- Symbolic Rees Algebras (with Alexandra Seceleanu)
We survey old and new approaches to the study of symbolic powers of ideals. Our focus is on the symbolic Rees algebra of an ideal, viewed both as a tool to investigate its symbolic powers and as a source of challenging problems in its own right. We provide an invitation to this area of investigation by stating several open questions.abstract
- Constructing non-proxy small test modules for the complete intersection property (with Ben Briggs and Josh Pollitz)
A local ring R is regular if and only if every finitely generated R-module has finite projective dimension. Moreover, the residue field k is a test module: R is regular if and only if k has finite projective dimension. This characterization can be extended to the bounded derived category Df(R), which contains only small objects if and only if R is regular.abstract
Recent results of Pollitz, completing work initiated by Dwyer-Greenlees-Iyengar, yield an analogous characterization for complete intersections: R is a complete intersection if and only if every object in Df(R) is proxy small. In this paper, we study a return to the world of R-modules, and search for finitely generated R-modules that are not proxy small whenever R is not a complete intersection. We give an algorithm to construct such modules in certain settings, including over equipresented rings and Stanley-Reisner rings.
- Demailly's conjecture and the containment problem (with Sankhaneel Bisui, Tài Huy Hà, and Thái Thành Nguyễn)
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 particular implies Demailly's bound, and prove that a general version of that containment holds for determinantal ideals and defining ideals of star configurations.abstract
- Expected resurgence of ideals defining Gorenstein rings (with Craig Huneke and Vivek Mukundan)
Building on previous work by the same authors, we show that certain ideals defining Gorenstein rings have expected resurgence, and thus satisfy the stable Harbourne Conjecture. In prime characteristic, we can take any radical ideal defining a Gorenstein ring in a regular ring, provided its symbolic powers are given by saturations with the maximal ideal. While this property is not suitable for reduction to characteristic p, we show that a similar result holds in equicharacteristic 0 under the additional hypothesis that the symbolic Rees algebra of I is noetherian.abstract
- Chudnovsky's Conjecture and the stable Harbourne—Huneke containment (with Sankhaneel Bisui, Tài Huy Hà, and Thái Thành Nguyễn)
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 Harbourne—Huneke containment conjectures for a general set of sufficiently many points.abstract
- Expected resurgences and symbolic powers of ideals (with Craig Huneke and Vivek Mukundan)
Journal of the London Mathematical Society (2) 102 (2020) 453-469.arXivjournal
We give explicit criteria that imply the resurgence of a self-radical ideal in a regular ring is strictly smaller than its codimension, which in turn implies that the stable version of Harbourne's conjecture holds for such ideals. This criterion is used to give several explicit families of such ideals, including the defining ideals of space monomial curves. Other results generalize known theorems concerning when the third symbolic power is in the square of an ideal, and a strong resurgence bound for some classes of space monomial curves.abstract
- A stable version of Harbourne's Conjecture and the containment problem for space monomial curves
J. Pure Appl. Algebra, Volume 224, Issue 12, 2020.arXivjournal
The symbolic powers I(n) of a radical ideal I in a polynomial ring consist of the functions that vanish up to order n in the variety defined by I. These do not necessarily coincide with the ordinary algebraic powers In, but it is natural to compare the two notions. The containment problem consists of determining the values of $n$ and $m$ for which I(n) ⊆ Im holds. When I is an ideal of height 2 in a regular ring, I(3) ⊆ I2 may fail, but we show that this containment does hold for the defining ideal of the space monomial curve (ta, tb tc). More generally, given a radical ideal I of big height h, while the containment I(hn-h+1) ⊆ In conjectured by Harbourne does not necessarily hold for all n, we give sufficient conditions to guarantee such containments for n sufficiently large.abstract
- Symbolic power containments in singular rings in positive characteristic (with Linquan Ma and Karl Schwede)
The containment problem for symbolic and ordinary powers of ideals asks for what values of a and b we have I(a) ⊆ Ib. Over a regular ring, a result by Ein-Lazarsfeld-Smith, Hochster-Huneke, and Ma-Schwede partially answers this question, but the containments it provides are not always best possible. In particular, a tighter containment conjectured by Harbourne has been shown to hold for interesting classes of ideals — although it does not hold in general. In this paper, we develop a Fedder (respectively, Glassbrenner) type criterion for F-purity (respectively, strong F-regularity) for ideals of finite projective dimension over F-finite Gorenstein rings and use our criteria to extend the prime characteristic results of Grifo-Huneke to singular ambient rings. For ideals of infinite projective dimension, we prove that a variation of the containment still holds, in the spirit of work by Hochster-Huneke and Takagi.abstract
- A Zariski-Nagata Theorem for smooth ℤ-algebras (with Alessandro De Stefani and Jack Jeffries)
J. Reine Angew. Math. (Crelles Journal) 2020 (761), 123—140.arXivjournal
In a polynomial ring over a perfect field, the symbolic powers of a prime ideal can be described via differential operators: a classical result by Zariski and Nagata says that the n-th symbolic power of a given prime ideal consists of the elements that vanish up to order n on the corresponding variety. However, this description fails in mixed characteristic. In this paper, we use p-derivations, a notion due to Buium and Joyal, to define a new kind of differential powers in mixed characteristic, and prove that this new object does coincide with the symbolic powers of prime ideals. This seems to be the first application of p-derivations to Commutative Algebra.abstract
- Symbolic powers of ideals defining F-pure and strongly F-regular rings (with Craig Huneke)
Int. Math. Res. Not. (IMRN) 2019, no. 10, 2999—3014.arXivjournal
Given a radical ideal I in a regular ring R, the containment problem of symbolic and ordinary powers of I consists of determining when the containment I(a) ⊆ Ib holds. By work of Ein-Lazersfeld-Smith and Hochster-Huneke, there is a uniform answer to this question, but the resulting containments are not necessarily best possible. We show that a conjecture of Harbourne holds when R/I is F-pure, and prove tighter containments in the case when R/I is strongly F-regular.abstract
- Symbolic powers of ideals (with Hailong Dao, Alessandro De Stefani, Craig Huneke and Luis Núñez-Betancourt)
in Singularities and foliations. Geometry, topology and applications, pp. 387-432, Springer Proc. Math. Stat., 222, Springer, Cham, 2018.arXivjournal
We survey classical and recent results on symbolic powers of ideals. We focus on properties and problems of symbolic powers over regular rings, on the comparison of symbolic and regular powers, and on the combinatorics of the symbolic powers of monomial ideals. In addition, we present some new results on these aspects of the subject.abstract
- Local cohomology and Lyubeznik numbers of F-pure rings (with Alessandro De Stefani and Luis Núñez-Betancourt)
To appear in the Journal of Algebra Special Issue in honor of Craig Huneke
In this article, we study certain local cohomology modules over F-pure rings. We give sufficient conditions for the vanishing of some Lyubeznik numbers, derive a formula for computing these invariants when the F-pure ring is standard graded and, by its means, we provide some new examples of Lyubeznik tables. We study associated primes of certain Ext-modules, showing that they are all compatible ideals. Finally, we focus on properties that Lyubeznik numbers detect over a globally F-split projective variety.abstract
- Lower bounds on projective levels of complexes (with Hannah Altmann, Jonathan Montaño, William Sanders, and Thanh Vu)
Journal of Algebra, 491C (2017), pp. 343-356.arXivjournal
For an associative ring R, the level of a perfect complex F is the smallest number of mapping cones needed to build F from add(R). We establish lower bounds of the level of F in terms of the vanishing of homology of F. We then use these bounds to derive an improved version of The New Intersection Theorem for level when R is a commutative local ring containing a field.abstract
- On the growth of deviations (with Adam Boocher, Alessio D'Alì, Jonathan Montaño, and Alessio Sammartano)
Proceedings of the American Mathematical Society 144 (2016), pp. 5049-5060.arXivjournal
The deviations of a graded algebra are a sequence of integers that determine the Poincaré series of its residue field and arise as the number of generators of certain DG algebras. In a sense, deviations measure how far a ring is from being a complete intersection. In this paper we study extremal deviations among those of algebras with a fixed Hilbert series. In this setting, we prove that, like the Betti numbers, deviations do not decrease when passing to an initial ideal and are maximized by the Lex-segment ideal. We also prove that deviations grow exponentially for Golod rings and for certain quadratic monomial algebras.abstract
- Edge ideals and DG algebra resolutions (with Adam Boocher, Alessio D'Alì, Jonathan Montaño, and Alessio Sammartano)
Le Matematiche 70 (2015), no.1, 215-237.arXivjournal
Let R=S/I where S=k[T_1,…,T_n] and I is a homogeneous ideal in S. The acyclic closure R of k over R is a DG algebra resolution obtained by means of Tate's process of adjoining variables to kill cycles. In a similar way one can obtain the minimal model S[X], a DG algebra resolution of R over S. By a theorem of Avramov there is a tight connection between these two resolutions. In this paper we study these two resolutions when I is the edge ideal of a path or a cycle. We determine the behavior of the deviations ε_i(R), which are the number of variables in R in homological degree i. We apply our results to the study of the k-algebra structure of the Koszul homology of R.abstract
- Computations involving symbolic powers (with Ben Drabkin, Alexandra Seceleanu, and Branden Stone)
Journal of Software for Algebra and Geometry 9 (2019), 71—80.arXivjournal
Symbolic powers are a classical commutative algebra topic that relates to primary decomposition, consisting, in some circumstances, of the functions that vanish up to a certain order on a given variety. However, these are notoriously difficult to compute, and there are seemingly simple questions related to symbolic powers that remain open even over polynomial rings. In this paper, we describe a Macaulay2 software package that allows for computations of symbolic powers of ideals and which can be used to study the equality and containment problems, among others.abstract
- The software package SpectralSequences (with Adam Boocher and Nathan Grieve)
We describe the computer algebra software package SpectralSequences for the computer algebra system Macaulay2. This package implements many data types, objects and algorithms which pertain to, among other things, filtered complexes, spectral sequence pages and maps therein. We illustrate some of the syntax and capabilities of SpectralSequences by way of several examples.abstract
Talks and notes
Escuela de otoño de Àlgebra Conmutativa at CIMAT in November 2019 organized by Luis Núñez-Betancourt and Abraham Martín del Campo.
Written for the RTG Advanced Mini-course in Commutative Algebra at the University of Utah in May 2018, organized by Anurag K. Singh and Srikanth B. Iyengar.
SoftwareI often use Macaulay2 in my research, and I have contributed to a few Macaulay2 packages.
(developed with David Berlekamp, Adam Boocher, Nathan Grieve, Gregory G. Smith, and Thanh Vu)
This package comes with Macaulay2 version ≥ 1.10.
This is a Macaulay2 package that provides tools for effective computation of the pages and differentials in spectral sequences obtained from many kinds of filtered chain complexes.
(with contributions from Ben Drabkin, Alexandra Seceleanu, and Branden Stone)
Version 2.0 comes with Macaulay2 version ≥ 1.14.
This is a Macaulay2 package that provides tools for computing symbolic powers.
Recent and future conferences
- May 2018: I spoke at the Utah RTG Summer School in Commutative Algebra.
- October 2018: Mathematisches Forschungsinstitut Oberwolfach Mini-workshop on Asymptotic Invariants of Homogeneous Ideals.
- October 2018: AMS meeting in Ann Arbor, MI. I spoke in the sessions on Advances in commutative algebra and Commutative algebra and complexity.
- December 2018: I spoke in the session on Symbolic and regular powers of ideals at the CMS Winter Meeting 2018.
- January 2019: I spoke at the conference Frobenius Actions in commutative algebra: recent developments, in Barcelona, Spain.
- February 2019: I spoke at the Mathematisches Forschungsinstitut Oberwolfach Workshop on Singularities and Homological Aspects of Commutative Algebra.
- March 2019: Patricia Klein and I organized an AMS Special Session on developments in Commutative Algebra in Auburn, AL.
- April 13 and 14, 2019: I spoke at the conference Morgantown Algebra Days, at West Virginia University.
- October, 2019: I participated in the Women in Commutative Algebra workshop at the Banff International Research Station.
- November 2, 2019: I spoke at the Undergraduate Mathematics Symposium, in the University of Illinois at Chicago.
- November, 2019: I gave a lecture series (in spanish) in the Fall School in Commutative Algebra, at CIMAT, Mexico.
- March, 2020: Sean Sather-Wagstaff and I organized a special session on commutative algebra at the AMS Sectional Meeting at UVa. Unfortunately, the meeting got canceled due to CORONAVIRUS concerns, but we hosted a zoom session on March 21 and 22. Click here for more information, including videos of some of the talks.
- April, 2020: I was going to speak in the special session on combinatorial techniques in commutative algebra at the AMS Sectional Meeting at Purdue. Unfortunately, the meeting got canceled due to the CORONAVIRUS crisis.
- May, 2020: I gave a talk at the Fellowship of the ring, worldwide Commutative Algebra Seminar, which was recorded and can be watched here.
- October, 2020: I spoke in the special Session on Homological Methods in Algebra, at the Penn State AMS Sectional Meeting.
- March, 2020: I will speak in the special Session on Commutative Algebra, at the Brown Virtual AMS Sectional Meeting.
- April, 2020: I will speak in the special Session on Commutative Algebra, at the Cincinnati Virtual AMS Sectional Meeting.
- May, 2020: Alessandra Costantini and I are organizing a virtual Special Session titled Women in Commutative Algebra - One hundred years of Idealtheorie in Ringbereichen, at the San Francisco State Virtual AMS Sectional Meeting.
- June, 2021: I will give a lecture series in BRIDGES, a conference aimed towards early graduate students and advanced undergraduate students, at the University of Utah. This was originally scheduled for May 2020 but has been postponed to 2021 due to the CORONAVIRUS crisis.
- August, 2021: I will speak at the ICERM workshop D-modules, Group Actions, and Frobenius: Computing on Singularities.
- October, 2022: Adam Boocher, Jenny Kenkel and I are organizing a special session on commutative algebra in the Utah AMS Sectional Meeting.
- I was going to speak at BrianFest, aka the Conference on unexpected and asymptotic properties of algebraic varieties, in honor of Brian Harbourne, at UNL. Unfortunately, the meeting has been postponed to a later date due to CORONAVIRUS concerns.
- Vou dar uma palestra no ENEMath 2020, no Instituto Superior Técnico, Portugal; a conferência foi adiada até uma data por determinar.
Some later date:
My research is supported by NSF grant DMS-2001445.