We introduce the class of principal symmetric ideals, which are ideals generated by the orbit of a single polynomial under the action of the symmetric group. Fixing the degree of the generating polynomial, this class of ideals is parametrized by points in a suitable projective space. We show that the minimal free resolution of a principal symmetric ideal is constant on a nonempty Zariski open subset of this projective space and we determine this resolution explicitly.

The starting point of this paper is a duality for sequences of natural numbers which interchanges superaditive and subaditive sequences. We show how this duality has natural algebraic consequences in conjunction with the theory of Macaulay inverse systems and how it underpins the reciprocity between two geometric invariants: the Seshadri constant and the asymptotic regularity of a finite set of points in projective space.

Symmetric shifted ideals are a class of monomial ideals which come equipped with an action of the symmetric group and share many of the properties of the well-studied strongly stable ideal from combinatorial commutative algebra. We study algebraic and combinatorial properties of these ideals: behavior under operations, primary decomposition, Rees algebra, and uncover interesting connections to the theory of discrete polymatroids, convex geometry, and permutohedral toric varieties.

We classify Artinian Gorenstein algebras in codimension two that have the Hodge-Riemann property in terms of higher Hessian matrices and total positivity of certain Toeplitz matrices.

We study sequences of Betti numbers of modules over complete intersection rings. It is known that the subsequences with even, respectively odd index I are given eventually by some polynomial in i. We study conditions under which the two polynomials agree and more generally give bounds on the degree of their difference.

We show that every standard graded codimension three Artinian Gorenstein algebra A having low maximum value of the Hilbert function, at most six, satisfies the strong Lefschetz property provided that the characteristic is zero. When the characteristic is greater than the socle degree of A, we show that A is almost strong Lefschetz.

We introduce a notion of sectional regularity for a homogeneous ideal I which measures the regularity of its generic sections with respect to linear spaces of various dimensions. This is related to axial constants defined as the intercepts on the coordinate axes of the set of exponents of monomials in the reverse lexicographic generic initial ideal of I.

We introduce the cohomological blow up of a graded Artinian Gorenstein algebra along a surjective map, which we term BUG (Blow Up Gorenstein) for short. This is intended to translate to an algebraic context the cohomology ring of a blow up of a projective manifold along a projective submanifold.

The aims of this work are to study Rees algebras of filtrations of monomial ideals associated to covering polyhedra of rational matrices with non-negative entries and non-zero columns using combinatorial optimization and integer programming, and to study powers of monomial ideals and their integral closures using irreducible decompositions and polyhedral geometry.

While it is customary for the exponentiation operation on ideals to consider natural powers, we extend this notion to powers where the exponent is a positive real number. Real powers of a monomial ideal generalize the integral closure operation and highlight many interesting connections to the theory of convex polytopes.

We study several consequences of the packing problem, a conjecture from combinatorial optimization, using algebraic invariants of square-free monomial ideals. While the packing problem is currently unresolved, we successfully settle the validity of its consequences.

Continuing a well-established tradition of associating convex bodies to monomial ideals, we initiate a program to construct asymptotic Newton polyhedra from decompositions of monomial ideals. This is achieved by forming a graded family of ideals based on a given decomposition. Based on irreducible decompositions, we introduce a novel family of irreducible powers which generalizes the notions of ordinary and symbolic powers.

We survey old and new approaches to the study of symbolic powers of ideals. Our focus is on the symbolic Rees algebra of an ideal, viewed both as a tool to investigate its symbolic powers and as a source of challenging problems in its own right. We provide an invitation to this area of investigation by stating several open questions.

We generalize Buchsbaum and Eisenbud’s resolutions for the powers of the maximal ideal of a polynomial ring to resolve powers of the homogeneous maximal ideal over graded Koszul algebras.

Using the method of idealization, we produce examples of graded Artinian Gorenstein algebras that are not Koszul, do not satisfy the subadditivity property for degree of syzygies and fail to satisfy the Lefschetz property.

We paper provide insights into the role of symmetry in studying polynomial functions vanishing to high order on an algebraic variety. The varieties we study are singular loci of hyperplane arrangements in projective space, with emphasis on arrangements arising from complex reflection groups. We provide minimal sets of equations for the radical ideals defining these singular loci and study containments between the ordinary and symbolic powers of these ideals.

We derive the implicit equations for certain parametric surfaces in three-dimensional projective space termed tensor product surfaces. Our method computes the implicit equation for such a surface based on the knowledge of the syzygies of the base point locus of the parametrization by means of constructing an explicit virtual projective resolution.

We introduce a new class of monomial ideals which we call symmetric shifted ideals. Symmetric shifted ideals are fixed by the natural action of the symmetric group and, within the class of monomial ideals fixed by this action, they can be considered as an analogue of stable monomial ideals within the class of monomial ideals. We show that a symmetric shifted ideal has linear quotients and compute its (equivariant) graded Betti numbers. As an application of this result, we obtain several consequences for graded Betti numbers of symbolic powers of defining ideals of star configurations.

A connected sum construction for local rings was introduced in a paper by H. Ananthnarayan, L. Avramov, and W.F. Moore. In the graded artinian Gorenstein case, this can be viewed as an algebraic analogue of the topological construction of the same name. We give two alternative description of this algebraic connected sum: the first uses algebraic analogues of Thom classes of vector bundles and Gysin homomorphisms, the second is in terms of Macaulay dual generators. We also investigate the extent to which the connected sum construction preserves the weak or strong Lefschetz property.

Motivated by notions from coding theory, we study the generalized minimum distance (GMD) function of a graded ideal I in a polynomial ring over an arbitrary field using commutative algebraic methods. It is shown that the GMD function is non-decreasing as a function of its first argument and non-increasing as a function of the second argument. For vanishing ideals over finite fields, we show that the GMD function is in fact strictly decreasing as a function of the second argument until it stabilizes. We also study algebraic invariants of Geramita ideals. Those ideals are graded, unmixed, 1-dimensional and their associated primes are generated by linear forms. We also examine GMD functions of complete intersections and show some special cases of two conjectures of Toh?neanu-Van Tuyl and Eisenbud-Green-Harris.

We introduce a new class of algebraic varieties which we call frieze varieties. Each frieze variety is determined by an acyclic quiver. The frieze variety is defined in an elementary recursive way by constructing a set of points in affine space. From a more conceptual viewpoint, the coordinates of these points are specializations of cluster variables in the cluster algebra associated to the quiver. We give a new characterization of the finite-tame-wild trichotomy for acyclic quivers in terms of their frieze varieties by showing that an acyclic quiver is representation finite, tame, or wild, respectively, if and only if the dimension of its frieze variety is 0,1, or ?2, respectively.

Symbolic powers are a classical commutative algebra topic that relates to primary decomposition, consisting, in some circumstances, of the functions that vanish up to a certain order on a given variety. However, these are notoriously difficult to compute, and there are seemingly simple questions related to symbolic powers that remain open even over polynomial rings. In this paper, we describe a Macaulay2 software package that allows for computations of symbolic powers of ideals and which can be used to study the equality and containment problems, among others.

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 self-intersection. We use the representation theory of the stabilizers of the singular points to discover several invariant curves of negative self-intersection on X, and use these curves to study Nagata-type questions for linear series on X.

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.

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.

Fermat ideals define planar point configurations that are closely related to the base locus of a specific pencil of curves. We describe the minimal generators and free resolutions of the ordinary powers and many of the symbolic powers of these ideals. We show that the symbolic Rees algebras of Fermat ideals are Noetherian.

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.

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.

J. Pure Appl. Algebra 222 (2018) no. 9, 2524-2551 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.

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 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.

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 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 an exponential function in d. Here is the Macaulay 2 code mentioned in the paper.

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.