Monomial localizations and polymatroidal ideals

In this paper we consider monomial localizations of monomial ideals and conjecture that a monomial ideal is polymatroidal if and only if all its monomial localizations have a linear resolution. The conjecture is proved for squarefree monomial ideals where it is equivalent to a well-known characterization of matroids. We prove our conjecture in many other special cases. We also introduce the concept of componentwise polymatroidal ideals and extend several of the results known for polymatroidal ideals to this new class of ideals.     

Certain variants of multipermutohedron ideals

Multipermutohedron ideals have rich combinatorial properties. An explicit combinatorial formula for the multigraded Betti numbers of a multipermutohedron ideal and their Alexander duals are known. Also, the dimension of the Artinian quotient of an Alexander dual of a multipermutohedron ideal is the number of generalized parking functions. In this paper, monomial ideals which are certain variants of multipermutohedron ideals are studied. Multigraded Betti numbers of these variant monomial ideals and their Alexander duals are obtained. Further, many interesting combinatorial properties of multipermutohedron ideals are extended to these variant monomial ideals.     

Tameness of local cohomology of monomial ideals with respect to monomial prime ideals

In this paper we consider the local cohomology of monomial ideals with respect to monomial prime ideals and show that all these local cohomology modules are tame.     

Betti-linear ideals

We introduce the notion of a Betti-linear monomial ideal, which generalizes the notion of lattice-linear monomial ideal introduced by Clark. We provide a characterization of Betti-linearity in terms of Tchernev's poset construction. As an application we obtain an explicit canonical construction for the minimal free resolutions of monomial ideals having pure resolutions.     

Cohen-Macaulay monomial ideals with given radical

We study the set of Cohen-Macaulay monomial ideals with a given radical. Among this set of ideals are the so-called Cohen-Macaulay modifications. Not all Cohen-Macaulay squarefree monomial ideals admit nontrivial Cohen-Macaulay modifications. It is shown that if there exists one such modification, then there exist indeed infinitely many.     

Principal Borel ideals and Gotzmann ideals

In this paper we characterize all principal Borel ideals with Borel generator up to degree 4 which are Gotzmann. We also classify principal Borel ideals with a Borel generator of degree d which are lexsegment and we describe the shadows of principal Borel ideals. Finally, we discuss the corresponding results for squarefree monomial ideals.Received: 10 May 2002     

Reverse lex ideals

We study reverse lex ideals in a polynomial ring, and compare their properties to those of lex ideals. In particular we provide an analogue of Green's Theorem for reverse lex ideals. We also compare the Betti numbers of strongly stable and square-free strongly stable monomial ideals to those of reverse lex ideals.     

On the stable set of associated prime ideals of a monomial ideal

It is shown that any set of nonzero monomial prime ideals can be realized as the stable set of associated prime ideals of a monomial ideal. Moreover, an algorithm is given to compute the stable set of associated prime ideals of a monomial ideal.     

Freiman ideals

In this paper we study the Freiman inequality for the least number of generators of the square of an equigenerated monomial ideal. Such an ideal is called a Freiman ideal if equality holds in the Freiman inequality. We classify all Freiman ideals of maximal height, the Freiman ideals of certain classes of principal Borel ideals, the Hibi ideals which are Freiman, and classes of Veronese type ideals which are Freiman.     

On the fiber cone of monomial ideals

Archiv der Mathematik - We consider the fiber cone of monomial ideals. It is shown that for monomial ideals $$I\subset K[x,y]$$ of height 2, generated by 3 elements, the fiber cone F(I) of I is a...     

Betti numbers of mixed product ideals

We compute the Betti numbers of the resolution of a special class of square-free monomial ideals, the ideals of mixed products. Moreover when these ideals are Cohen-Macaulay we calculate their type. Received: 9 March 2008     

Log canonical thresholds of binomial ideals

We prove that the log canonical thresholds of a large class of binomial ideals, such as complete intersection binomial ideals and the defining ideals of space monomial curves, are computable by linear programming. Dedicated to Professor Toshiyuki Katsura on the occasion of his sixtieth birthday     

Convex bodies and asymptotic invariants for powers of monomial ideals

Continuing a well established tradition of associating convex bodies to monomial ideals, we initiate a program to construct asymptotic Newton polyhedra from decompositions of monomial ideals. This is achieved by forming a graded family of ideals based on a given decomposition. We term these graded families powers since they generalize the notions of ordinary and symbolic powers. Asymptotic invariants for these graded families are expressed as solutions to linear optimization problems on the respective convex bodies. This allows to establish a lower bound on the Waldschmidt constant of a monomial ideal by means of a more easily computable invariant, which we introduce under the name of naive Waldschmidt constant.     

Componentwise linear ideals with minimal or maximal Betti numbers

We characterize componentwise linear monomial ideals with minimal Taylor resolution and consider the lower bound for the Betti numbers of componentwise linear ideals.     

Monomial ideals with tiny squares and Freiman ideals

Czechoslovak Mathematical Journal - We provide a construction of monomial ideals in R = K[x, y] such that μ(I2) < μ(I), where μ denotes the least number of generators. This...     

Specializations of Ferrers ideals

We introduce a specialization technique in order to study monomial ideals that are generated in degree two by using our earlier results about Ferrers ideals. It allows us to describe explicitly a cellular minimal free resolution of various ideals including any strongly stable and any squarefree strongly stable ideal whose minimal generators have degree two. In particular, this shows that threshold graphs can be obtained as specializations of Ferrers graphs, which explains their similar properties.     

Some algebraic invariants related to mixed product ideals

We compute some algebraic invariants (e.g. depth, Castelnuovo-Mumford regularity) for a special class of monomial ideals, namely the ideals of mixed products. As a consequence, we characterize the Cohen-Macaulay ideals of mixed products. Received: 25 October 2007     

The core of zero-dimensional monomial ideals

The core of an ideal is the intersection of all its reductions. We describe the core of a zero-dimensional monomial ideal I as the largest monomial ideal contained in a general reduction of I. This provides a new interpretation of the core in the monomial case as well as an efficient algorithm for computing it. We relate the core to adjoints and first coefficient ideals, and in dimension two and three we give explicit formulas.     

Toric rings arising from vertex cover ideals

We extend the sortability concept to monomial ideals which are not necessarily generated in one degree and as an application we obtain normal Cohen-Macaulay toric rings attached to vertex cover ideals of graphs. Moreover, we consider a construction on a graph called a clique multi-whiskering which always produces vertex cover ideals with componentwise linear powers.