Toric ideals and diagonal 2-minors

Let $G$ be a simple graph on the vertex set $\{1,\ldots,n\}$ with $m$ edges. An algebraic object attached to $G$ is the ideal $P_{G}$ generated by diagonal 2-minors of an $n \times n$ matrix of variables. In this paper we prove that if $G$ is bipartite, then every initial ideal of $P_{G}$ is generated by squarefree monomials of degree at most $\left \lfloor{\frac{m+n+1}{2}} \right \rfloor$. Furthermore, we completely characterize all connected graphs $G$ for which $P_{G}$ is the toric ideal associated to a finite simple graph. Finally we compute in certain cases the universal Gr{\"o}bner basis of $P_{G}$.Comment: To appear in Acta Mathematica Hungaric

Projections of cones and the arithmetical rank of toric varieties

Let $I_M$ and $I_N$ be defining ideals of toric varieties such that $I_M$ is a projection of $I_N$, i.e. $I_N \subseteq I_M$. We give necessary and sufficient conditions for the equality $I_M=rad(I_N+(f_1,...,f_s))$, where $f_1,...,f_s$ belong to $I_M$. Also a method for finding toric varieties which are set-theoretic complete intersection is given. Finally we apply our method in the computation of the arithmetical rank of certain toric varieties and provide the defining equations of the above toric varieties.Comment: To appear in the Journal of Pure and Applied Algebr

Arithmetical rank and cohomological dimension of generalized binomial edge ideals

Let $G$ be a connected and simple graph on the vertex set $[n]$. To the graph $G$ one can associate the generalized binomial edge ideal $J_{m}(G)$ in the polynomial ring $R=K[x_{ij}: i \in [m], j \in [n]]$. We provide a lower bound for the cohomological dimension of $J_{m}(G)$. We also study when $J_{m}(G)$ is a cohomologically complete intersection. Finally, we show that the arithmetical rank of $J_{2}(G)$ equals the projective dimension of $R/J_{2}(G)$ in several cases.Comment: Journal of Algebra and its Applications (to appear

On the binomial arithmetical rank of lattice ideals

To any lattice $L \subset \mathbb{Z}^{m}$ one can associate the lattice ideal $I_{L} \subset K[x_{1},...,x_{m}]$. This paper concerns the study of the relation between the binomial arithmetical rank and the minimal number of generators of $I_{L}$. We provide lower bounds for the binomial arithmetical rank and the $\mathcal{A}$-homogeneous arithmetical rank of $I_{L}$. Furthermore, in certain cases we show that the binomial arithmetical rank equals the minimal number of generators of $I_{L}$. Finally we consider a class of determinantal lattice ideals and study some algebraic properties of them.Comment: 22 page

An indispensable classification of monomial curves in \mathbb{A}^4(\mathbbmss{k})

In this paper a new classification of monomial curves in \mathbb{A}^4(\mathbbmss{k}) is given. Our classification relies on the detection of those binomials and monomials that have to appear in every system of binomial generators of the defining ideal of the monomial curve; these special binomials and monomials are called indispensable in the literature. This way to proceed has the advantage of producing a natural necessary and sufficient condition for the definining ideal of a monomial curve in \mathbb{A}^4(\mathbbmss{k}) to have a unique minimal system of binomial generators. Furthermore, some other interesting results on more general classes of binomial ideals with unique minimal system of binomial generators are obtained.Comment: 17 pages; fixed typos, added some clarifying remarks, minor corrections to the original version. Accepted for publication in Pacific Journal of Mathematic

Binomial generation of the radical of a lattice ideal

Let $I_{L, \rho}$ be a lattice ideal. We provide a necessary and sufficient criterion under which a set of binomials in $I_{L, \rho}$ generate the radical of $I_{L, \rho}$ up to radical. We apply our results to the problem of determining the minimal number of generators of $I_{L, \rho}$ or of the $rad(I_{L, \rho})$ up to radical.Comment: 14 pages, to appear in Journal of Algebr

Minimal systems of binomial generators and the indispensable complex of a toric ideal

Let $A=\{{\bf a}_1,...,{\bf a}_m\} \subset \mathbb{Z}^n$ be a vector configuration and $I_A \subset K[x_1,...,x_m]$ its corresponding toric ideal. The paper consists of two parts. In the first part we completely determine the number of different minimal systems of binomial generators of $I_A$. We also prove that generic toric ideals are generated by indispensable binomials. In the second part we associate to $A$ a simplicial complex \Delta _{\ind(A)}. We show that the vertices of \Delta_{\ind(A)} correspond to the indispensable monomials of the toric ideal $I_A$, while one dimensional facets of \Delta_{\ind(A)} with minimal binomial $A$-degree correspond to the indispensable binomials of $I_{A}$