

Teaching – Fall 2014
Math 189H
 The Joy of Numbers (freshman honors seminar) A guided exploration into number theory
from Euclid’s proof of the infinitude of primes to applications in public key
cryptography. 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: 

