

Teaching (Fall 2017)
Fundamental concepts of commutative ring
theory: primary decomposition, filtrations and completions, dimension theory,
integral extensions, homological methods, regular rings.
An introduction to mathematical reasoning,
construction of proofs, and careful mathematical writing in the context of
continuous mathematics and calculus. For past teaching at UNL click here. All course materials can be found on Blackboard or Canvas. I am a faculty advisor for the UNL Math Club and the
organizer of the new lecture series Career Perspectives in
Mathematics. Recent and
upcoming conferences
Research and
Publications
My research is in commutative algebra, with an interest
in computational algebra, homological methods and connections to algebraic
geometry. 1. 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) 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. 2. 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) 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. 3.
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.
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. 5.
Determinants of incidence and Hessian
matrices arising from the vector space lattice (with S. Nasseh and J.
Watanabe) J. Commutative Algebra 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. 6.
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. 7.
A tight bound on the projective
dimension of four quadrics (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. 8. 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. 9.
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. 10. 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. 11. 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. 12. 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. 13. 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). 14. 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. 15. 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. 16. 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. 17. 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. 18. 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 and
Videos from Selected Talks
Software
I have
authored or coauthored several packages for the computer algebra program Macaulay2: 

