– 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. (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 Castelnuovo-Mumford 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 Chudnovsky-like lower bound on this constant, thus verifying a conjecture of Cooper-Embree-Ha-Hoefel for monomial ideals in the squarefree case.
J. Pure Applied Algebra 219 (2015) no.11, 4857-4871
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
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 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 ) 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.
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 Cohen-Macaulay property (with C. Huneke, P. Mantero and J. McCullough)
Proc. Amer. Math. Soc. 143 (2015) no.6, 2365-2377
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.
8. Containment counterexamples for ideals of various configurations of points in P^N (with B. Harbourne) J. Pure Appl. Algebra 219 (2015), 1062-1072
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), 170-186
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 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.
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, 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).
13. Syzygies and singularities of tensor product surfaces of bidegree (2,1) (with H. Schenck and J. Validashti), Math. Comp. 83 (2014), 1337-1372
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.
14. Syzygy theorems via comparison of order ideals on a hypersurface (with P. A. Griffith) J. Pure Appl. Algebra 216 (2012), no. 2, 468-479
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.
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, 712-730
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.
16. Ideals with Large(r) Projective Dimension and Regularity (with J. Beder, J. McCullough, L. Nunez-Betancourt, B. Snapp, B. Stone)
J. Symbolic Comput. 46 (2011), no. 10, 1105-1113
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, 2335-2339
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.
I have authored or co-authored several packages for the computer algebra program Macaulay2: