Introduction to Jung's method of resolution of singularities

On higher dimensional Hirzebruch-Jung singularities

A germ of normal complex analytical surface is called a Hirzebruch-Jung singularity if it is analytically isomorphic to the germ at the 0-dimensional orbit of an affine toric surface. Two such germs are known to be isomorphic if and only if the toric surfaces corresponding to them are equivariantly isomorphic. We extend this result to higher-dimensional Hirzebruch-Jung singularities, which we define to be the germs analytically isomorphic to the germ at the 0-dimensional orbit of an affine toric variety determined by a lattice and a simplicial cone of maximal dimension. We deduce a normalizati…

Equisingularity and simultaneous resolution of singularities

The purpose, mainly expository and speculative, of this paper---an outgrowth of a survey lecture at the September 1997 Obergurgl working week---is to indicate some (not all) of the efforts that have been made to interpret equisingularity, and connections among them; and to suggest directions for further exploration. Zariski defined equisingularity on an n-dimensional hypersurface V via stratification by ``dimensionality type," an integer associated to a point by means of a generic local projection to affine n-space. A possibly more intuitive concept of equisingularity can be based on stratific…

Local Resolution of Ideals Subordinated to a Foliation

Let $M$ be a complex- or real-analytic manifold, $θ$ be a singular distribution and $\mathcal{I}$ a coherent ideal sheaf defined on $M$. We prove the existence of a local resolution of singularities of $\mathcal{I}$ that preserves the class of singularities of $θ$, under the hypothesis that the considered class of singularities is invariant by $θ$-admissible blowings-up. In particular, if $θ$ is monomial, we prove the existence of a local resolution of singularities of $\mathcal{I}$ that preserves the monomiality of the singular distribution $θ$.

除极小奇点外的消解 II：四元变量情形

Resolution except for minimal singularities II. The case of four variables

本文是奇点消解研究续篇，刻画了四元变量下无法通过避免吹胀正规交叉奇点的双有理态射消除的、作为三重正规交叉奇点极限的最小奇点类。

Lojasiewicz exponents and resolution of singularities

We show an effective method to compute the Łojasiewicz exponent of an arbitrary sheaf of ideals of $\OO_X$, where $X$ is a non-singular scheme. This method is based on the algorithm of resolution of singularities.

Resolution except for minimal singularities I

The philosophy of the article is that the desingularization invariant together with natural geometric information can be used to compute local normal forms of singularities. The idea is used in two related problems: (1) We give a proof of resolution of singularities of a variety or a divisor, except for simple normal crossings (i.e., which avoids blowing up simple normal crossings, and ends up with a variety or a divisor having only simple normal crossings singularities). (2) For more general normal crossings (in a local analytic or formal sense), such a result does not hold. We find the small…