On Symbolic Powers of Ideals

Conference on Unexpected and Asymptotic Properties of Algebraic Varieties (aka BrianFest)

Mike Janssen

2023-08-11

Welcome and Introduction

April 11, 2013

First, I’d like to thank everyone for coming, and I’d especially like to thank the organizers for inviting me to speak.

Brian has been such a wonderful teacher and mentor to so many of us, and it’s delightful to be here–in part–to represent his Ph.D. students.

The photo here is unfortunately rather blurry, but is from the day of my defense, which, it seems, was successful.

Over a decade after graduating, a vivid memory I have of the courses I took from Brian is the way he would motivate and introduce new ideas–very naturally and in a way that was accessible to his audience. And in my role as a faculty member at a teaching-oriented undergraduate institution, it’s one of the very concrete ways in which Brian has had a lasting impact on my career.

That is also my goal for today. My hope is that even if you’re a Ph.D. student who hasn’t given a lot of thought to these questions that you’ll find this an accessible introduction to questions about symbolic powers of ideals.

So, let’s get started with a definition.

Exploring Symbolic Powers

General Definition

Definition 1 Let \(I\) be an ideal in a Noetherian ring \(R\), and \(m\ge 1\). Then the \(m\)-th symbolic power of \(I\), denoted \(I^{(m)}\), is the ideal \[ I^{(m)} = \bigcap\limits_{P\in\text{Ass}(I)} (I^m R_P \cap R), \] where \(R_P\) denotes the localization of \(R\) at the prime ideal \(P\).

Theorem 1 Let \(I\) be a radical ideal in a Noetherian ring \(R\) with minimal primes \(P_1, P_2, \ldots, P_s\). Then \(I = P_1 \cap P_2 \cap \cdots \cap P_s\), and \[ I^{(m)} = P_1^{(m)} \cap P_2^{(m)} \cap \cdots \cap P_s^{(m)}. \]

Theorem and an Example

Theorem 2 Let \(R\) be noetherian and suppose \(I\subseteq R\) is an ideal generated by a regular sequence. Then \(I^{(m)} = I^m\) for all \(m\ge 1\).

Example 1 (A point in \(\mathbb{P}^2\)) Let \(R = k[\mathbb{P}^2] = k[x, y, z]\) and \(p\in \mathbb{P}^2\). Then \(I = I(p)\) can be taken to be \(I = (x,y)\), and

\[ I^{(m)} = (x,y)^{(m)} = (x,y)^m. \]

Geometric Interpretation

Theorem 3 (Zariski, Nagata) Let \(k\) be a perfect field, \(R = k[x_0, x_1, \ldots, x_N]\), \(I\subseteq R\) a radical ideal, and \(X\subseteq \mathbb{P}^N\) the variety corresponding to \(I\). Then \(I^{(m)}\) is the ideal generated by forms vanishing to order at least \(m\) on \(X\).

Two Contexts

For the remainder, we’ll consider two types of ideals:

Ideals of (fat) points
Squarefree monomial ideals

The Containment Problem and Ideals of Points

Ideals of Points

Definition 2 If \(p_i\in \mathbb{P}^N\) and \(Z = m_1 p_1 + m_2 p_2 + \cdots m_s p_s\) is a fat points subscheme with \(I = I(Z)\), then

\[ I(Z) = I(p_1)^{m_1} \cap I(p_2)^{m_2} \cap \cdots \cap I(p_s)^{m_s}. \]

The symbolic powers of \(I = I(Z)\) are therefore

\[ I^{(m)} = I(mZ) = I(p_1)^{m m_1} \cap I(p_2)^{m m_2} \cap \cdots \cap I(p_s)^{m m_s}. \]

Our Question (First Draft)

Given a nontrivial homogeneous ideal \(I\subseteq k[x_0, \ldots, x_n]\), how do \(I^{(m)}\) and \(I^r\) compare?

Comparing Powers

Theorem 4 Let \(I\) be an ideal in a Noetherian ring \(R\). Then:

\(I^m \subseteq I^r\) if and only if \(m\ge r\).
\(I^{(m)} \subseteq I^{(r)}\) if and only if \(m\ge r\).
if \(R\) is a domain, \(I^m \subseteq I^{(r)}\) if and only if \(m\ge r\).
\(I^{(m)}\subseteq I^r\) implies \(m \ge r\), but the converse need not hold.

Our (General) Question (Final Draft)

Containment Problem. Given a nontrivial homogeneous ideal \(I\subseteq k[x_0, x_1, x_2, \ldots, x_N]\), for which \(m,r\) do we have \(I^{(m)}\subseteq I^r\)?

A Uniform Bound

Theorem 5 (Ein-Lazarsfeld-Smith (2001), Hochster-Huneke (2002), Ma-Schwede (2017), Murayama (2021)) Let \(R\) be a regular ring and \(I\) a radical ideal in \(R\) of big height \(e\). Then for all \(r\ge 1\), \(I^{(er)} \subseteq I^r\).

Corollary 1 Let \(I\) be a nontrivial homogeneous ideal in \(k[\mathbb{P}^N]\). If \(m \ge Nr\), then \(I^{(m)}\subseteq I^r\).

Question (Huneke).

When \(I = I(S)\) is the ideal defining any finite set \(S\) of points in \(\mathbb{P}^2\), is it true that \(I^{(3)}\subseteq I^2\)?

The first result I became aware of, back when I started on what would become my dissertation (this was approximately 2008-2009), was this result, which, at the time, had Lawrence Ein, Robert Lazarsfeld, and Karen Smith’s names (in characteristic 0) and Mel Hochster and Craig Huneke’s names (in equal characteristic \(p\)) attached to it. It has since been generalized, first by Linquan Ma and Karl Schwede and, more recently, Takumi Murayama, to the lovely result that we see here.

Note that this implies a uniform result for all ideals in an \(N\)-dimensional ring \(R\), as the big-height of any ideal is at most the dimension of the ring (really, for non-maximal ideals, at most 1 less than the dimension of the ring, since ordinary and symbolic powers of maximal ideals coincide).

We thus arrive at Huneke’s question: in the setting in which this is framed, we know that \(I^{(4)}\subseteq I^2\), and can find plenty of examples for which the ordinary and symbolic squares differ, so at the time the question was posed it was quite natural.

We further note that different versions of Huneke’s question has appeared in the literature, and I don’t doubt that they were all asked at one point or another.

So this was the setting in which Annika Denkert and I were starting to work in 2009-2010 or so. And I want to highlight some of the work Brian was doing around that time. This should not be taken to be in any way exhaustive, but hopefully will help orient us around the problem.

2000: I. Swanson investigated linear equivalence of \(P\)-adic and \(P\)-symbolic topologies.

This led to the work of Ein-Lazarsfeld-Smith (in equal characteristic 0), Hochster-Huneke (in equal characteristic \(p\)), Ma and Schwede (for radical ideals in excellent regular rings of mixed characteristic), and recently, Murayama in arbitrary regular rings of all characteristics.

Comparing Powers and Symbolic Powers of Ideals (2010; with C. Bocci)

Answered Huneke’s question in the affirmative for \(I(S)\) when \(S\) is a finite generic set of points in \(\mathbb{P}^2\).
Introduced the resurgence, \(\rho(I)\), the supremum of the ratios \(m/r\) for which \(I^{(m)}\not\subseteq I^r\), and calculated \(\rho\) for ideals of various point configurations in \(\mathbb{P}^2\).
Obtained bounds on \(\rho(I(Z))\) in terms of other invariants of \(I(Z)\).
Used these bounds to establish the sharpness of the uniform bound of Theorem 5.

One of the first papers I read when I was getting into my thesis work.

Brian and Cristiano answered Huneke’s question affirmatively for the ideal of a finite set of generic points in \(\mathbb{P}^2\).

They also introduced the resurgence of the ideal, which is an upper bound on ratios of integers for which the containment fails, and they constructed the first non-monomial example for which the resurgence was greater than 1. I think Cristiano may have more to say on this tomorrow.

At the same time, as calculating \(\rho\) is done more or less ad hoc, they obtained bounds on \(\rho\) in terms of other well-understood invariants such as the initial degree and Waldschmidt constants.

They furthermore showed that the uniform result of ELS-HH was sharp by constructing a sequence of schemes whose resurgences converge to the dimension of the ring. The schemes they construct are generalizations of the star configuration we looked at earlier.

Postulational invariants: “invariants that are determined by the Hilbert functions of \(I(Z)^{(m)}\). Thus these bounds are the same for any \(Z\) for which the Hilbert functions of \(I(Z)\) and its symbolic powers remain the same. This is useful since postulational data is reasonably accessible.” Made assumptions like \(r\alpha(I) > \alpha(I^{(m)})\), etc.

Postulational invariants used:

Initial degree, \(\alpha(I)\)
Castelnuovo-Mumford regularity \(\text{reg}(I)\)
Waldschmidt constant \(\widehat\alpha(I)\) (therein denoted with \(\gamma\))

The Resurgence of Ideals of Points and the Containment Problem (2010; with C. Bocci)

Theorem 6 (3.4) Assume the points \(p_1, \ldots, p_n\) lie on a smooth conic curve. Let \(I = I(Z)\) where \(Z = p_1 + \cdots + p_n\). Let \(m, r > 0\).

If \(n\) is even or \(n=1\), then \(I^{(m)}\subseteq I^r\) if and only if \(m\ge r\). In particular, \(\rho(I) = 1\).
If \(n > 1\) is odd, then \(I^{(m)}\subseteq I^r\) if and only if \((n+1)r - 1 \le nm\); in particular, \(\rho(I) = (n+1)/n\).

Conjecture 1 (B. Harbourne) Let \(I\subseteq k[\mathbb{P}^N]\) be a homogeneous ideal. Then \(I^{(m)}\subseteq I^r\) if \(m\ge rN - (N-1)\).

The theorem displayed is a representative example of others in the paper, and this line of inquiry overall. Statement 1 follows from the fact that such points form a complete intersection. The proof of Statement 2 relies on a few lemmata which are built on properties of linear systems on the variety obtained by blowing up the points.
The paper concludes by calculating \(\rho(I)\) for \(n\) points in general position, where \(6\le n \le 9\) (when considering \(n\le 5\) general points, they lie on a conic).
In joint work with Annika Denkert, which formed parts of our dissertations, we explored cases of points on a reducible conic.

Brian’s conjecture here is interesting. It’s a very natural extension of Huneke’s question It was verified in many cases:

Points in general position (in this paper)
Generic points in \(\mathbb{P}^2\) and \(\mathbb{P}^3\)
Codimension \(c\) star configurations in \(\mathbb{P}^N\)
Squarefree monomial ideals

Note, too, that it’s a very natural extension of Huneke’s question!

A counter-example to a question by Huneke and Harbourne (2013)

Dumnicki, Szemberg, and Tutaj-Gasińska construct a radical ideal \(\mathcal{I}\) of 12 points in \(\mathbb{P}^2\) for which \(\mathcal{I}^{(3)}\not\subseteq \mathcal{I}^2\).
The configuration is in the family of Fermat configurations over \(\mathbb{C}\).
This negatively answers Conjecture 1 that \(I^{(m)}\subseteq I^r\) if \(m\ge rN - (N-1)\) for homogeneous ideals in \(k[\mathbb{P}^N]\).

Their construction consists of 12 points formed using third roots of unity; the points are the intersection points of 9 lines such that exactly 3 lines pass through each of the points and exactly 4 points lie on a line.

S-G: every finite set of points in the Euclidean plane has a line that passes through exactly two of the points or a line that passes through all of them.

The additional points of the projective plane cannot help create non-Euclidean finite point sets with no ordinary line, as any finite point set in the projective plane can be transformed into a Euclidean point set with the same combinatorial pattern of point-line incidences. Therefore, any pattern of finitely many intersecting points and lines that exists in one of these two types of plane also exists in the other.

A real counterexample was announced shortly after this one.

Additionally, Brian and Alexandra then built on this work and developed a family of counterexamples in positive characteristic.

The answer to this version of Huneke’s question is thus ‘no’. But another version has him asking: if \(P\) is a prime ideal of height 2 in a regular local ring \(R\), is \(P^{(3)}\subseteq P^2\)?

With our remaining time, I want to say a bit about some related questions involving symbolic powers of squarefree monomial ideals.

Squarefree Monomial Ideals

Oberwolfach Mini-Workshop: Ideals of Linear Subspaces, Their Symbolic Powers and Waring Problems (2015)

Example and Definition

Definition 3 Let \(I\subseteq k[x_0, \ldots, x_N]\) be homogeneous. The initial degree of \(I\), denoted \(\alpha(I)\), is the least degree of a nonzero \(f\in I\).

Definition 4 The Waldschmidt constant, denoted \(\widehat\alpha(I)\), is the limit

\[ \widehat\alpha(I) := \lim\limits_{m\to\infty} \frac{\alpha(I^{(m)})}{m}. \]

Example

Example 2 (A (squarefree) monomial ideal) Let \(R = k[x,y,z]\) and set \(I = (xy, yz, xz) = (x,y) \cap (x,z) \cap (y,z)\). It turns out that \[ I^{(m)} = (x,y)^m \cap (x,z)^m \cap (y,z)^m. \]

Example 3 Given \(I = (x,y) \cap (x,z) \cap (y,z)\subseteq k[x,y,z]\):

\[ \begin{align*} \alpha(I) &= 2\\ \alpha(I^{(2)}) &= 3\\ \alpha(I^{(3)}) &= 5\\ \alpha(I^{(4)}) &= 6\\ &\vdots \end{align*} \] In fact, \(\widehat\alpha(I) = \frac{3}{2}\).

Symbolic Powers of Squarefree Monomial Ideals

Theorem 7 Let \(I\) be a squarefree monomial ideal in \(k[x_1, \ldots, x_N]\).

There exist unique prime ideals of the form \(P_i = (x_{i,1}, \ldots, x_{i,t_i})\) such that \(I = P_1 \cap \cdots \cap P_s\).
With the \(P_i\)’s as above, we have \[ I^{(m)} = P_1^m \cap \cdots \cap P_s^m. \]
For all \(m\ge 1\), \[ \alpha(I^{(m)}) = \min\{a_1 + \cdots + a_N\mid x_1^{a_1} \cdots x_N^{a_N} \in I^{(m)}\}. \]
We have \(x_1^{a_1} \cdots x_N^{a_N} \in I^{(m)}\) if and only if \(a_{i,1} + \cdots + a_{i,t_i} \ge m\) for \(i = 1, \ldots, s\).

Example

Example 4 Let \(I = (x_1 x_3 x_5, x_2 x_3 x_4, x_1 x_2 x_4 x_5, x_3 x_4 x_5)\subseteq k[x_1, x_2, \ldots, x_5]\). Then

\[ \begin{align*} I^{(m)} &= (x_1, x_3)^m \cap (x_2, x_3)^m \cap (x_1, x_4)^m \cap (x_3, x_4)^m\\ & \cap (x_2, x_5)^m \cap (x_3, x_5)^m \cap (x_4,x_5)^m. \end{align*} \]

Determining if \(x_1^{a_1} x_2^{a_2} x_3^{a_3} x_4^{a_4} x_5^{a_5}\in I^{(m)}\) is equivalent to determining if the following system of inequalities are satisfied:

\[ \begin{align*} a_1 + a_3 & \ge m \leftrightarrow x_1^{a_1} x_2^{a_2} x_3^{a_3} x_4^{a_4} x_5^{a_5} \in (x_1, x_3)^m\\ a_2 + a_3 & \ge m \leftrightarrow x_1^{a_1} x_2^{a_2} x_3^{a_3} x_4^{a_4} x_5^{a_5} \in (x_2, x_3)^m\\ a_1 + a_4 & \ge m \leftrightarrow x_1^{a_1} x_2^{a_2} x_3^{a_3} x_4^{a_4} x_5^{a_5} \in (x_1, x_4)^m\\ a_3 + a_4 & \ge m \leftrightarrow x_1^{a_1} x_2^{a_2} x_3^{a_3} x_4^{a_4} x_5^{a_5} \in (x_3, x_4)^m\\ a_2 + a_5 & \ge m \leftrightarrow x_1^{a_1} x_2^{a_2} x_3^{a_3} x_4^{a_4} x_5^{a_5} \in (x_2, x_5)^m\\ a_3 + a_5 & \ge m \leftrightarrow x_1^{a_1} x_2^{a_2} x_3^{a_3} x_4^{a_4} x_5^{a_5} \in (x_3, x_5)^m\\ a_4 + a_5 & \ge m \leftrightarrow x_1^{a_1} x_2^{a_2} x_3^{a_3} x_4^{a_4} x_5^{a_5} \in (x_4, x_5)^m \end{align*} \]

To calculate \(\alpha(I^{(m)})\), we wish to minimize \(a_1 + a_2 + a_3 + a_4 + a_5\) subject to the above constraints.

A Linear Program for \(\widehat\alpha\)

Theorem 8 (Bocci et al. (2016)) Let \(I\subseteq k[x_1, \ldots, x_N]\) be a squarefree monomial ideal with minimal primary decomposition \(I = P_1 \cap \cdots \cap P_s\) with \(P_i = (x_{i,1}, \ldots, x_{i,t_i})\) for \(i = 1, \ldots, s\). Let \(A\) be the \(s\times n\) matrix where \[ A_{i,j} = \begin{cases} 1 & \text{if } x_j \in P_i\\ 0 & \text{if } x_j\notin P_i. \end{cases} \] Consider the following linear program (LP):

minimize \(\mathbf{1}^T \mathbf{y}\)
subject to \(A \mathbf{y}\ge \mathbf{1}\) and \(\mathbf{y}\ge \mathbf{0}\)

and suppose \(\mathbf{y^*}\) is a feasible solution that realizes the optimal value. Then \[ \widehat\alpha(I) = \mathbf{1}^T \mathbf{y^*}. \] That is, \(\widehat\alpha(I)\) is the optimal value of the LP.

Application to Edge Ideals

Definition 5 Let \(G\) be a (finite, simple) graph with vertices \(x_1, x_2, \ldots, x_N\). The edge ideal \(I(G)\) is the ideal in \(k[x_1, \ldots, x_N]\) generated by the set \[ \{x_i x_j\mid \{x_i, x_j\}\in E(G)\}. \]

Theorem 9 (Bocci et al. (2016)) Let \(G\) be a finite simple graph with edge ideal \(I(G)\). Then \[ \widehat\alpha(I(G)) = \frac{\chi_f(G)}{\chi_f(G)-1}, \] where \(\chi_f(G)\) denotes the fractional chromatic number of \(G\).

\(\widehat\alpha\) for Edge Ideals

Theorem 10 (Bocci et al. (2016)) Let \(G\) be a nonempty graph.

If \(\chi(G) = \omega(G)\), then \(\widehat\alpha(I(G)) = \frac{\chi(G)}{\chi(G) - 1}\).
If \(G\) is \(k\)-partite, then \(\widehat\alpha(I(G)) \ge \frac{k}{k-1}\). When \(G\) is complete \(k\)-partite, \(\widehat\alpha(I(G)) = \frac{k}{k-1}\).
If \(G\) is bipartite, \(\widehat\alpha(I(G)) = 2\).
If \(G = C_{2n+1}\) is an odd cycle, then \(\widehat\alpha(I(C_{2n+1})) = \frac{2n+1}{n+1}\).
If \(G = C_{2n+1}^c\), then \(\widehat\alpha(I(C_{2n+1}^c)) = \frac{2n+1}{2n-1}\).

Comparing Powers of Edge Ideals

Theorem 11 (J—, Kamp, and Vander Woude (2019)) Let \(I\) be the edge ideal of an odd cycle on \(2n+1\) vertices. Then:

\(I^{(m)} = I^m\) for \(1\le m\le n\).
\(\rho(I) = \frac{2n+2}{2n+1}\)

Furthermore, \(I^{(n+1)} = I^{n+1} + (x_1 x_2 \cdots x_{2n+1})\).

Extended in 2023 by Cooper and Noteboom: Symbolic power decompositions of disjoint cycle graphs

This is joint with Thomas Kamp (now an actuary, i think), and Jason VW, who finished a joint PhD in math/CS at UNL this past spring.

We focused on edge ideals of odd cycles, and noticed that the ordinary and symbolic powers coincided up to a certain threshold, and then the symbolic power had some extra stuff in it. We were able to develop a method of factoring the monomials ‘along’ edges, and then sorting the monomials into the ordinary power, or the extra stuff.

We were able to use our method to calculate the resurgence and symbolic defect.

Cooper and Noteboom built on our ideas in a paper published earlier this year in considering the edge ideals of what they call disjoint cycle graphs, which are connected graphs in which no two odd cycles share an edge; they’re able to extend our work by giving a more general decomposition of the edge ideal of such graphs.

I’d like to close with a few thoughts.

Closing Thoughts

Resources

Symbolic Powers of Ideals (2017) by Dao et al. arXiv:1708.03010
Eloísa Grifo’s lecture notes
SymbolicPowers M2 package

First, if you haven’t thought about these questions before, or maybe just not for a while, I hope you were able to find something interesting to think about.

If you’re interested in learning more I want to suggest a few resources.

A resource I found very helpful as I organized this talk was Eloisa Grifo’s lecture notes from a Spring 2022 topics course on symbolic powers.

Another helpful survey is due to Dao, De Stefani, Grifo, Huneke, and Núñez-Betancourt, and explores questions about symbolic powers that I didn’t even touch on.

Finally, I wanted to mention IPPI, which grew out of the 2017 edition of the PRAGMATIC research school at the University of Catania in Italy.

From the website: “First book to contain a summary of known results on the associated primes of edge ideals First book to contain a summary of known results on the regularity of powers of monomial ideals Contains open problems for graduate students With a chapter for early career researchers on”how to do mathematics research””

Regularly share Chapter 20: The Art of Research, with my students. Lots of great advice, not necessarily unique to Brian, but I definitely remember some of it.

Last line, which is in a section on sharing your research ends thus: “And finally, ending a little early is OK, but never go over your allotted time!” And so I won’t. Thanks for your attention.

Thanks!

mkjanssen.org/talks/brianfest/Janssen-BrianFest.html

Email: mike.janssen@dordt.edu

Appendices

On the Fractional Chromatic Number

Definition 6 A \(b\)-fold coloring of a graph \(G\) is an assignment of \(b\) colors to each vertex such that adjacent vertices receive different colors.

Definition 7 The fractional chromatic number is then \[ \chi_f(G) := \lim\limits_{b\to\infty} \frac{\chi_b(G)}{b}. \]

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