Related papers: Singularities in any Characteristic
The main purpose of this article is to lay the foundations for a classification of isolated hypersurface singularities in positive characteristic. Although our article is in the spirit of Arnol'd who classified real an complex hypersurfaces…
We present a hypersurface singularity in positive characteristic which is defined by a purely inseparable power series, and a sequence of point blowups so that, after applying the blowups to the singularity, the same type of singularity…
The paper is motivated on the open problem of resolution of singularities in positive characteristic. The aim is to present a form of induction which is different from that used by Hironaka. In characteristic zero induction is formulated by…
In this survey paper we give an overview on some aspects of singularities of algebraic varieties over an algebraically closed field of arbitrary characteristic. We review in particular results on equisingularity of plane curve…
Assume that there exists a hypersurface singularity which cannot be resolved by iterated monoidal transformations in positive characteristic. We show that in the set of defining functions of hypersurface singularities which cannot be…
The objective of this paper is to discuss invariants of singularities of algebraic schemes over fields of positive characteristic, and to show how they yield the simplification of singularities. We focus here on invariants which arise in an…
We classify isolated hypersurface singularities $f\in K[[x_1,..., x_n]]$, $K$ an algebraically closed field of characteristic $p>0$, which are simple w.r.t. right equivalence, that is, which have no moduli up to analytic coordinate change.…
We introduce an upper semi-continuous function that stratifies the highest multiplicity locus of a hypersurface in arbitrary characteristic (over a perfect field). The blow-up along the maximum stratum defined by this function leads to a…
We survey some recent results concerning the behaviour of the contact structure defined on the boundary of a complex isolated hypersurface singularity or on the boundary at infinity of a complex polynomial.
We present an application of elimination theory to the study of singularities over arbitrary fields, particularly to the open problem of resolution. A partial extension of a function, defining resolution of singularities over fields of…
We survey determinantal singularities, their deformations, and their topology. This class of singularities generalizes the well studied case of complete intersections in several different aspects, but exhibits a plethora of new phenomena…
This is the abstract prepared for Workshop on Topology and Geometry (Zhang jiang, China, October 1994), and is a review of my recent works. What kinds of combinations of singularities can appear in small deformation fibers of a fixed…
Optical singularities, which are positions within an electromagnetic field where certain field parameters become undefined, hold significant potential for applications in areas such as super-resolution microscopy, sensing, and…
The purpose of this paper is to show how Rees algebras can be applied in the study of singularities embedded in smooth schemes over perfect fields. In particular, we will study situations in which the multiplicity of a hypersurface is a…
By the famous ADE classification rational double points are simple. Rational triple points are also simple. We conjecture that the simple normal surface singularities are exactly those rational singularities, whose resolution graph can be…
We establish a one-to-one correspondence between the singularity categories of rational double points and the simply-laced Dynkin graphs in arbitrary characteristic. This correspondence is well-known in characteristic zero since the…
A sequence of monoidal transformations is defined, in terms of invariants, for a singular hypersurface embedded in a smooth scheme of positive characteristic. Some examples are added to illustrate the improvement of singularities by this…
Over $\C$, Henry Laufer classified all taut surface singularities. We adapt and extent his transcendental methods to positive characteristic. With this we show that if a normal surface singularity is taut over $\C$, then the normal surface…
We generalize the notions of singularities and ordinary points from linear ordinary differential equations to D-finite systems. Ordinary points of a D-finite system are characterized in terms of its formal power series solutions. We also…
In this note, we provide a complete classification for entire area maximizing hypersurfaces having an isolated singularity. We also construct an interesting illustrated example. For area maximizing hypersurfaces over exterior domains, we…