We establish a criterion for the failure of the
containment of the symbolic cube in the square for 3-generated 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

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 n-dimensional
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.

(with M. Dumnicki, B. Harbourne, U.
Nagel, T. Szemberg and H. Tutaj-Gasińska)

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.

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.

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 Cohen-Macaulay 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 Cohen-Macaulay property.
Applications to quasi-Gorenstein rings and almost
complete intersection ideals are given.

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.

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.

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 Cohen-Macaulay 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.

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, Springer-Verlag
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 n-1. 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).

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 Buse-Jouanolou, Buse-Chardin, Botbol and Botbol-Dickenstein-Dohm
on the approximation complex. In particular this allows us to completely
describe the implicit equation and singular locus of the image.

We introduce a weak order ideal property that suffices
for establishing the Evans-Griffith 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 Evans-Griffith syzygy
conjecture over local rings of unramified mixed
characteristic p, with the case of syzygies of prime ideals of Cohen-Macaulay
local rings of unramified mixed characteristic
being noted.

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 Migliore-Mirò-Roig-Nagel.

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.