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Jorge Neves

Publications and source records attributed to Jorge Neves.

16 recordsLinked to original sources
arXiv2024

On the minimal generating sets of the Eulerian ideal

We study the minimal homogeneous generating sets of the Eulerian ideal associated with a simple graph and its maximal generating degree. We show that the Eulerian ideal is a lattice ideal and use this to give a characterization of binomials belonging to a minimal homogeneous generating set. In this way, we obtain an explicit minimal homogeneous generating set. We find an upper bound for the maximal generating degree in terms of the graph. This invariant is half the number of edges of a largest Eulerian subgraph of even cardinality without even-chords. We show that for bipartite graphs this invariant is the maximal generating degree. In particular, we prove that if the graph is bipartite, the Eulerian ideal is generated in degree $2$ if and only if the graph is chordal. Furthermore, we show that the maximal generating degree is also $2$ when the graph is a complete graph.

math.AC↗
arXiv2023

On the socle of Artinian algebras associated to graphs

Given a simple graph, consider the polynomial ring with coefficients in a field and variables identified with the edges of the graph. Given a non-empty even cardinality Eulerian subgraph and a choice of half of its edges, consider the homogeneous binomial obtained by taking the product of these edges minus the product of the remaining edges of the subgraph. We define a homogeneous ideal by taking as generators all binomials obtained in this way, varying the Eulerian subgraph and the choice of half of its edges, together with the squares of the variables of the ring. This ideal is related to the Eulerian ideal, introduced by Neves, Vaz Pinto and Villarreal. We call the corresponding quotient the Eulerian Artinian algebra associated to the graph. The goal of the present work is to study the socle of these algebras through the lens of graph theory. Our main results include a combinatorial characterization of a monomial basis of the socle, a characterization of Gorenstein Eulerian Artinian algebras in the case of bipartite graphs and the computation of the h-vector and socle degrees in the cases of a complete graph and a complete bipartite graph.

math.AC↗
arXiv2021

Parameterized codes over graphs

In this article we review known results on parameterized linear codes over graphs, introduced by Renter\'ia, Simis and Villarreal in 2011. Very little is known about their basic parameters and invariants. We review in detail the parameters dimension, regularity and minimum distance. As regards the parameter dimension, we explore the connection to Eulerian ideals in the ternary case and we give new combinatorial formulas.

math.AC↗
arXiv2020

Eulerian ideals

Let $G$ be a simple graph and $I(X_G)=\varphi^{-1}(x_i^2-x_j^2 : i,j\in V_G)$, where $\varphi \colon K[E_G]\to K[V_G]$ is the homomorphism that sends an edge to the product of its vertices. The ideal $I(X_G)$ is Cohen--Macaulay, one-dimensional and binomial. If $G$ is bipartite, it is known that the Castelnuovo--Mumford regularity of $I(X_G)$ is equal to the maximum cardinality of a set of edges having no more than half of the edges of any Eulerian subgraph of $G$. Here, with respect to the grevlex order associated to an ordering of the edge set of $G$, we describe a Gr\"obner basis for $I(X_G)$, and we characterize the standard monomials of the ideal $(I(X_G),t_e)$ in terms of even sets of vertices marked with a parity. Using these results, we give a combinatorial interpretation of the degree of $I(X_G)$, via the set of even sets of vertices of $G$; and we show that the Castelnuovo--Mumford regularity of $I(X_G)$, for any graph, is the maximum cardinality of a set of edges having no more than half of the edges of any \emph{even} Eulerian subgraph of $G$ or, equivalently, the maximum cardinality of a minimum fixed parity $T$-join.

math.CO↗
arXiv2019

Joins, Ears and Castelnuovo-Mumford regularity

We introduce a new class of polynomial ideals associated to a simple graph, $G$. Let $K[E_G]$ be the polynomial ring on the edges of $G$ and $K[V_G]$ the polynomial ring on the vertices of $G$. We associate to $G$ an ideal, $I(X_G)$, defined as the preimage of $(x_i^2-x_j^2 : i,j\in V_G)\subseteq K[V_G]$ by the map $K[E_G]\to K[V_G]$ which sends a variable, $t_e$, associated to an edge $e=\{i,j\}$, to the product $x_ix_j$ of the variables associated to its vertices. We show that $K[E_G]/I(X_G)$ is a one-dimensional, Cohen-Macaulay, graded ring, that $I(X_G)$ is a binomial ideal and that, with respect to a fixed monomial order, its initial ideal has a generating set independent of the field $K$. We focus on the Castelnuovo-Mumford regularity of $I(X_G)$ providing the following sharp upper and lower bounds: $$ \mu(G) \leq \operatorname{reg} I(X_G) \leq |V_G|-b_0(G)+1, $$ where $\mu(G)$ is the maximum vertex join number of the graph and $b_0(G)$ is the number of its connected components. We show that the lower bound is attained for a bipartite graph and use this to derive a new combinatorial result on the number of even length ears of nested ear decomposition.

math.AC↗
arXiv2018

Regularity of the vanishing ideal over a bipartite nested ear decomposition

We study the Castelnuovo-Mumford regularity of the vanishing ideal over a bipartite graph endowed with a decomposition of its edge set. We prove that, under certain conditions, the regularity of the vanishing ideal over a bipartite graph obtained from a graph by attaching a path of length $\ell$ increases by $\lfloor \frac{\ell}{2}\rfloor (q-2)$, where $q$ is the order of the field of coefficients. We use this result to show that the regularity of the vanishing ideal over a bipartite graph, $G$, endowed with a weak nested ear decomposition is equal to $$\textstyle \frac{|V_G|+ \epsilon -3}{2}(q-2),$$ where $\epsilon$ is the number of even length ears and pendant edges of the decomposition. As a corollary, we show that for bipartite graph, the number of even length ears in a nested ear decomposition starting from a vertex is constant.

math.AC↗
arXiv2016

Regularity of the vanishing ideal over a parallel composition of paths

Let G be a graph obtained by taking r>=2 paths and identifying all first vertices and identifying all the last vertices. We compute the Castelnuovo--Mumford regularity of the quotient S/I(X), where S is the polynomial ring on the edges of G and I(X) is the vanishing ideal of the projective toric subset parameterized by G. The case we consider is the first case where the regularity was unknown, following earlier computations (by several authors) of the regularity when G is a tree, cycle, complete graph or complete bipartite graph, but specially in light of the reduction of the computation of the regularity in the bipartite case to the computation of the regularity of the blocks of G. We also prove new inequalities relating the Castelnuovo--Mumford regularity of S/I(X) with the combinatorial structure of G, for a general graph.

math.AC↗
arXiv2013

Vanishing ideals over complete multipartite graphs

We study the vanishing ideal of the parametrized algebraic toric associated to the complete multipartite graph $\G=\mathcal{K}_{α_1,...,α_r}$ over a finite field of order $q$. We give an explicit family of binomial generators for this lattice ideal, consisting of the generators of the ideal of the torus, (referred to as type I generators), a set of quadratic binomials corresponding to the cycles of length 4 in $\G$ and which generate the \emph{toric algebra of $\G$} (type II generators) and a set of binomials of degree $q-1$ obtained combinatorially from $\G$ (type III generators). Using this explicit family of generators of the ideal, we show that its Castelnuovo--Mumford regularity is equal to $\max\set{α_1(q-2),...,α_r(q-2), \lceil (n-1)(q-2)/2\rceil}$, where $n=α_1+... + α_r$.

math.AC↗
arXiv2013

Codes over a weighted torus

We define weighted projective Reed-Muller codes over a subset of weighted projective space over a finite field. We focus on the case when the set X is a projective weighted torus. We show that the vanishing ideal of X is a lattice ideal and relate it with the lattice ideal of a minimal presentation of the semigroup algebra of Q, the numerical semigroup generated by the weights of the projective space. We compute the index of regularity of the vanishing ideal as function of the weights and the Frobenius number of Q. We compute the basic parameters of weighted projective Reed-Muller codes over a 1-dimensional weighted torus and prove they are maximum distance separable codes.

math.AC↗
arXiv2013

New examples of Calabi-Yau threefolds and genus zero surfaces

We classify the subgroups of the automorphism group of the product of 4 projective lines admitting an invariant anticanonical smooth divisor on which the action is free. As a first application, we describe new examples of Calabi-Yau 3-folds with small Hodge numbers. In particular, the Picard number is 1 and the number of moduli is 5. Furthermore, the fundamental group is non-trivial. We also construct a new family of minimal surfaces of general type with geometric genus zero, K^2=3 and fundamental group of order 16. We show that this family dominates an irreducible component of dimension 4 of the moduli space of the surfaces of general type.

math.AG↗
arXiv2013

Regularity and algebraic properties of certain lattice ideals

We study the regularity and the algebraic properties of certain lattice ideals. We establish a map I --> I\~ between the family of graded lattice ideals in an N-graded polynomial ring over a field K and the family of graded lattice ideals in a polynomial ring with the standard grading. This map is shown to preserve the complete intersection property and the regularity of I but not the degree. We relate the Hilbert series and the generators of I and I\~. If dim(I)=1, we relate the degrees of I and I\~. It is shown that the regularity of certain lattice ideals is additive in a certain sense. Then, we give some applications. For finite fields, we give a formula for the regularity of the vanishing ideal of a degenerate torus in terms of the Frobenius number of a semigroup. We construct vanishing ideals, over finite fields, with prescribed regularity and degree of a certain type. Let X be a subset of a projective space over a field K. It is shown that the vanishing ideal of X is a lattice ideal of dimension 1 if and only if X is a finite subgroup of a projective torus. For finite fields, it is shown that X is a subgroup of a projective torus if and only if X is parameterized by monomials. We express the regularity of the vanishing ideal over a bipartie graph in terms of the regularities of the vanishing ideals of the blocks of the graph.

math.AC↗
arXiv2011

Vanishing ideals over graphs and even cycles

Let X be an algebraic toric set in a projective space over a finite field. We study the vanishing ideal, I(X), of X and show some useful degree bounds for a minimal set of generators of I(X). We give an explicit description of a set of generators of I(X), when X is the algebraic toric set associated to an even cycle or to a connected bipartite graph with pairwise disjoint even cycles. In this case, a fomula for the regularity of I(X) is given. We show an upper bound for this invariant, when X is associated to a (not necessarily connected) bipartite graph. The upper bound is sharp if the graph is connected. We are able to show a formula for the length of the parameterized linear code associated with any graph, in terms of the number of bipartite and non-bipartite components.

math.AC↗
arXiv2011

Unprojection and deformations of tertiary Burniat surfaces

We construct a 4-dimensional family of surfaces of general type with p_g=0 and K^2=3 and fundamental group Z/2xQ_8, where Q_8 is the quaternion group. The family constructed contains the Burniat surfaces with K^2=3. Additionally, we construct the universal coverings of the surfaces in our family as complete intersections on (\PP^1)^4 and we also give an action of Z/2xQ_8 on (\PP^1)^4 lifting the natural action on the surfaces. The strategy is the following. We consider an \'etale (Z/2)^3-cover T of a surface with p_g=0 and K^2=3 and assume that it may be embedded in a Fano 3-fold V. We construct V by using the theory of parallel unprojection. Since V is an Enriques--Fano 3-fold, considering its Fano cover yields the simple description of the universal covers above.

math.AG↗
arXiv2010

Parallel Kustin--Miller unprojection with an application to Calabi--Yau geometry

Kustin--Miller unprojection constructs more complicated Gorenstein rings from simpler ones. Geometrically, it inverts certain projections, and appears in the constructions of explicit birational geometry. However, it is often desirable to perform not only one but a series of unprojections. The main aim of the present paper is to develop a theory, which we call parallel Kustin--Miller unprojection, that applies when all the unprojection ideals of a series of unprojections correspond to ideals already present in the initial ring. As an application of the theory, we explicitly construct 7 families of Calabi--Yau 3-folds of high codimensions.

math.AG↗
arXiv2007

A construction of numerical Campedelli Surfaces with \Z/6 torsion group

We produce a family of numerical Campedelli surfaces with \Z/6 torsion by constructing the (Gorenstein codimension 5) canonical ring of the étale six to one cover using serial unprojection. In Section 2 we develop the necessary algebraic machinery. Section 3 contains the numerical Campedelli surface construction, while Section 4 contains remarks and open questions.

math.AG↗