Desingularization of surface maps

Genus-one Stable Maps, Local Equations, and Vakil-zinger’s Desingularization

We describe an algebro-geometric approach to Vakil-Zinger’s desingularization of the main component of the moduli of genus one stable maps to P [6, 7]. The new approach provides complete local structural results for this moduli space as well as for the desingularization of the entire moduli space and should fully extend to higher genera.

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.

A Desingularization of ( the Main Component of ) the Moduli Space of Genus - One Stable Maps into

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.

Desingularization of Hyperkähler Varieties I Desingularization of Singular Hyperkähler Varieties I

• In Section 4, we define locally homogeneous singularities. A space with locally homogeneous singularities (SLHS) is an analytic space X such that for all x ∈ X , the x-completion of a local ring OxX is isomorphic to an x-completion of associated graded ring (OxX)gr. We show that a complex variety is SLHS if and only if the underlying real analytic variety is SLHS. This allows us to define inv...