This paper is devoted to a detailed study of certain remarkable posets which form a natural partition of all abelian ideals of a Borel subalgebra. Our main result is a nice uniform formula for the dimension of maximal ideals in these posets. We also obtain results on ad-nilpotent ideals which complete the analysis started in [CP2], [CP3].

We define the reduced horseshoe resolution and the notion of conjoined pairs of ideals in order to study the minimal graded free resolution of a class of p-Borel ideals and recover Pardue’s regularity formula for them. It will follow from our technique that the graded betti numbers of these ideals do not depend on the characteristic of the base field k.

With every σ-ideal I on a Polish space we associate the σ-ideal generated by closed sets in I. We study the forcing notions of Borel sets modulo the respective σ-ideals and find connections between their forcing properties. To this end, we associate to a σ-ideal on a Polish space an ideal on a countable set and show how forcing properties of the forcing depend on combinatorial properties of the...

Using results obtained from a study of homogeneous ideals sharing the same initial ideal with respect to some term order, we prove the singularity of the point corresponding to a segment ideal with respect to a degreverse term order (as, for example, the degrevlex order) in the Hilbert scheme of points in Pn. In this context, we look into the properties of several types of ‘‘segment’’ ideals th...

Fixed any term-ordering < on the set T(n) of terms in n variables x1, . . . , xn, we study homogeneous ideal a ⊆ P(n) of the ring of polynomials in n variables over a field k, via the associated order-ideal N (a), consisting of all the terms which are not maximal terms of elements of a and called sous-éscalier of a [7, 12, 14]. In particular we will focus our attention on the following combinat...