

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. 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. 2.
Ordinary and symbolic Rees algebras for
ideals of Fermat point configurations (with U. Nagel) J. Algebra 468 (2016), 80102 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. 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 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 from
Selected Talks
Software
I have
authored or coauthored several packages for the computer algebra program Macaulay2: 

