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Motivated by using combinatorics to study jets of monomial ideals, we extend a definition of jets from graphs to clutters. We offer some structural results on their vertex covers, and show an interesting connection between the cover ideal of the jets of a clutter and the symbolic powers of the cover ideal of the original clutter. We use this connection to prove that jets of very well-covered graphs are very well-covered. Next, we turn our attention to principal jets of monomial ideals, describing their primary decomposition and minimal generating sets. Finally, we give formulas to compute various algebraic invariants of principal jets of monomial ideals, including their Hilbert series, Betti numbers, multiplicity and regularity., Comment: 29 pages

The connected sum construction, which takes as input Gorenstein rings and produces new Gorenstein rings, can be considered as an algebraic analogue for the topological construction having the same name. We determine the graded Betti numbers for connected sums of graded Artinian Gorenstein algebras. Along the way, we find the graded Betti numbers for fiber products of graded rings; an analogous result was obtained in the local case by Geller. We relate the connected sum construction to the doubling construction, which also produces Gorenstein rings. Specifically, we show that a connected sum of doublings is the doubling of a fiber product ring.

We introduce a Macaulay2 package for working with jet schemes. The main method constructs jets of ideals, polynomial rings and their quotients, ring homomorphisms, affine varieties, and (hyper)graphs. The package also includes additional methods to compute principal components and radicals of jets of monomial ideals., Comment: 8 pages, 1 figure

Finite group actions on free resolutions and modules arise naturally in many interesting examples. Understanding these actions amounts to describing the terms of a free resolution or the graded components of a module as group representations which, in the non modular case, are completely determined by their characters. With this goal in mind, we introduce a Macaulay2 package for computing characters of finite groups on free resolutions and graded components of finitely generated graded modules over polynomial rings., Comment: 7 pages

Under reasonable assumptions, a group action on a module extends to the minimal free resolutions of the module. Explicit descriptions of these actions can lead to a better understanding of free resolutions by providing, for example, convenient expressions for their differentials or alternative characterizations of their Betti numbers. This article introduces an algorithm for computing characters of finite groups acting on minimal free resolutions of finitely generated graded modules over polynomial rings., Comment: 11 pages, minor edits

We define an operation of jets on graphs inspired by the corresponding notion in commutative algebra and algebraic geometry. We examine a few graph theoretic properties and invariants of this construction, including chromatic numbers, co-chordality, and vertex covers., Comment: fixed some typos, cleaned up notations, figures, and examples

We describe a significant update to the existing InvariantRing package for Macaulay2. In addition to expanding and improving the methods of the existing package for actions of finite groups, the updated package adds functionality for computing invariants of diagonal actions of tori and finite abelian groups as well as invariants of arbitrary linearly reductive group actions. The implementation of the package has been completely overhauled with the aim of serving as a unified resource for invariant theory computations in Macaulay2., Comment: 8 pages, submitted to Journal of Software for Algebra and Geometry. v2: Added a link to the GitHub repository for the package

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., Comment: Corrected typo in Example 5.8

We compute the graded Betti numbers for the toric ideal of a family of graphs constructed by adjoining a cycle to a complete bipartite graph. The key observation is that this family admits an initial ideal which has linear quotients. As a corollary, we compute the Hilbert series and $h$-vector for all the toric ideals of graphs in this family., Comment: 12 pages; revised and corrected; to appear in Journal of Algebra and its Applications

A $\Bbbk$-configuration is a set of points $\mathbb{X}$ in $\mathbb{P}^2$ that satisfies a number of geometric conditions. Associated to a $\Bbbk$-configuration is a sequence $(d_1,\ldots,d_s)$ of positive integers, called its type, which encodes many of its homological invariants. We distinguish $\Bbbk$-configurations by counting the number of lines that contain $d_s$ points of $\mathbb{X}$. In particular, we show that for all integers $m \gg 0$, the number of such lines is precisely the value of $\Delta \mathbf{H}_{m\mathbb{X}}(m d_s -1)$. Here, $\Delta \mathbf{H}_{m\mathbb{X}}(-)$ is the first difference of the Hilbert function of the fat points of multiplicity $m$ supported on $\mathbb{X}$., Comment: Revised version of paper; most changes minor except the proof of Lemma 4.1 which has been rewritten; to appear in Illinois Journal of Mathematics

In this paper, we study the geometry of various Hessenberg varieties in type A, as well as families thereof, with the additional goal of laying the groundwork for future computations of Newton-Okounkov bodies of Hessenberg varieties. Our main results are as follows. We find explicit and computationally convenient generators for the local defining ideals of indecomposable regular nilpotent Hessenberg varieties, and then show that all regular nilpotent Hessenberg varieties are local complete intersections. We also show that certain families of Hessenberg varieties, whose generic fibers are regular semisimple Hessenberg varieties and the special fiber is a regular nilpotent Hessenberg variety, are flat and have reduced fibres. This result further allows us to give a computationally effective formula for the degree of a regular nilpotent Hessenberg variety with respect to a Pl\"ucker embedding. Furthermore, we construct certain flags of subvarieties of a regular nilpotent Hessenberg variety, obtained by intersecting with Schubert varieties, which are suitable for computing Newton-Okounkov bodies. As an application of our results, we explicitly compute many Newton-Okounkov bodies of the two-dimensional Peterson variety with respect to Pl\"ucker embeddings., Comment: 25 pages. Corrected minor errors in Example 3.5 and in Sections 6 and 7. The main results remain the same

We consider ideals in a polynomial ring that are generated by regular sequences of homogeneous polynomials and are stable under the action of the symmetric group permuting the variables. In previous work, we determined the possible isomorphism types for these ideals. Following up on that work, we now analyze the possible degrees of the elements in such regular sequences. For each case of our classification, we provide some criteria guaranteeing the existence of regular sequences in certain degrees., Comment: 22 pages

Let $I$ be a homogeneous ideal of $\Bbbk[x_0,\ldots,x_n]$. To compare $I^{(m)}$, the $m$-th symbolic power of $I$, with $I^m$, the regular $m$-th power, we introduce the $m$-th symbolic defect of $I$, denoted $\operatorname{sdefect}(I,m)$. Precisely, $\operatorname{sdefect}(I,m)$ is the minimal number of generators of the $R$-module $I^{(m)}/I^m$, or equivalently, the minimal number of generators one must add to $I^m$ to make $I^{(m)}$. In this paper, we take the first step towards understanding the symbolic defect by considering the case that $I$ is either the defining ideal of a star configuration or the ideal associated to a finite set of points in $\mathbb{P}^2$. We are specifically interested in identifying ideals $I$ with $\operatorname{sdefect}(I,2) = 1$., Comment: To appear in Journal of Pure and Applied Algebra; revised at referees' suggestion. Fixed typos and clarified writing, included additional references, shortened proof of Thm 6.3

An explicit construction is given of a minimal free resolution of the ideal generated by all squarefree monomials of a given degree. The construction relies upon and exhibits the natural action of the symmetric group on the syzygy modules. The resolution is obtained over an arbitrary coefficient ring; in particular, it is characteristic free. Two applications are given: an equivariant resolution of De Concini-Procesi rings indexed by hook partitions, and a resolution of FI-modules., Comment: Updated references

We consider complete intersection ideals in a polynomial ring over a field of characteristic zero that are stable under the action of the symmetric group permuting the variables. We determine the possible representation types for these ideals, and describe formulas for the graded characters of the corresponding quotient rings., Comment: 14 pages, includes referee's suggestions for publication in Communications in Algebra

The action of a torus on a graded module over a polynomial ring extends to the entire minimal free resolution of the module. We explain how to determine the action of the torus on the free modules in the resolution, when the resolution can be calculated explicitly. The problem is reduced to analyzing how the weights of a torus propagate along an equivariant map of free modules. The results obtained are used to design algorithms which have been implemented in the software system Macaulay2., Comment: Includes various corrections, a new introductory example and examples for the algorithms

The irreducible representations of complex semisimple algebraic groups with finitely many orbits are parametrized by graded simple Lie algebras. For the exceptional Lie algebras, Kraskiewicz and Weyman exhibit the Hilbert polynomials and the expected minimal free resolutions of the normalization of the orbit closures. We present an interactive method to construct explicitly these and related resolutions in Macaulay2. The method is then used in the cases of the Lie algebras of type $E_6$, $F_4$ and $G_2$, to confirm the shape of the expected resolutions as well as some geometric properties of the orbit closures., Comment: v2: fixed section 5.1 v3: added some references, updated links

Adem and Reichstein introduced the ideal of truncated symmetric polynomials to present the permutation invariant subring in the cohomology of a finite product of projective spaces. Building upon their work, I describe a generating set of the ideal of truncated symmetric polynomials in arbitrary positive characteristic, and offer a conjecture for minimal generators., Comment: 11 pages, contains referee suggested edits for publication in the Journal of Pure and Applied Algebra