Initial Ideals Research Articles

Unmixed bipartite graphs and sublattices of the Boolean lattices

The correspondence between unmixed bipartite graphs and sublattices of the oolean lattice is discussed. By using this correspondence, we show the existence of squarefree quadratic initial ideals of toric ideals arising from minimal vertex covers of unmixed bipartite graphs.

Hilbert function of local Artinian level rings in codimension two

In this paper we characterize the possible Hilbert functions of a local Artinian level ring of given type and socle degree in the codimension two case. In particular, given an “admissible” numerical function h, an effective method is given to produce a zero-dimensional ideal I of R = k 〚 x , y 〛 such that R / I is a local Artinian level ring having h as Hilbert function. As a consequence we recover the well-known result proved by Briançon and Iarrobino concerning the Hilbert functions of a complete intersection of height two. We extend notions of standard or Gröbner bases from the graded to the local algebra context. The principal tools concern the concepts of enhanced standard basis and generic initial ideals which are defined for an ideal in the formal power series ring R = k 〚 x 1 , … , x n 〛 .

Initial ideals of tangent cones to Schubert varieties in orthogonal Grassmannians

We compute the initial ideals, with respect to certain conveniently chosen term orders, of ideals of tangent cones at torus fixed points to Schubert varieties in orthogonal Grassmannians. The initial ideals turn out to be square-free monomial ideals and therefore define Stanley–Reisner face rings of simplicial complexes. We describe these complexes. The maximal faces of these complexes encode certain sets of non-intersecting lattice paths.

Enacting the vision for sustainable development

Utopia, as a concept, is now used with varying intentions, engendering semantic confusion. When substituted with the notion of visioning, it can be also problematic, if sometimes for different reasons. An implicit engagement in visioning is examined in a community-based futures process for creating a strategy for the sustainable management of natural resources in the Mary River catchment, in Queensland, Australia. It is found that visioning, if it can be binding and impelling for common-interest groups, can be just as unrealistically utopian, or as slippery as the scaly creatures it is trying to protect, when applied afresh to a community exploring change from a variety of initial ideals and needs. It may be seen as an emergent property of active learning.

Gröbner bases and Betti numbers of monoidal complexes

In this note we consider monoidal complexes and their associated algebras, called toric face rings. These rings generalize Stanley-Reisner rings and affine monoid algebras. We compute initial ideals of the presentation ideal of a toric face ring, and determine its graded Betti numbers. Our results generalize celebrated theorems of Hochster in combinatorial commutative algebra.

A Swift European Literary Response to Castro's Revolution: Juan Goytisolo's Cuban Travel Book

In December 1961, almost three years after the downfall of the Batista regime, Juan Goytisolo arrived in Cuba and spent ten weeks travelling around the island. His motives for visiting post-revolutionary Cuba were political and also deeply personal. In 1963 Pueblo en marcha, the record of part of these travels, was published in Paris. In this travel book, the author provides an enthusiastic account of the aims of the Cuban Revolution, the socio-economic achievements of its early years, and the widespread support for Fidel Castro. However, later in the 1960s, convinced that the leadership had abandoned its initial ideals of social justice and had developed a repressive, Soviet-style regime, Goytisolo became completely disillusioned and rejected what he saw as a betrayal of the original revolutionary movement.

An algorithm for lifting points in a tropical variety

The aim of this paper is to give a constructive proof of one of the basic theorems of tropical geometry: given a point on a tropical variety (defined using initial ideals), there exists a Puiseuxvalued “lift” of this point in the algebraic variety. This theorem is so fundamental because it justifies why a tropical variety (defined combinatorially using initial ideals) carries information about algebraic varieties: it is the image of an algebraic variety over the Puiseux series under the valuation map. We have implemented the “lifting algorithm” usingSingular and Gfan if the base field is ℚ. As a byproduct we get an algorithm to compute the Puiseux expansion of a space curve singularity in (K n+1, 0).

Finiteness of Hilbert functions and bounds for Castelnuovo-Mumford regularity of initial ideals

Bounds for the Castelnuovo-Mumford regularity and Hilbert coefficients are given in terms of the arithmetic degree (if the ring is reduced) or in terms of the defining degrees. From this it follows that there exists only a finite number of Hilbert functions associated with reduced algebras over an algebraically closed field with a given arithmetic degree and dimension. A good bound is also given for the Castelnuovo-Mumford regularity of initial ideals which depends neither on term orders nor on the coordinates and holds for any field.

Conditions for generic initial ideals to be almost reverse lexicographic

Let I be a homogeneous Artinian ideal in a polynomial ring R = k [ x 1 , … , x n ] over a field k of characteristic 0. We study an equivalent condition for the generic initial ideal gin ( I ) with respect to reverse lexicographic order to be almost reverse lexicographic. As a result, we show that the Moreno-Socias conjecture implies the Fröberg conjecture and the Pardue conjecture. And for the case codim I ⩽ 3 , we show that R / I has the strong Lefschetz property if and only if gin ( I ) is almost reverse lexicographic. Finally we give a positive partial answer to the Moreno-Socias conjecture, and hence to the Pardue conjecture.

Rigidity of Linear Strands and Generic Initial Ideals

Let K be a field, S a polynomial ring and E an exterior algebra over K, both in a finite set of variables. We study rigidity properties of the graded Betti numbers of graded ideals in S and E when passing to their generic initial ideals. First, we prove that if the graded Betti numbers for some i > 1 and k ≥ 0, then for all q ≥ i, where I ⊂ S is a graded ideal. Second, we show that if for some i > 1 and k ≥ 0, then for all q ≥ 1, where I ⊂ E is a graded ideal. In addition, it will be shown that the graded Betti numbers for all i ≥ 1 if and only if I(k) and I(k+1) have a linear resolution. Here I(d) is the ideal generated by all homogeneous elements in I of degree d, and R can be either the polynomial ring or the exterior algebra.

Generic initial ideals of Artinian ideals having Lefschetz properties or the strong Stanley property

For a standard Artinian k-algebra A = R / I , we give equivalent conditions for A to have the weak (or strong) Lefschetz property or the strong Stanley property in terms of the minimal system of generators of gin ( I ) . Using the equivalent condition for the weak Lefschetz property, we show that some graded Betti numbers of gin ( I ) are determined just by the Hilbert function of I if A has the weak Lefschetz property. Furthermore, for the case that A is a standard Artinian k-algebra of codimension 3, we show that every graded Betti number of gin ( I ) is determined by the graded Betti numbers of I if A has the weak Lefschetz property. And if A has the strong Lefschetz (respectively Stanley) property, then we show that the minimal system of generators of gin ( I ) is determined by the graded Betti numbers (respectively by the Hilbert function) of I.

A spherical initial ideal for Pfaffians

We determine a term order on the monomials in the variables X-ij, 1 <= i <= j <= n, such that corresponding initial ideal of the ideal of Pfaffians of degree r of a generic n by n skew-symmetric matrix is the Stanley-Reisner ideal of a join of a simplicial sphere and a simplex. Moreover, we demonstrate that the Pfaffians of the 2r by 2r skew-symmetric submatrices form a, Grobner basis for the given term order. The same methods and similar term orders as for the Pfaffians also yield squarefree initial ideals for certain determinantal ideals. Yet, in contrast to the case of Pfaffians, the corresponding simplicial complexes are balls that do not decompose into a join as above.

Generic Initial Ideals of Arithmetically Cohen–Macaulay Projective Subschemes

For an arithmetically Cohen–Macaulay closed subscheme X in ℙ n with the saturated defining ideal I X ⊂ k[x 0,…, x n ], let gin(I X ) be the generic initial ideal under the reverse lexicographic order. In this article, we find the minimal system of generators of gin (I X ) in terms of characters f i ′s, which will be defined in (3.1). Then we obtain the formula for deg(X) in terms of characters. For a curve C in ℙ n , we get the formula for the arithmetic genus in terms of characters of a general hyperplane section and the number of sporadic zeros. As an application, we give a new proof of the following upper bound on the regularity given in Ahn and Migliore (2007) and Nagel (1989): if X ⊂ ℙ n is an arithmetically Cohen–Macaulay closed subscheme, then where s = {l | (I X ) l ≠ 0}. Furthermore we give an equivalent condition for an arithmetically Cohen–Macaulay closed subscheme X ⊂ ℙ n to satisfy the equality from the view point of the generic initial ideal of I X , this shows that the upper bound is sharp.

Hilbert functions of d-regular ideals

In the present paper, we characterize all possible Hilbert functions of graded ideals in a polynomial ring whose regularity is smaller than or equal to d, where d is a positive integer. In addition, we prove the following result which is a generalization of Bigatti, Hulett and Pardue's result: Let p ⩾ 0 and d > 0 be integers. If the base field is a field of characteristic 0 and there is a graded ideal I whose projective dimension proj dim ( I ) is smaller than or equal to p and whose regularity reg ( I ) is smaller than or equal to d, then there exists a monomial ideal L having the maximal graded Betti numbers among graded ideals J which have the same Hilbert function as I and which satisfy proj dim ( J ) ⩽ p and reg ( J ) ⩽ d . We also prove the same fact for squarefree monomial ideals. The main methods for proofs are generic initial ideals and combinatorics on strongly stable ideals.

Hilbert scheme strata defined by bounding cohomology

Let Hilb p be the Hilbert scheme parametrizing the closed subschemes of P K n with Hilbert polynomial p ∈ Q [ t ] over a field K of characteristic zero. By bounding below the cohomological Hilbert functions of the points of Hilb p we define locally closed subspaces of the Hilbert scheme. The aim of this paper is to show that some of these subspaces are connected. For this we exploit the edge ideals constructed by D. Mall in [D. Mall, Connectedness of Hilbert function strata and other connectedness results, J. Pure Appl. Algebra 150 (2000) 175–205]. It turns out that these ideals are sequentially Cohen–Macaulay and that their initial ideals with respect to the reverse lexicographic term order are generic initial ideals.

Generic initial ideals and squeezed spheres

In 1988 Kalai constructed a large class of simplicial spheres, called squeezed spheres, and in 1991 presented a conjecture about generic initial ideals of Stanley–Reisner ideals of squeezed spheres. In the present paper this conjecture will be proved. In order to prove Kalai's conjecture, based on the fact that every squeezed ( d − 1 ) -sphere is the boundary of a certain d-ball, called a squeezed d-ball, generic initial ideals of Stanley–Reisner ideals of squeezed balls will be determined. In addition, generic initial ideals of exterior face ideals of squeezed balls are determined. On the other hand, we study the squeezing operation, which assigns to each Gorenstein* complex Γ having the weak Lefschetz property a squeezed sphere Sq ( Γ ) , and show that this operation increases graded Betti numbers.

Generic initial ideals and exterior algebraic shifting of the join of simplicial complexes

In this paper, the relation between algebraic shifting and join which was conjectured by Eran Nevo will be proved. Let σ and τ be simplicial complexes and σ*τ be their join. Let Jσ be the exterior face ideal of σ and Δ(σ) the exterior algebraic shifted complex of σ. Assume that σ*τ is a simplicial complex on [n]={1,2,..., n}. For any d-subset S⊂[n], let $m_{\preceq_{\textrm{rev}}S}(\sigma)$ denote the number of d-subsets R∈σ which are equal to or smaller than S with respect to the reverse lexicographic order. We will prove that $m_{\preceq_{\textrm{rev}}S}(\Delta(\sigma*\tau))\geq m_{\preceq_{\textup{rev}}S}(\Delta(\Delta(\sigma) *\Delta(\tau)))$ for all S⊂[n]. To prove this fact, we also prove that $m_{\preceq_{\textrm{rev}}S}(\Delta(\sigma))\geq m_{\preceq_{\textup{rev}}S}(\Delta(\Delta_{\varphi}(\sigma)))$ for all S⊂[n] and for all nonsingular matrices ϕ, where Δϕ(σ) is the simplicial complex defined by $J_{\Delta_{\varphi}(\sigma)}=\textup{in}(\varphi(J_{\sigma}))$.

Generic initial ideals and graded Artinian-level algebras not having the Weak-Lefschetz Property

We find a sufficient condition that H is not level based on a reduction number. In particular, we prove that a graded Artinian algebra of codimension 3 with Hilbert function H = ( h 0 , h 1 , … , h d − 1 > h d = h d + 1 ) cannot be level if h d ≤ 2 d + 3 , and that there exists a level O-sequence of codimension 3 of type H for h d ≥ 2 d + k for k ≥ 4 . Furthermore, we show that H is not level if β 1 , d + 2 ( I lex ) = β 2 , d + 2 ( I lex ) , and also prove that any codimension 3 Artinian graded algebra A = R / I cannot be level if β 1 , d + 2 ( Gin ( I ) ) = β 2 , d + 2 ( Gin ( I ) ) . In this case, the Hilbert function of A does not have to satisfy the condition h d − 1 > h d = h d + 1 . Moreover, we show that every codimension n graded Artinian level algebra having the Weak-Lefschetz Property has a strictly unimodal Hilbert function having a growth condition on ( h d − 1 − h d ) ≤ ( n − 1 ) ( h d − h d + 1 ) for every d > θ where h 0 < h 1 < ⋯ < h α = ⋯ = h θ > ⋯ > h s − 1 > h s . In particular, we show that if A is of codimension 3, then ( h d − 1 − h d ) < 2 ( h d − h d + 1 ) for every θ < d < s and h s − 1 ≤ 3 h s , and prove that if A is a codimension 3 Artinian algebra with an h -vector ( 1 , 3 , h 2 , … , h s ) such that h d − 1 − h d = 2 ( h d − h d + 1 ) > 0 and soc ( A ) d − 1 = 0 for some r 1 ( A ) < d < s , then ( I ≤ d + 1 ) is ( d + 1 ) -regular and dim k soc ( A ) d = h d − h d + 1 .

Some geometric results arising from the Borel fixed property

In this paper, we will give some geometric results using generic initial ideals for the degree reverse lex order. The main goal of the paper is to improve on results of Bigatti, Geramita and Migliore concerning geometric consequences of maximal growth of the Hilbert function of the Artinian reduction of a set of points. When the points have the Uniform Position Property, the consequences are even more striking. Here we weaken the growth condition, assuming only that the values of the Hilbert function of the Artinian reduction are equal in two consecutive degrees, and that the first of these degrees is greater than the second reduction number of the points. We continue to get nice geometric consequences even from this weaker assumption. However, we have surprising examples to show that imposing the Uniform Position Property on the points does not give the striking consequences that one might expect. This leads to a better understanding of the Hilbert function, and the ideal itself, of a set of points with the Uniform Position Property, which is an important open question. In the last section we describe the role played by the Weak Lefschetz Property (WLP) in this theory, and we show that the general hyperplane section of a smooth curve may not have the WLP.

Face Numbers and Nongeneric Initial Ideals

Certain necessary conditions on the face numbers and Betti numbers of simplicial complexes endowed with a proper action of a prime order cyclic group are established. A notion of colored algebraic shifting is defined and its properties are studied. As an application a new simple proof of the characterization of the flag face numbers of balanced Cohen-Macaulay complexes originally due to Stanley (necessity) and Björner, Frankl, and Stanley (sufficiency) is given. The necessity portion of their result is generalized to certain conditions on the face numbers and Betti numbers of balanced Buchsbaum complexes.