The core of zero-dimensional monomial ideals

Abstract

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.

Multiplicity of the special fiber of blowups

“…This statement is analogous to the characterization of the integral closure I of I as the largest ideal containing I with the same multiplicity e 0 . However, our interest inǏ has also been motivated by a recent connection, made in [18], with the computation of the core of I (we recall that core(I) is the intersection of all the reductions of I). The core of an ideal has lately been under much scrutiny: One of the classical motivations to study the core comes from the Briançon-Skoda theorem, but more recently Hyry and Smith have shown that Kawamata's well-known conjecture on the existence of sections of line bundles is equivalent to a statement about the core of certain ideals in section rings.…”

Joint reductions of monomial ideals and multiplicity of complex analytic maps

“…, x n ]], reduces to compute the intersection of all ideals J of O n such that Γ(I) = Γ(J) and J is Newton non-degenerate. We remark that the study of the core of an ideal is quite an active research topic in commutative algebra (see for instance [8] or [12]). …”

Monomial and toric ideals associated to Ferrers graphs

“…We end by saying that the numerical data we explicitly determine also allow us to compute the reduction number of Ferrers ideals, which is a key ingredient in determining the structure of the core of ideals. Indeed, some classes of monomial ideals possess a surprising interpretation for their core (see [44]). Preliminary investigations also show that the core of a Ferrers ideal has a remarkable structure related to the shape of the corresponding Ferrers tableau.…”