Related papers: Toric ideals of simple surface singularities
In this article, we investigate the strongly robust property of toric ideals associated with weighted oriented graphs. We establish that the toric ideals of a broad class of monomial ideals are strongly robust; this class encompasses the…
We investigate, using the notion of linear quotients, significative classes of connected graphs whose monomial edge ideals, not necessarily squarefree, have linear resolution, in order to compute standard algebraic invariants of the…
The universal Gr\"obner basis of an ideal is a Gr\"obner basis with respect to all term orders simultaneously. The aim of this paper is to present an algorithmic approach to compute the universal Gr\"obner basis for the toric ideal…
An ideal I is robust if its universal Gr\"obner basis is a minimal generating set for this ideal. In this paper, we generalize the meaning of robust ideals. An ideal is defined as generalized robust if its universal Gr\"obner basis is equal…
Given a finite, simple, vertex-weighted graph, we construct a graded associative (non-commutative) algebra, whose generators correspond to vertices and whose ideal of relations has generators that are graded commutators corresponding to…
In the first part of this article we show for some examples of surfaces of general type in toric 3-folds how to construct minimal and canonical models by toric methods explicitly. The examples we study turn out to be surfaces of general…
We compute the initial ideals, with respect to certain conveniently chosen term orders, of ideals of tangent cones at torus fixed points to Schubert varieties in orthogonal Grassmannians. The initial ideals turn out to be square-free…
The goal of this paper is to explicitly describe a minimal binomial generating set of a class of lattice ideals, namely the ideal of certain Veronese $3$-fold projections. More precisely, for any integer $d\ge 4$ and any $d$-th root $e$ of…
First we solve the problem of finding minimal degree families on toric surfaces by reducing it to lattice geometry. Then we describe how to find minimal degree families on, more generally, rational complex projective surfaces.
We investigate the algebra and geometry of general interventions in discrete DAG models. To this end, we introduce a theory for modeling soft interventions in the more general family of staged tree models and develop the formalism to study…
This is a survey article on Gorenstein initial complexes of extensively studied ideals in commutative algebra and algebraic geometry. These include defining ideals of Segre and Veronese varieties, toric deformations of flag varieties known…
We encode the binomials belonging to the toric ideal $I_A$ associated with an integral $d \times n$ matrix $A$ using a short sum of rational functions as introduced by Barvinok \cite{bar,newbar}. Under the assumption that $d,n$ are fixed,…
We identify several classes of monomial ideals that possess minimal generalized Barile-Macchia resolutions. These classes of ideals include generic monomial ideals, monomial ideals with linear quotients, and edge ideals of hypertrees. We…
In this paper we prove the existence of a special order on the set of minimal monomial generators of powers of edge ideals of arbitrary graphs. Using this order we find new upper bounds on the regularity of powers of edge ideals of graphs…
This paper is concerned with the combinatorial description of the graded minimal free resolution of certain monomial algebras which includes toric rings. Concretely, we explicitly describe how the graded minimal free resolution of those…
We describe the ideals, especially the prime ideals, of semirings of polynomials over layered domains, and in particular over supertropical domains. Since there are so many of them, special attention is paid to the ideals arising from…
In this paper we describe the equations defining the multi-Rees algebra $k[x_1,\dots,x_n][I_1^{a_1}t_1,\dots,I_r^{a_r}t_r]$, where the ideals are generated by subsets of $x_1,\dots,x_n$. We also show that a family of binomials whose leading…
We prove that for any toric ideal of a graph the degree of any element of Graver basis is bounded above by an exponential function of the maximal degree of a circuit.
In this paper, we are interested in the generic initial ideals of \textit{singular} projective curves with respect to the graded lexicographic order. Let $C$ be a \textit{singular} irreducible projective curve of degree $d\geq 5$ with the…
Let $(R,\mathfrak{m},\Bbbk)$ be a regular local ring of dimension 3. Let $I$ be a Gorenstein ideal of $R$ of grade 3. Buchsbaum and Eisenbud proved that there is a skew-symmetric matrix of odd size such that $I$ is generated by the…