Related papers: Toric ideals of simple surface singularities

We provide a new combinatorial approach to study the minimal free resolutions of edge ideals, that is, quadratic square-free monomial ideals. With this method we can recover most of the known results on resolutions of edge ideals with…

To integer programming problems, computational algebraic approaches using Grobner bases or standard pairs via the discreteness of toric ideals have been studied in recent years. Although these approaches have not given improved time…

This paper investigates the Poisson geometry associated to a cluster algebra over the complex numbers, and its relationship to compatible torus actions. We show, under some assumptions, that each Noetherian cluster algebra has only finitely…

This work concerns the study of properties of a group of Koszul algebras coming from the toric ideals of a chordal bipartite infinite family of graphs (alternately, these rings may be interpreted as coming from determinants of certain…

Let C be an irreducible projective curve of degree d in Pn(K), where K is an algebraically closed field, and let I be the associated homogeneous prime ideal. We wish to compute generators for I, assuming we are given sufficiently many…

We study points of moderately low degree on a curve $C$ over a number field, which is embedded on a nice toric surface $S$. Recently, Smith and Vogt related the linear equivalence classes of such points to intersections of $C$ with curves…

The generating functions of the Severi degrees for sufficiently ample line bundles on algebraic surfaces are multiplicative in the topological invariants of the surface and the line bundle. Recently new proofs of this fact were given for…

Let K be a finite field and let X* be an affine algebraic toric set parameterized by monomials. We give an algebraic method, using Groebner bases, to compute the length and the dimension of C_X*(d), the parameterized affine code of degree d…

In this paper, it is proved that, if a toric ideal possesses a fundamental binomial none of whose monomials is squarefree, then the corresponding semigroup ring is not very ample. Moreover, very ample semigroup rings of Lawrence type are…

The distance-reduction property for a generating set, i.e., a Markov basis, of a toric ideal is a condition that ensures tight connectivity of its fibres. In this paper, we study the distance-reduction property for toric ideals of graphs…

In the first part of this paper we study scrollers and linearly joined varieties. A particular class of varieties, of important interest in classical Geometry are Cohen--Macaulay varieties of minimal degree. They appear naturally studying…

Recently, it was shown that a binary linear code can be associated to a binomial ideal given as the sum of a toric ideal and a non-prime ideal. Since then two different generalizations have been provided which coincide for the binary case.…

We survey some recent results on the minimal graded free resolution of a square-free monomial ideal. The theme uniting these results is the point-of-view that the generators of a monomial ideal correspond to the maximal faces (the facets)…

Let $R=k[x,y,z]$ be a standard graded $3$-variable polynomial ring, where $k$ denotes any field. We study grade $3$ homogeneous ideals $I \subseteq R$ defining compressed rings with socle $k(-s) \oplus k(-2s+1)$, where $s \geq3$ is some…

Let $I = ( f_1, \dots, f_n )$ be a homogeneous ideal in the polynomial ring $K[x_1, \dots,x_n]$ over a field $K$ generated by generic polynomials. Using an incremental approach based on a method by Gao, Guan and Volny, and properties of the…

It has been shown previously that a large class of monomial maps equivariant under the action of an infinite symmetric group have finitely generated kernels up to the symmetric action. We prove that these symmetric toric ideals also have…

We introduce and study vertex cover algebras of weighted simplicial complexes. These algebras are special classes of symbolic Rees algebras. We show that symbolic Rees algebras of monomial ideals are finitely generated and that such an…

Let $S=K[x_1,\dots,x_n]$ be the polynomial ring over a field $K$ and $I\subset S$ be a squarefree monomial ideal generated in degree $n-2$. Motivated by the remarkable behavior of the powers of $I$ when $I$ admits a linear resolution, as…

We obtain tight bounds for the minimal number of generators of an ideal with bounded-degree generators in a polynomial ring $K[X_1,\dots,X_n],$ as well as a sharp quantification of the maximum possible size of a minimal generating set of…

We enumerate complex curves on toric surfaces of any given degree and genus, having a single cusp and nodes as their singularities, and matching appropriately many point constraints. The solution is obtained via tropical enumerative…