相关论文: Betti numbers of ideals generated by $n+1$ powers …

Let $M$ be a finitely generated module over a local ring $(R,\mathfrak{m})$. By $\mathcal{S}_j(M)$, we denote the $j$th symmetric power of $M$ ($j$th graded component of the symmetric algebra $\mathcal{S}_R(M)$). The purpose of this paper…

We prove that for a finite set of points $X$ in the projective $n$-space over any field, the Betti number $\beta_{n,n+1}$ of the coordinate ring of $X$ is non-zero if and only if $X$ lies on the union of two planes whose sum of dimension is…

We give a necessary and sufficient condition for a standard graded Artinian ring defined by an m-full ideal, to have the weak Lefschetz property in terms of graded Betti numbers. This is a generalization of a theorem of Wiebe for…

In a paper by Lin an interesting family of semipermutations comes out to index the elements of a cohomology basis of a Hessenberg type variety. The corresponding Betti numbers are a generalization of Eulerian numbers. We show three…

Inspired by recent work in the theory of central projections onto hypersurfaces, we characterize self-linked perfect ideals of grade 2 as those with a Hilbert--Burch matrix that has a maximal symmetric subblock. We also prove that every…

Let $J$ be a quadratically presented grade three Gorenstein ideal in the standard graded polynomial ring $R= k[x,y,z]$, where $k$ is a field. Assume that $R/J$ satisfies the weak Lefschetz property. We give the presentation matrix for $J$…

We investigate products J of ideals of "row initial" minors in the polynomial ring K[X] defined by a generic m-by-n matrix. Such ideals are shown to be generated by a certain set of standard bitableaux that we call superstandard. These…

A minor is principal means it is defined by the same row and column indices. Let $X$ be a square generic matrix, $K[X]$ the polynomial ring in entries of $X$, over an algebraically closed field, $K$. For fixed $t\leq n$, let $\mathfrak P_t$…

We give an explicit set of generators for the semigroup of the Gr\"obner degeneration of a toric ideal. This set of generators is used to study algebraic properties of the semigroup it generates: approximation of semigroups,…

Consider ideals $I$ of the form \[ I=(x_1^2,\dots, x_n^2)+\mathrm{RLex}(x_ix_j) \] where $\mathrm{RLex}(x_ix_j)$ is the ideal generated by all the square-free monomials which are greater than or equal to $x_ix_j$ in the reverse…

We give a sufficient condition for a monomial ideal to have a nonzero Betti number in each multidegree. In the case of facet ideals of simplicial forests, this condition becomes a necessary one and it allows us to characterize Betti…

Let $(R,\mathfrak{m},\Bbbk)$ be a regular local ring of dimension 3. Let $I$ be a Gorenstein ideal of $R$ of grade 3. It follows from a result of Buchsbaum and Eisenbud that there is a skew-symmetric matrix of odd size such that $I$ is…

In the first part of the paper we answer (positively) a question raised by the first author which has to do with some sort of rigity of the tail of resolution of an ideal. Let $I$ be a homogeneous ideal in a polynomial ring over a field of…

Fix a pair of positive integers d and n. We create a ring R and a complex G of R-modules with the following universal property. Let P be a polynomial ring in d variables over a field and let I be a grade d Gorenstein ideal in P which is…

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…

The conjecture of Kalai, Kleinschmidt, and Lee on the number of empty simplices of a simplicial polytope is established by relating it to the first graded Betti numbers of the polytope. The proof allows us to derive explicit optimal bounds…

Polygon spaces like $M_\ell=\{(u_1,...,u_n)\in S^1\times... S^1 ;\ \sum_{i=1}^n l_iu_i=0\}/SO(2)$ or they three dimensional analogues $N_\ell$ play an important r\^ole in geometry and topology, and are also of interest in robotics where the…

The generators of the algebra $gl_{n+1}$ in a form of differential operators of the first order acting on ${\bf R}^n$ with matrix coefficients are explicitly written. The algebraic Hamiltonians for a matrix generalization of $3-$body…

We use purely combinatorial arguments to give a formula to compute all graded Betti numbers of path ideals of line graphs and cycles. As a consequence we can give new and short proofs for the known formulas of regularity and projective…

Let $J_{n,m} = (x_1\cdots x_{m},x_2 \cdots x_{m+1},\ldots,x_{n}x_1\cdots x_{m-1})$ be the $m$-path ideal of a cycle of length $n \ge 5$ over a polynomial ring $S = k[x_1,\ldots,x_n]$. Let $t\geq 1$ be an integer. We show that $J_{n,m}^t$…