The generic initial ideals of powers of a 2-complete intersection

Abstract

We compute the reverse lexicographic generic initial ideals of the powers of a 2-complete intersection ideal I. In particular, we give six algorithms to compute these generic initial ideals, the choice of which depends on the power and on the relative degrees of the minimal generators of I. 1 arXiv:1206.5750v2 [math.AC] 30 Sep 2012 λ nα−1 = β − α + 1.Note that we can write λ 0 and λ k−1 in terms of l := β − α and α as follows:3.2. The Hilbert function of gin(I n ). The following result tells us that the invariants of gin(I n ) are completely determined by H gin(I n ) (t); this observation will be the key to computing these invariants.Lemma 3.6. Suppose that we have an ideal J of the formwhere the µ i s are strictly decreasing. If H J (t) = H I n (t) for a type (α, β) complete intersection ideal I then gin(I n ) = J.This lemma is an immediate consequence of the following well-known result. Although it is used in the literature (for example, it has the same content as Lemma 4.2 of [Gre98]), we record a complete proof here. 8 SARAH MAYES Lemma 3.7. An ideal of the form J = (x k , x k−1 y λ k−1 , . . . , xy λ 1 , y λ 0 )where λ 0 > λ 1 > · · · > λ k−1 is uniquely determined by its Hilbert function.Proof. The key observation here is thatis the Hilbert function of an ideal J as in the statement of the lemma.First note that x k is the smallest degree element of J so that k = min{t|H J (t) = 0}.Consider the ideal L k = (x k ) ⊂ J and its Hilbert function H L k (t). Set S k = min{t|H J (t) = H L k (t)} so that the smallest degree monomial that is in J but not in L k is of degree

The Limiting Shape of the Generic Initial System of a Complete Intersection

“…, d r . For explicit descriptions of the generators of gin(I n ) see [May12] for the case where r = 2 and [Cim06] for the case where n = 1, r = 3, and I is strongly Lefschetz. The generic initial system of a complete intersection, then, follows the philosophy guiding the study of asymptotic objects: the ideals gin(I n ) are complex but uniformity is gained in the limit.…”

Axial constants and sectional regularity of homogeneous ideals

“…Indeed our terminology -axial constants -is borrowed from his work. These ideas were motivated by the construction in [19,20,21] of asymptotic convex bodies as limits of Newton polyhedra for the generic initial systems, that is, families of monomial ideals {gin rev (I (n) )} n∈N , where I is a homogeneous ideal. This body of work provided the original motivation for our present investigations.…”