

Teaching – Spring 2015
Math
310  Introduction to Modern Algebra An introduction to proofs course designed for mathematics majors and
preservice secondary education majors, covering mathematical induction,
elementary number theory, the Fundamental Theorem of Arithmetic, modular
arithmetic and elementary notions about rings. For
past teaching at UNL click here. Research and
Publications
My research is in commutative algebra, with an interest
in computational algebra, homological methods and connections to algebraic
geometry. 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 2.
Determinants of incidence and Hessian
matrices arising from the vector space lattice (with S. Nasseh and J. Watanabe) 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. 3.
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 ) 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. 4.
Ideals
generated by four quadric polynomials (with
C. Huneke, P. Mantero and
J. McCullough) 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 9. 5. A multiplicity bound for graded rings
and a criterion for the CohenMacaulay property (with C. Huneke, P. Mantero and J. McCullough) to appear in Proc.
Amer. Math. Soc. 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. 6.
Containment counterexamples for ideals
of various configurations of points in P^N (with B. Harbourne)
to appear in J. Pure Appl. Algebra 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. 7. 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. 8.
Multiple structures with arbitrarily
large projective dimension on linear subspaces (with C. Huneke, P. Mantero and J.
McCullough) 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. 9.
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. 10.
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). 11.
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. 12.
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. 13.
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. 14.
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 as. Here is
the Macaulay 2 code mentioned in the paper. 15.
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. Slides from
Selected Talks
Software
I have
authored or coauthored several packages for the computer algebra program Macaulay2: 

