Related papers: Rees algebras of complementary edge ideals

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 study the defining equations of the Rees algebra of square-free monomial ideals in a polynomial ring over a field. We determine that when an ideal $I$ is generated by $n$ square-free monomials of the same degree then $I$ has relation…

Let $M$ and $N$ be two monomials of the same degree, and let $I$ be the smallest Borel ideal containing $M$ and $N$. We show that the toric ring of $I$ is Koszul by constructing a quadratic Gr\"obner basis for the associated toric ideal.…

In this paper we describe the defining equations of the Rees algebra and the special fiber ring of a truncation I of a complete intersection ideal in a polynomial ring over a field with homogeneous maximal ideal m. To describe explicitly…

Let $S={\sf k}[X_1,\dots, X_n]$ be a polynomial ring, where ${\sf k}$ is a field. This article deals with the defining ideal of the Rees algebra of squarefree monomial ideal generated in degree $n-2$. As a consequence, we prove that Betti…

We study the Koszul property of a standard graded $K$-algebra $R$ defined by the binomial edge ideal of a pair of graphs $(G_1,G_2)$. We show that the following statements are equivalent: (i) $R$ is Koszul; (ii) the defining ideal…

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 $k$ be a field of characteristic zero, and $R=k[x_1, \ldots, x_d]$ with $d \geq 3$ be a polynomial ring in $d$ variables. Let $\m=(x_1, \ldots, x_d)$ be the homogeneous maximal ideal of $R$. Let $\mathcal{K}$ be the kernel of the…

Let $G$ be a simple graph on $n$ vertices and $J_G$ denote the binomial edge ideal of $G$ in the polynomial ring $S = \mathbb{K}[x_1, \ldots, x_n, y_1, \ldots, y_n].$ In this article, we compute the second graded Betti numbers of $J_G$, and…

The aims of this work are to study Rees algebras of filtrations of monomial ideals associated to covering polyhedra of rational matrices with non-negative entries and non-zero columns using combinatorial optimization and integer…

Generalizing techniques that prove that Veronese subrings are Koszul, we show that Rees and multi-Rees algebras of certain types of principal strongly stable ideals are Koszul. We provide explicit Gr\"obner basis for the defining ideals of…

We investigate the Rees algebra and the toric ring of the squarefree monomial ideal associated to the three-dimensional Ferrers diagram. Under the projection property condition, we describe explicitly the presentation ideals of the Rees…

An almost complete intersection ideal can be seen as a $d$-sequence ideal with the minimal number of generators being one more than its height. In this paper, we give exact formulas for the regularity of powers of graded almost complete…

Let S=K[x_1,...,x_n] be a polynomial ring over a field K and I a homogeneous ideal in S generated by a regular sequence f_1,f_2,...,f_k of homogeneous forms of degree d. We study a generalization of a result of Conca, Herzog, Trung, and…

Given a skew-symmetric matrix $X$, the Pfaffian of $X$ is defined as the square root of the determinant of $X$. In this article, we give the explicit defining equations of the Rees algebra of a Pfaffian ideal $I$ generated by the maximal…

Let $(A,\mathfrak{m})$ be an excellent local complete intersection ring and let $I = (a_1, \ldots, a_r)$ be an ideal of positive height. Let $\mathcal{R}(I) = A[It]$ be the Rees algebra of $I$. Consider the map $\psi \colon S = A[X_1,…

Let I be the ideal of minors of a 2 by n matrix of linear forms with the expected codimension. In this paper we prove that the Rees algebra of I and its special fiber ring are Cohen-Macaulay and Koszul; in particular, they are quadratic…

Let $S=K[x_1,\ldots,x_n]$ be the polynomial ring in $n$ variables over a field $K$ and $I\subset S$ a squarefree monomial ideal. In the present paper we are interested in the monomials $u \in S$ belonging to the socle $\Soc(S/I^{k})$ of…

There is a natural epimorphism from the symmetric algebra to the Rees algebra of an ideal. When this epimorphism is an isomorphism, we say that the ideal is of linear type. Given two determinantal rings over a field, we consider the…

Let ${\sf k}$ be a field, $S$ be a bigraded ${\sf k}$-algebra, and $S_\Delta$ denote the diagonal subalgebra of $S$ corresponding to $\Delta = \{ (cs,es) \; | \; s \in \mathbb{Z} \}$. It is know that the $S_\Delta$ is Koszul for $c,e \gg…