Gotzmann lexsegment ideals

Linear resolutions of $t$-spread lexsegment ideals via Betti splittings

Let $S=K[x_1,\dots,x_n]$ be a polynomial ring in $n$ variables with coefficients over a field $K$. A $t$-spread lexsegment ideal $I$ of $S$ is a monomial ideal generated by a $t$-spread lexsegment set. We determine all $t$-spread lexsegment ideals with linear resolution by means of Betti splittings. As applications we provide formulas for the Betti numbers of such a class of ideals and furthermore we characterize all incompletely $t$-spread lexsegment ideals with linear quotients.

The Lefschetz Property for Componentwise Linear Ideals and Gotzmann Ideals

For standard graded Artinian $K$-algebras defined by componentwise linear ideals and Gotzmann ideals, we give conditions for the weak Lefschetz property in terms of numerical invariants of the defining ideals.

Normally torsion-free lexsegment ideals

In this paper we characterize all the lexsegment ideals which are normally torsion-free. Our characterization is given in terms of the ends of the lexsegment. We also prove that the property of being normally torsion-free is equivalent to the property of the depth function of being constant.

Powers of lexsegment ideals with linear resolution

All powers of lexsegment ideals with linear resolution (equivalently, with linear quotients) have linear quotients with respect to suitable orders of the minimal monomial generators. For a large subclass of the lexsegment ideals the corresponding Rees algebra has a quadratic Gröbner basis, thus it is Koszul. We also find other classes of monomial ideals with linear quotients whose powers have linear quotients too.

Arithmetical rank of lexsegment edge ideals

Let $I\subset S=K[x_1,...,x_n]$ be a lexsegment edge ideal or the Alexander dual of such an ideal. In both cases it turns out that the arithmetical rank of $I$ is equal to the projective dimension of $S/I.$

Algebraic properties of universal squarefree lexsegment ideals

Let K be a field and let A be the polynomial ring in n variables with coefficients in the field K We study the universal squarefree lexsegment ideals in A. We put our attention on their combinatorics computing some invariants. Moreover we study the link between such special class of squarefree lexsegment ideals and the so called s-sequences.

Lexsegment ideals are sequentially Cohen-Macaulay

The associated primes of an arbitrary lexsegment ideal $I\subset S=K[x_1,...,x_n]$ are determined. As application it is shown that $S/I$ is a pretty clean module, therefore, $S/I$ is sequentially Cohen-Macaulay and satisfies Stanley's conjecture.

Gotzmann ideals of the polynomial ring

Let $A = K[x_1, ..., x_n]$ denote the polynomial ring in $n$ variables over a field $K$. We will classify all the Gotzmann ideals of $A$ with at most $n$ generators. In addition, we will study Hilbert functions $H$ for which all homogeneous ideals of $A$ with the Hilbert function $H$ have the same graded Betti numbers. These Hilbert functions will be called inflexible Hilbert functions. We introduce the notion of segmentwise critical Hilbert function and show that segmentwise critical Hilbert functions are inflexible.

Lexsegment ideals and their h-polynomials

Let $S = K[x_1, \ldots, x_n]$ denote the polynomial ring in $n$ variables over a field $K$ with each $\mathrm{deg}\ x_i = 1$ and $I \subset S$ a homogeneous ideal of $S$ with $\dim S/I = d$. The Hilbert series of $S/I$ is of the form $h_{S/I}(λ)/(1 - λ)^d$, where $h_{S/I}(λ) = h_0 + h_1λ+ h_2λ^2 + \cdots + h_sλ^s$ with $h_s \neq 0$ is the $h$-polynomial of $S/I$. Given arbitrary integers $r \geq 1$ and $s \geq 1$, a lexsegment ideal $I$ of $S = K[x_1, \ldots, x_n]$, where $n \leq \max\{r, s\} + 2$, satisfying $\mathrm{reg}(S/I) = r$ and $ \mathrm{deg}\ h_{S/I}(λ) = s$ will be constructed.

Monomial ideals under ideal operations

In this paper, we show for a monomial ideal $I$ of $K[x_1,x_2,\ldots,x_n]$ that the integral closure $\ol{I}$ is a monomial ideal of Borel type (Borel-fixed, strongly stable, lexsegment, or universal lexsegment respectively), if $I$ has the same property. We also show that the $k^{th}$ symbolic power $I^{(k)}$ of $I$ preserves the properties of Borel type, Borel-fixed and strongly stable, and $I^{(k)}$ is lexsegment if $I$ is stably lexsegment. For a monomial ideal $I$ and a monomial prime ideal $P$, a new ideal $J(I, P)$ is studied, which also gives a clear description of the primary decompos…

A combinatorial proof of Gotzmann's persistence theorem for monomial ideals

Gotzmann proved the persistence for minimal growth for ideals. His theorem is called Gotzmann's persistence theorem. In this paper, based on the combinatorics on binomial coefficients, a simple combinatorial proof of Gotzmann's persistence theorem in the special case of monomial ideals is given.

Classes of sequentially Cohen-Macaulay squarefree monomial ideals

We compute the minimal primary decomposition for completely squarefree lexsegment ideals. We show that critical squarefree monomial ideals are sequentially Cohen-Macaulay. As an application, we give a complete characterization of the completely squarefree lexsegment ideals which are sequentially Cohen-Macaulay and we also derive formulas for some homological invariants of this class of ideals.

Gotzmann Edge Ideals

Let P = k[x_1, ..., x_n] be the polynomial ring in n variables. A homogeneous ideal I of P generated in degree d is called Gotzmann if it has the smallest possible Hilbert function out of all homogeneous ideals with the same dimension in degree d. The edge ideal of a simple graph G on vertices x_1, ..., x_n is the quadratic square-free monomial ideal generated by all x_i x_j where {x_i,x_j} is an edge of G. The only edge ideals that are Gotzmann are those edge ideals corresponding to star graphs.

Lexsegment ideals of Hilbert depth 1

Let $I\subset S=\KK[x_1,...,x_n]$ be a lexsegment ideal, generated by monomials of degree $d$. The main aim of this paper is to characterize when the Hilbert depth of $I$ will be 1, in the standard graded case. In addition to this, we will give an estimate of depth of squarefree monomial ideals, generalizing a result of Popescu. We will also show that Stanley conjecture holds for squarefree stable ideals, in the multigraded case.

A local version of Gotzmann's Persistence

Gotzmann's Persistence states that the growth of an arbitrary ideal can be controlled by comparing it to the growth of the lexicographic ideal. This is used, for instance, in finding equations which cut out the Hilbert scheme (of subschemes of $\mathbf{P}^n$ with fixed Hilbert polynomial) sitting inside an appropriate Grassmannian. We introduce the notion of an {\it extremal ideal} which extends the notion of the lex ideal to other term orders. We then state and prove a version of Gotzmann's theorem for these ideals, valid in an open subset of a Grassmannian.

Gotzmann Monomials In Four Variables

It is a widely open problem to determine which monomials in the n-variable polynomial ring $K[x_1,...,x_n]$ over a field $K$ have the Gotzmann property, i.e. induce a Borel-stable Gotzmann monomial ideal. Since 2007, only the case $n \le 3$ was known. Here we solve the problem for the case $n = 4$. The solution involves a surprisingly intricate characterization.

Classes of Monomial Ideals

In this thesis, we focus on the study of some classes of monomial ideals, namely lexsegment ideals and monomial ideals with linear quotients.

关于单项式的Gotzmann阈值

On the Gotzmann threshold of monomials

本文提出一种有效算法，确定任意 n 元多项式环中单项式的 Gotzmann 阈值，并应用于 x_2^d 情形，给出 n=5 的精确结果及 n≥6 的猜想。

Chordality, $d$-collapsibility, and componentwise linear ideals

Using the concept of $d$-collapsibility from combinatorial topology, we define chordal simplicial complexes and show that their Stanley-Reisner ideals are componentwise linear. Our construction is inspired by and an extension of "chordal clutters'' which was defined by Bigdeli, Yazdan Pour and Zaare-Nahandi in 2017, and characterizes Betti tables of all ideals with linear resolution in a polynomial ring. We show $d$-collapsible and $d$-representable complexes produce componentwise linear ideals for appropriate $d$. Along the way, we prove that there are generators that when added to the ideal,…