

Teaching Spring 2016
Math 918 – Topics in Algebra: The Geometry of Syzygies An introduction to graded free resolutions
viewed from a geometric perspective, following the book by the same title by
David Eisenbud. 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. 1.
Ordinary and symbolic Rees algebras for
ideals of Fermat point configurations (with U. Nagel) We give a systematic treatment of the family of Fermat
ideals, describing explicitly the minimal generators and the minimal free
resolutions of all their ordinary powers as well as many symbolic powers. We
use these to study the ordinary and the symbolic Rees algebra of Fermat
ideals. Specifically, we show that the symbolic Rees algebras of Fermat
ideals are Noetherian. Along the way, we give formulas for the
CastelnuovoMumford regularity of the powers of Fermat ideals and we determine
their reduction ideals. 2.
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) 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.
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 4.
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. 5.
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 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. 6.
Ideals
generated by four quadric polynomials (with
C. Huneke, P. Mantero and J. McCullough) with accompanying Macaulay 2 code. 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. 7. 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. 8.
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. 9.
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. 10.
Multiple structures with arbitrarily
large projective dimension on linear subspaces (with C. Huneke, P.
Mantero and J. McCullough) J. Algebra 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. 11.
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. 12.
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). 13.
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. 14.
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. 15.
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. 16.
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. 17.
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: 

