of key problems followed by afternoon work sessions in which researchers split into groups to collaborate especially with some mathematicians from different research areas, maintaining flexibility so that people could float between groups if desired. Moreover, we would invite a number of mathematicians who are experts in computer algebra systems and can write code quickly, allowing the group to test conjectures and get evidence for different lines of attack on the spot.