Related papers: On some ideals with linear free resolutions

Given $\Sigma\subset R:=\mathbb K[x_1,\ldots,x_k]$, where $\mathbb K$ is a field of characteristic 0, any finite collection of linear forms, some possibly proportional, and any $1\leq a\leq |\Sigma|$, we prove that $I_a(\Sigma)$, the ideal…

Let $\Sigma$ be a finite collection of linear forms in $\mathbb K[x_0,\ldots,x_n]$, where $\mathbb K$ is a field. Denote ${\rm Supp}(\Sigma)$ to be the set of all nonproportional elements of $\Sigma$, and suppose ${\rm Supp}(\Sigma)$ is…

Let $\mathbb K$ be a field of characteristic 0. Given $n$ linear forms in $R=\mathbb K[x_1,\ldots,x_k]$, with no two proportional, in one of our main results we show that the ideal $I\subset R$ generated by all $(n-2)$-fold products of…

Let $R = k[x_1, \dotsc , x_n]$ denote the standard graded polynomial ring over a field $k$. We study certain classes of equigenerated monomial ideals with the property that the so-called complementary ideal has no linear relations on the…

Let $I$ be a graded ideal of $K[x_1,\ldots,x_n]$ generated by homogeneous polynomials of a same degree $d$, and assume that $I$ has linear quotients. In this note, we use Horseshoe Lemma to give a relatively direct inductive construction of…

An ideal $I \subset \mathbb{k}[x_1, \ldots, x_n]$ is said to have linear powers if $I^k$ has a linear minimal free resolution, for all $k$. In this paper we study the Betti numbers of $I^k$, for ideals $I$ with linear powers. The Betti…

The ideals generated by fold products of linear forms are generalizations of powers of defining ideals of star configurations, or of Veronese type ideals, and in this paper we study their Betti numbers. In earlier work, the authors together…

We study monomial ideals with linear presentation or partially linear resolution. We give combinatorial characterizations of linear presentation for square-free ideals of degree 3, and for primary ideals whose resolutions are linear except…

A graded ideal $I$ in $\mathbb{K}[x_1,\ldots,x_n]$, where $\mathbb{K}$ is a field, is said to have almost maximal finite index if its minimal free resolution is linear up to the homological degree $\mathrm{pd}(I)-2$, while it is not linear…

For a graded ideal I in a graded ring, the deviation of I is defined as the difference between the minimal number of generators of I and its grade. In this article, we provide bigraded free resolutions of the symmetric algebras for specific…

Let I(G) be the edge ideal associated to a simple graph G. We study the graded Betti numbers that appear in the linear strand of the minimal free resolution of I(G).

Given a collection of $t$ subspaces in an $n$-dimensional $\mathbb{K} $-vector space $W$ we can associate to them $t$ vanishing ideals in the symmetric algebra $\mathcal{S}(W^*) = \mathbb{K}[x_1,x_2,\dots,x_n]$. As a subspace is defined by…

Let $I_1,\dots,I_n$ be ideals generated by linear forms in a polynomial ring over an infinite field and let $J = I_1 \cdots I_n$. We describe a minimal free resolution of $J$ and show that it is supported on a polymatroid obtained from the…

Consider an ideal $I\subset K[x_1,..., x_n]$, with $K$ an arbitrary field, generated by monomials of degree two. Assuming that $I$ does not have a linear resolution, we determine the step $s$ of the minimal graded free resolution of $I$…

In recent years, the combinatorial properties of monomials ideals and binomial ideals have been widely studied. In particular, combinatorial interpretations of free resolution algorithms have been given in both cases. In this present work,…

We construct two families of free resolutions that resolve the ideals of certain opposite Schubert varieties restricted to the big open cell. We conjecture that these examples have genericity properties translating to structure theorems for…

Let $S = K[x_1, \dots, x_n]$ be the standard graded polynomial ring over a field $K$. In this paper, we address and completely solve two fundamental open questions in Commutative Algebra: (i) For which degrees $d$, does there exist a…

Let $ X $ be an $ m \times n $ matrix of distinct indeterminates over a field $ K $, where $ m \le n $. Set the polynomial ring $ K[X] := K[X_{ij} : 1 \le i \le m, 1 \le j \le n] $. Let $ 1 \le k < l \le n $ be such that $ l - k + 1 \ge m…

We show that if $G$ is a gap-free and diamond-free graph, then $I(G)^s$ has a linear minimal free resolution for every $s\geq 2$.

Semidefinite programming is based on optimization of linear functionals over convex sets defined by linear matrix inequalities, namely, inequalities of the form $$L_A(X)=I-A_1X_1-\dots-A_g X_g\succeq0.$$ Here the $X_j$ are real numbers and…