On constructive desingularization

An Introduction to Constructive Desingularization.

Constructive General Neron Desingularization for one dimensional local rings

An algorithmic proof of General Neron Desingularization is given here for one dimensional local rings and it is implemented in Singular. Also a theorem recalling Greenberg’ strong approximation theorem is presented for one dimensional local rings.

Desingularization of Ore Operators

We show that Ore operators can be desingularized by calculating a least common left multiple with a random operator of appropriate order. Our result generalizes a classical result about apparent singularities of linear differential equations, and it gives rise to a surprisingly simple desingularization algorithm.

Embedded desingularization of toric varieties

We present a new method to achieve an embedded desingularization of a toric variety. Let W be a regular toric variety defined by a fan Σ and X ⊂ W be a toric embedding. We construct a finite sequence of combinatorial blowing-ups such that the final strict transforms X ′ ⊂ W ′ are regular and X ′ has normal crossing with the exceptional divisor.

On a desingularization of the moduli space of noncommutative tori

It is shown that the moduli space of the noncommutative tori Aθ admits a natural desingularization by the group Ext (Aθ, Aθ). Namely, we prove that the moduli space of pairs (Aθ, Ext (Aθ, Aθ)) is homeomorphic to a punctured two-dimensional sphere. The proof is based on a correspondence (a covariant functor) between the complex and noncommutative tori.