Related papers: On the Rees algebra of certain codimension two per…

In this paper we investigate the Rees algebras of squarefree monomial ideals $I \subset S=K[x_1,\dots,x_n]$ generated in degree $n-2$, where $K$ is a field. Every such ideal arises as the complementary edge ideal $I_c(G)$ of a finite simple…

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 $\mathfrak P_t$…

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…

For an ideal $I$ in a Noetherian ring $R$, we introduce and study its conductor as a tool to explore the Rees algebra of $I$. The conductor of $I$ is an ideal $C(I)\subset R$ obtained from the defining ideals of the Rees algebra and the…

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…

Using linear algebra methods we study certain algebraic properties of monomial rings and matroids. Let I be a monomial ideal in a polynomial ring over an arbitrary field. If the Rees cone of I is quasi-ideal, we express the normalization of…

Consider a rational projective plane curve C parameterized by three homogeneous forms h1,h2,h3 of the same degree d in the polynomial ring R=k[x,y] over the field k. Extracting a common factor, we may harmlessly assume that the ideal…

Let $I \subset R = \mathbb{F}[x_1,x_2]$ be a height two ideal minimally generated by three homogeneous polynomials of the same degree $d$, where $\mathbb{F}$ is a field of characteristic zero. We use the theory of $D$-modules to deduce…

The goal of this paper is the fine structure of the ideals in the title, with emphasis on the properties of the associated Rees algebra and the special fiber. The watershed between the present approach and some of the previous work in the…

Let G be a perfect graph and let J be its ideal of vertex covers. We show that the Rees algebra of J is normal and that this algebra is Gorenstein if G is unmixed. Then we give a description--in terms of cliques--of the symbolic Rees…

Finite generation of the symbolic Rees ring of a space monomial prime ideal of a 3-dimensional weighted polynomial ring is a very interesting problem. Negative curves play important roles in finite generation of these rings. We are…

Let $V$ be a smooth scheme over a field $k$, and let $\{I_n, n\geq 0\}$ be a filtration of sheaves of ideals in $\calo_V$, such that $I_0=\calo_V$, and $I_s\cdot I_t\subset I_{s+t}$. In such case $\bigoplus I_n$ is called a Rees algebra. A…

Given a symbolic power of a homogeneous ideal in a polynomial ring, we study the problem of determining which powers of the ideal contain it. For ideals defining 0-dimensional subschemes of projective space, as an immediate corollary of our…

In this thesis we are interested in studying algebraic properties of monomial algebras, that can be linked to combinatorial structures, such as graphs and clutters, and to optimization problems. A goal here is to establish bridges between…

We study ideals $\mathcal{I}$ on $\mathbb{N}$ satisfying the following Baire-type property: if $X$ is a complete metric space and $\{X_{A} \colon A \in \mathcal{I} \}$ is a family of nowhere dense subsets of $X$ with $X_{A} \subset X_{B}$…

The question of when the Rees algebra ${\mathcal R} (I)= \bigoplus_{n \ge 0}I^n$ of $I$ is an almost Gorenstein graded ring is explored, where $R$ is a two-dimensional regular local ring and $I$ a contracted ideal of $R$. It is known that…

Given an affine algebra $R=K[x_1,\dots,x_n]/I$ over a field $K$, where $I$ is an ideal in the polynomial ring $P=K[x_1,\dots,x_n]$, we examine the task of effectively calculating re-embeddings of $I$, i.e., of presentations $R=P'/I'$ such…

Let A be a local ring with maximal ideal m. For an arbitrary ideal I of A, we define the generalized Hilbert coefficients j_k(I) \in Z^{k+1} (k=0,1,...,dim A). When the ideal I is m-primary, j_k(I)=(0,...,0,(-1)^k e_k(I)), where e_k(I) is…

In this paper, we study arithmetic Macaulayfication of projective schemes and Rees algebras of ideals. We give a necessary and sufficient condition for a nonsingular projective scheme to have an arithmetic Macaulayfication. If the scheme is…

Let $R$ be a polynomial ring in $N$ variables over an arbitrary field $K$ and let $I$ be an ideal of $R$ generated by $n$ polynomials of degree at most 2. We show that there is a bound on the projective dimension of $R/I$ that depends only…