About
Me
I am an associate professor
of mathematics at the University of NebraskaLincoln. My research focuses on
commutative algebra with a geometric and computational flavor.
I have been awarded the
2018 Harold & Esther
Edgerton Junior Faculty Award for creative research,
extraordinary teaching abilities, and academic promise.
I coorganize a special session on Commutative Algebra and a conference on Unexpected and Asymptotic Properties of Algebraic
Varieties
(postponed to 2023).
In the news: When life gives you lemons, make mathematicians, an article in the Notices
of the AMS about the Polymath program  a large scale online REU I
mentored.
My Students
PhD students:
 Ben Drabkin
– PhD
2020
 Andrew Conner –MA 2020
 Erica
Hopkins (coadvised
with Mark Walker) – PhD 2021
 Michael
DeBellevue (coadvised with Mark Walker) – Outstanding
Qualifying Exams award (20182019), Linda Bors
fellowship (20202021)
 Shah
Roshan Zamir
Undergraduate
students:
 Joey
Becker
– Chair’s
prize for best graduating math major (2015)
 Diana
(Xuehua) Zhong – graduate student at North
Carolina State University (since 2018)
Undergraduate
Activities
I have mentored a team of students working in the Polymath Jr programs of
2020 and 2021. This wonderful group of students produced three of the
papers listed below.
In April 2018 I organized the Central States Mathematics
Undergraduate Research Conference (CeSMUR
2018) at UNL. CeSMUR 2019 took place at Kansas
State University.
Videos from Selected Talks
 Reflection arrangements, syzygies, and
the containment problem (video)
Fellowship of the Ring
national
commutative algebra seminar
organized by MSRI, Jul 2020
 Connected
sums of graded artinian Gorenstein
algebras and the Lefschetz properties (video) BIRS workshop New
Trends in Syzygies in Banff, June 2018
 Combinatorial methods for symbolic powers (video) CMO Workshop on Symbolic
and Ordinary Powers of Ideals in Oaxaca, May 2017
 A handson approach to tensor product surfaces of
bidegree (2,1) (video) CMO Workshop on
Computational Algebra and geometric Modeling in Oaxaca, Aug 2016
Research and
Publications
My research is in commutative algebra, with an interest
in computational algebra, homological methods and connections to algebraic
geometry.
1. Cohomological blow ips of graded artinian
Gorenstein algebras along surjective maps (with A. Iarrobino, P Macias
Marques, C. McDaniel, J. Watanabe)
We introduce the cohomological blow up of a
graded Artinian Gorenstein algebra along a
surjective map, which we term BUG (Blow Up Gorenstein) for short. This is
intended to translate to an algebraic context the cohomology ring of a blow
up of a projective manifold along a projective submanifold.
2.
Rees algebras of filtrations of
covering polyhedra and integral closure of powers
of monomial ideals (with G. Grisalde, R. Villarreal)
The aims of this work are to study Rees
algebras of filtrations of monomial ideals associated to covering polyhedra of rational matrices with nonnegative entries
and nonzero columns using combinatorial optimization and integer
programming, and to study powers of monomial ideals and their integral
closures using irreducible decompositions and polyhedral geometry.
3.
Real
powers of monomial ideals (with Polymath)
While it is customary for the
exponentiation operation on ideals to consider natural powers, we extend this
notion to powers where the exponent is a positive real number. Real powers of
a monomial ideal generalize the integral closure operation and highlight many
interesting connections to the theory of convex polytopes.
4.
Consequences of the packing
problem (with Polymath) J. Algebraic Combinatorics, forthcoming
We study several consequences of the
packing problem, a conjecture from combinatorial optimization, using
algebraic invariants of squarefree monomial ideals. While the packing
problem is currently unresolved, we successfully settle the validity of its consequences.
5.
Convex bodies and asymptotic
invariants for powers of monomial ideals (with Polymath)
Continuing a wellestablished tradition of
associating convex bodies to monomial ideals, we initiate a program to
construct asymptotic Newton polyhedra from
decompositions of monomial ideals. This is achieved by forming a graded
family of ideals based on a given decomposition. Based on irreducible
decompositions, we introduce a novel family of irreducible powers which
generalizes the notions of ordinary and symbolic powers.
6.
Symbolic Rees
algebras (with E. Grifo)
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.
7.
Canonical resolutions over Koszul algebras (with E. Faber, M. JuhnkeKubitzke,
H. Lindo, C. Miller, R. R.G.) Proceedings of the WICA Conference, forthcoming
We generalize Buchsbaum
and Eisenbud's resolutions for the powers of the
maximal ideal of a polynomial ring to resolve powers of the homogeneous
maximal ideal over graded Koszul algebras.
8.
Quadratic Gorenstein
algebras with many surprising properties (with J. McCullough) Arch. Math. (Basel)
115 (2020), no. 5, 509521
Using the method of idealization, we
produce examples of graded Artinian Gorenstein
algebras that are not Koszul, do not satisfy the
subadditivity property for degree of syzygies and fail to satisfy the Lefschetz property.
9.
Singular loci of reflection
arrangements and the containment problem (with B. Drabkin) Math. Z., forthcoming
We paper provide insights into the role of
symmetry in studying polynomial functions vanishing to high order on an
algebraic variety. The varieties we study are singular loci of hyperplane
arrangements in projective space, with emphasis on arrangements arising from
complex reflection groups. We provide minimal sets of equations for the
radical ideals defining these singular loci and study containments between
the ordinary and symbolic powers of these ideals.
10.
Implicitization
of tensor product surfaces via virtual projective resolutions (with E. Duarte) Mathematics
of Computation 89 (2020), no. 326, 30233056
We derive the implicit equations for
certain parametric surfaces in threedimensional projective space termed
tensor product surfaces. Our method computes the implicit equation for such a
surface based on the knowledge of the syzygies of the base point locus of the
parametrization by means of constructing an explicit virtual projective
resolution.
11. Betti
numbers of symmetric shifted ideals (with J. Biermann, H. de Alba, F. Galetto,
S. Murai, U. Nagel, A. O’Keefe, T. Römer) J. Algebra
560 (2020), 312–342
We introduce a new class of monomial ideals
which we call symmetric shifted ideals. Symmetric shifted ideals are fixed by
the natural action of the symmetric group and, within the class of monomial
ideals fixed by this action, they can be considered as an analogue of stable
monomial ideals within the class of monomial ideals. We show that a symmetric
shifted ideal has linear quotients and compute its (equivariant) graded Betti numbers. As an application of this result, we
obtain several consequences for graded Betti
numbers of symbolic powers of defining ideals of star configurations.
12. Connected sums of graded artinian Gorenstein algebras
and Lefschetz properties (with A. Iarrobino and C.
McDaniel) J. Pure Applied Algebra, forthcoming
A connected sum construction for local rings
was introduced in a paper by H. Ananthnarayan, L. Avramov, and W.F. Moore. In the graded artinian Gorenstein case, this
can be viewed as an algebraic analogue of the topological construction of the
same name. We give two alternative description of this algebraic connected
sum: the first uses algebraic analogues of Thom classes of vector bundles and
Gysin homomorphisms, the second is in terms of Macaulay dual generators. We
also investigate the extent to which the connected sum construction preserves
the weak or strong Lefschetz property.
13. General minimum distance
functions and algebraic invariants of Geramita
ideals (with S. Cooper, Ș.
Tohăneanu, M. Vaz
Pinto and R. Villarreal),
Advances
in Applied Mathematics 112 (2020) 101940
Motivated by
notions from coding theory, we study the generalized minimum distance (GMD)
function of a graded ideal I in a
polynomial ring over an arbitrary field using commutative algebraic methods.
It is shown that the GMD function is nondecreasing as a function of its
first argument and nonincreasing as a function of the second argument. For
vanishing ideals over finite fields, we show that the GMD function is in fact
strictly decreasing as a function of the second argument until it stabilizes.
We also study algebraic invariants of Geramita
ideals. Those ideals are graded, unmixed, 1dimensional and their associated
primes are generated by linear forms. We also examine GMD functions of
complete intersections and show some special cases of two conjectures of TohăneanuVan Tuyl and EisenbudGreenHarris.
14. Frieze varieties: A characterization
of the finitetamewild trichotomy for acyclic quivers (with K. Li, L. Li, M. Mills and R. Schiffler) Advances in Math. 367 (2020)
We introduce a
new class of algebraic varieties which we call frieze varieties. Each frieze
variety is determined by an acyclic quiver. The frieze variety is defined in
an elementary recursive way by constructing a set of points in affine space.
From a more conceptual viewpoint, the coordinates of these points are
specializations of cluster variables in the cluster algebra associated to the
quiver. We give a new characterization of the finitetamewild trichotomy for
acyclic quivers in terms of their frieze varieties by showing that an acyclic
quiver is representation finite, tame, or wild, respectively, if and only if
the dimension of its frieze variety is 0,1, or ≥2, respectively.
15. Computations involving
symbolic powers (with B. Drabkin, E. Grifo, B. Stone) and the Macaulay2
package SymbolicPowers
J. Software for
Algebra and Geometry vol. 9, no.1 (2019), 71–80
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.
16.
Negative
curves on symmetric blowups of the projective plane, resurgences and Waldschmidt constants
(with Th. Bauer,
S. Di Rocco, B. Harbourne, J. Huizenga, T. Szemberg) International
Math. Research Notices IMRN 2019, no. 24,
7459–7514
In this paper
we study the surface X obtained by blowing up the projective plane in the
singular points of one of two highly symmetric line configurations known as
the Klein configuration and the Wiman configuration
respectively. We study invariant curves on X in detail, with a particular
emphasis on curves of negative selfintersection. We use the representation
theory of the stabilizers of the singular points to discover several
invariant curves of negative selfintersection on X, and use these curves to
study Nagatatype questions for linear series on X.
17. Symbolic powers of codimension two
CohenMacaulay ideals (with S. Cooper, G. Fatabbi,
E. Guardo, A. Lorenzini,
J. Migliore, U. Nagel, J. Szpond,
A. Van Tuyl)
Comm. Algebra 48
(2020), no.11, 46634680
Under the
additional hypothesis that X is a local complete intersection, we classify
when I(X)^(m) = I(X)^m for all m >= 1. The key tool to prove this
classification is the ability to construct a graded minimal free resolution
of I^m under these hypotheses. Among our
applications are significantly simplified proofs for known results about
symbolic powers of ideals of points in P^1 x P^1.
18. The Waldschmidt
constant for squarefree monomial ideals
(with C. Bocci, S. Cooper, E. Guardo, B.
Harbourne, M. Janssen, U. Nagel, A. Van Tuyl, T. Vu) J. Algebraic Combinatorics 44 (2016) no. 4, 875904
Given a squarefree monomial ideal I, we show that the Waldschmidt constant of I can be expressed as the optimal
solution to a linear program constructed from the primary decomposition of I.
By applying results from fractional graph theory, we can then express the Waldschmidt constant in terms of the fractional chromatic
number of a hypergraph also constructed from the primary decomposition of I.
Moreover, we prove a Chudnovskylike lower bound on
this constant, thus verifying a conjecture of CooperEmbreeHaHoefel for monomial ideals in the squarefree
case.
19. A homological
criterion for the failure of containment of the symbolic cube in the square
of some ideals of points in P^2
J. Pure Applied Algebra
219 (2015) no.11, 48574871
We establish a
criterion for the failure of the containment of the symbolic cube in the
square for 3generated ideals I defining reduced sets of points in P^2. Our
criterion arises from studying the minimal free resolutions of the powers of
I, specifically the minimal free resolutions for I^2 and I^3. We apply this
criterion to two point configurations that have
recently arisen as counterexamples to a question of B. Harbourne
and C. Huneke: the Fermat configuration and the
Klein configuration.
20. Determinants
of incidence and Hessian matrices arising from the vector space lattice (with
S. Nasseh and J. Watanabe)
J. Commut. Algebra 11 (2019) no. 1, 131154
We give
explicit formulas for the determinants of the incidence and Hessian matrices
arising from the interaction between the rank 1 and rank n−1 level sets
of the subspace lattice of an ndimensional finite vector space. Our
exploration is motivated by the fact that both of these matrices arise
naturally in the study of the combinatorial and algebraic Lefschetz
properties.
21. Resurgences for ideals of special point
configurations in P^N coming from hyperplane arrangements
(with M. Dumnicki, B. Harbourne, U.
Nagel, T. Szemberg and H. TutajGasińska ) J. Algebra 443 (2015), 383394
It had been expected for several years that
I^(Nr−N+1)⊆ I^r should hold for the ideal I of any finite set of
points in P^N for all r>0, but in the last year various counterexamples
have now been constructed, all involving point sets coming from hyperplane
arrangements. In the present work, we compute their resurgences and obtain in
particular the first examples where the resurgence and the asymptotic
resurgence are not equal.
22. A tight bound
on the projective dimension of four quadrics (with C. Huneke, P. Mantero and J. McCullough) with accompanying Macaulay 2 code.
J. Pure Appl. Algebra 222 (2018) no. 9, 25242551
Motivated by a question posed by Mike Stillman,
we show that the projective dimension of an ideal generated by four quadric
forms in a polynomial ring has projective dimension at most 6.
23. A
multiplicity bound for graded rings and a criterion for the CohenMacaulay
property (with
C. Huneke, P. Mantero and
J. McCullough)
Proc.
Amer. Math. Soc. 143 (2015) no.6, 23652377
We prove an
upper bound for the multiplicity of R/I, where I is a homogeneous ideal of
the form I=J+(F) and J is a CohenMacaulay ideal. The bound is given in terms
of invariants of R/J and the degree of F. We show that ideals achieving this
bound have high depth and deduce a numerical criterion for the CohenMacaulay
property. Applications to quasiGorenstein rings
and almost complete intersection ideals are given.
24. Containment
counterexamples for ideals of various configurations of points in P^N
(with B. Harbourne) J. Pure Appl. Algebra 219
(2015), 10621072
We provide counterexamples to a conjecture of Harbourne
and Huneke regarding containments between regular
powers and symbolic powers of ideals of points in projective space P^N. We
show that the conjecture fails in every prime characteristic p>2 when N=2
and we provide additional counterexamples for higher dimensional projective
spaces.
25. The projective
dimension of codimension two algebra presented by quadrics (with C. Huneke, P. Mantero and J.
McCullough) J. Algebra 393 (2013),
170186
We prove a sharp
upper bound for the projective dimension of ideals of height two generated by
quadrics in a polynomial ring with arbitrary large number of variables.
26. Multiple structures
with arbitrarily large projective dimension on linear subspaces (with C. Huneke, P. Mantero and J.
McCullough)
J. Algebra 447 (2016), 183205
We show that no finite characterization of multiple structures on linear
spaces is possible if one only assumes Serre’s S_1 property holds by by constructing structures with arbitrarily large
projective dimension. Our methods build upon a family of ideals with large
projective dimension using linkage. The result is in stark contrast to Manolache's characterization of CohenMacaulay multiple
structures in codimension 2 and multiplicity at most 4 and also to Engheta's characterization of unmixed ideals of height 2
and multiplicity 2.
27. Computations
in intersection rings of flag bundles (with D. Grayson and M. Stillman)
This paper arose out of an observation that was made while I was working
on the “Symmetric polynomials” package for Macaulay2. Intersection rings of
flag varieties and of isotropic flag varieties are generated by Chern classes of the tautological bundles modulo the
relations coming from multiplicativity of total Chern
classes. In this paper we describe the Gröbner
bases of the ideals of relations and give applications to computation of intersections,
as implemented in Macaulay2.
28. Bounding
projective dimension (with J. McCullough) a chapter in the book Commutative Algebra, SpringerVerlag
London, 2013
The celebrated Hilbert Syzygy Theorem states that the projective
dimension of any ideal in a polynomial ring on n variables is at most n1.
This paper surveys recent progress on Stillman’s
question, asking whether the degrees of a set of homogeneous polynomials
suffice in order to bound the projective dimension of the ideal they
generate, without prior knowledge of the ambient polynomial ring (hence
without using the number of variables).
29. Syzygies and
singularities of tensor product surfaces of bidegree (2,1) (with H.
Schenck and J. Validashti), Math. Comp. 83 (2014), 13371372
We study the associated ideal of a bigraded parametrization of a surface
in P^3 from the standpoint of commutative algebra, proving that there are
exactly six numerical types of possible bigraded minimal free resolution.
These resolutions play a key role in determining the implicit equation of the
image, via work of BuseJouanolou, BuseChardin, Botbol and BotbolDickensteinDohm on the approximation complex. In
particular this allows us to completely describe the implicit equation and
singular locus of the image.
30. Syzygy
theorems via comparison of order ideals on a hypersurface (with P. A.
Griffith) J. Pure Appl. Algebra 216 (2012), no. 2, 468479
We introduce a weak order ideal property that suffices for establishing
the EvansGriffith Syzygy Theorem. We study this weak order ideal property in
settings that allow for comparison between homological algebra over a local
ring R versus a hypersurface ring R/(x^n). Consequently we solve some relevant cases of the
EvansGriffith syzygy conjecture over local rings of unramified mixed
characteristic p, with the case of syzygies of prime ideals of CohenMacaulay
local rings of unramified mixed characteristic being noted.
31. Inverse
systems, fat points and the weak Lefschetz property
(with B. Harbourne and H. Schenck) J. Lond. Math. Soc. (2) 84 (2011), no. 3, 712730
We use the inverse system dictionary to connect ideals generated by
powers of linear forms to ideals of fat points and show that failure of WLP
for powers of linear forms in at least four variables is connected to the
geometry of the associated fat point scheme. This is in stark contrast with
the situation described in the paper “The Weak Lefschetz
Property and powers of linear forms in K[x, y, z]”
below. Closely related results can be found in the paper See the paper
"On the weak Lefschetz property for powers of
linear forms" by MiglioreMiròRoigNagel.
32. Ideals
with Large(r) Projective Dimension and Regularity (with J. Beder, J. McCullough, L. NunezBetancourt, B. Snapp, B.
Stone)
J.
Symbolic Comput. 46 (2011), no. 10, 11051113
This paper is an outcome of the Mathematical Research Communities
program. I am grateful to AMS and the organizers for this wonderful
opportunity. We define a family of homogeneous ideals with large projective
dimension and regularity relative to the number of generators and their
common degree. This family subsumes and improves upon constructions given by Caviglia and McCullough In
particular, we describe a family of homogeneous ideals with three generators
of degree d in arbitrary characteristic whose projective dimension grows
asymptotically an exponential function in d.
Here is the Macaulay 2 code mentioned in the paper.
33. The
Weak Lefschetz Property and powers of linear forms
in K[x, y, z] (with H. Schenck) Proc. Amer. Math. Soc.
138 (2010), no. 7, 23352339
We show that
any artinian quotient of K[x,
y, z] by an ideal I generated by powers of linear forms has the Weak Lefschetz Property.
Software
I have
authored or coauthored several packages for the computer algebra program Macaulay2:
