The resolution property holds away from codimension three

On the Weak Lefschetz Property for Artinian Gorenstein Algebras of Codimension Three

We study the problem of whether an arbitrary codimension three graded artinian Gorenstein algebra has the Weak Lefschetz Property. We reduce this problem to checking whether it holds for all compressed Gorenstein algebras of odd socle degree. In the first open case, namely Hilbert function (1, 3, 6, 6, 3, 1), we give a complete answer in every characteristic by translating the problem to one of...

The strong uniform Artin-Rees property in codimension one

This result is a variation of a theorem of Duncan and O’Carroll [DO]: maximal ideals are replaced for any ideal using the unavoidable hypothesis dim(M/N) ≤ 1 (as an Example of Wang shows [W1]). Moreover it provides a partial positive answer to the question raised by Huneke in Conjecture 1.3 [H1]. We begin by recalling what is called uniform Artin-Rees properties. Let A be a noetherian ring, I b...

Codimension-Three Bundle Singularities in F-Theory

We study new nonperturbative phenomena in N = 1 heterotic string vacua corresponding to pointlike bundle singularities in codimension three. These degenerations result in new four-dimensional infrared physics characterized by light solitonic states whose origin is explained in the dual F-theory model. We also show that such phenomena appear generically in E7 → E6 Higgsing and describe in detail...

Codimension two and three Kneser Transversals

Let k, d, λ > 1 be integers with d > λ and let X ⊂ R be a finite set. A (d−λ)-plane L transversal to the convex hull of all k-sets of X is called Kneser transversal. If in addition L contains (d− λ) + 1 points of X, then L is called complete Kneser transversal. In this paper, we present various results on the existence of (complete) Kneser transversals for λ = 2, 3. In order to do this, we intr...

A Personal Reflection on Measurement from Three Decades Away.