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Gert-Martin Greuel

Publications and source records attributed to Gert-Martin Greuel.

At least 19 recordsLinked to original sources
arXiv2026

Grauert's Approximation Theorem in any Characteristic and Applications

In his seminal Inventiones paper from 1972 Grauert proved the existence of a semiuniversal deformation of an arbitrary complex analytic isolated singularity. For the proof he invented an approximation theorem for solving a system of "nested" analytic equations, which is now called Grauert's approximation theorem. To prove this, Grauert introduced standard bases for ideals in power series rings and proved a generalized Weiertrass division theorem. All this was done for convergent power series over the complex numbers. The purpose of this article is to extend Grauert's division and approximation theorem to convergent power series over arbitrary real valued fields of any characteristic. As an application, which was actually the motivation for this article, we derive the existence of a convergent semiuniversal deformation for an isolated singularity and a splitting lemma for not necessarily isolated hypersurface singularities over any real valued field.

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arXiv2026

Singularities in any Characteristic

We give an overview of the fundamental definitions and results concerning hypersurface singularities, defined by convergent power series over an arbitrary real valued field. This approach combines, on the one hand, the classical case of analytic power series over the complex numbers with formal power series over arbitrary fields, but on the other hand, it goes significantly beyond that. Besides general definitions and basic results, we report on the classification of contact simple and right simple singularities in positive characteristic. Some of the results are new in this general setting, for which we provide complete proofs.

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arXiv2025

The Splitting Lemma in any Characteristic

We give a simple proof of the splitting lemma in singularity theory, also known as generalized Morse lemma, for formal power series over arbitrary fields. Our proof for the uniqueness of the residual part in any characteristic is new and was previously unknown in characteristic two. Beyond the formal case, we give proofs for algebraic power series and for convergent real and complex analytic power series, which are new for non-isolated singularities.

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arXiv2025

Classification of simple 0-dimensional isolated complete intersection singularities

The aim of this article is the classification of simple 0-dimensional isolated complete intersection singularities in positive characteristic. As usual, a singularity is called simple or 0-modal if there are only finitely many isomorphism classes of singularities into which the given singularity can deform. The notion of simpleness was introduced by V. I. Arnold and the classification of low modality singularities has become a fundamental task in singularity theory. Simple complex analytic isolated complete intersection singularities (ICIS) were classified by M. Giusti. However, the classification in positive characteristic requires different methods and is much more involved. The final result is nevertheless similar to the classification in characteristic 0 with some additional normal forms in low characteristic. The theoretical results in this paper mainly concern families of ICIS that are formal in the fiber and algebraic in the base (formal deformation theory is not sufficient). In particular, we give a definition of modality in this situation and prove its semicontinuity.

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arXiv2025

Classification of Unimodal Parametric Plane Curve Singularities in Positive Characteristic

In 2011, Hefez and Hernandes completed Zariski's analytic classification of plane branches belonging to a certain equisingularity class by creating "very short" parameterizations over the complex numbers. Their results were used by Mehmood and Pfister to classify unimodal plane branches in characteristic 0 by constructing lists of normal forms. The goal of this paper is to give a complete classification of unimodal plane branches over an algebraically closed field of positive characteristic. Since the methods of Hefez and Hernandes are not applicable in positive characteristic, we use a different approach and, for some sporadic singularities in small characteristic, computations with SINGULAR. Our methods are characteristic-independent and provide a different proof for the classification in characteristic 0, showing at the same time that this classification holds also in large characteristic. The main theoretical ingredients are the semicontinuity of the semigroup and the modality, which we prove.

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arXiv2022

The Delta Invariant and Simultaneous Normalization for Families of Isolated Non-Normal Singularities

We consider families of schemes over arbitrary fields resp. analytic varieties with finitely many (not necessarily reduced) isolated non-normal singularities, in particular families of generically reduced curves. We define a modified delta invariant for isolated non-normal singularities of any dimension that takes care of embedded points and prove that it behaves upper semicontinuous in flat families parametrized by an arbitrary principal ideal domain. Moreover, if the fibers contain no isolated points, then the familly admits a fiberwise normalization iff the delta invariant is locally constant. The results generalize results by Teissier and Chiang-Hsieh--Lipman for families of reduced curve singularities and provide possible improvements for algorithms to compute the genus of a curve.

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arXiv2021

Pairs of Lie-type and large orbits of group actions on filtered modules. (A characteristic-free approach to finite determinacy.)

Finite determinacy for mappings has been classically thoroughly studied in numerous scenarios in the real- and complex-analytic category and in the differentiable case. It means that the map-germ is determined, up to a given equivalence relation, by a finite part of its Taylor expansion. The equivalence relation is usually given by a group action and the first step is always to reduce the determinacy question to an "infinitesimal determinacy", i.e., to the tangent spaces at the orbits of the group action. In this work we formulate a universal, characteristic-free approach to finite determinacy, not necessarily over a field, and for a large class of group actions. We do not restrict to pro-algebraic or Lie groups, rather we introduce the notion of "pairs of (weak) Lie type", which are groups together with a substitute for the tangent space to the orbit such that the orbit is locally approximated by its tangent space, in a precise sense. This construction may be considered as a kind of replacement of the exponential resp. logarithmic maps. It is of independent interest as it provides a general method to pass from the tangent space to the orbit of a group action in any characteristic. In this generality we establish the "determinacy versus infinitesimal determinacy" criteria, a far reaching generalization of numerous classical and recent results, together with some new applications.

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arXiv2021

Using Semicontinuity for Standard Bases Computations

We present new results and an algorithm for standard basis computations of a 0-dimensional ideal I in a power series ring or in the localization of a polynomial ring in finitely many variables over a field K. The algorithm provides a significant speed up if K is the quotient field of a Noetherian integral domain A, when coefficient swell occurs. The most important special cases are perhaps when A is the ring of integers resp. when A is a polynomial ring over some field in finitely many parameters. Given I as an ideal in the polynomial ring over A, we compute first a standard basis modulo a prime number p, resp. by specializing the parameter to a constant. We then use the "highest corner" of the specialized ideal to cut off high order terms from the polynomials during the standard basis computation over K to get the speed up. An important fact is that we can choose p as an arbitrary prime resp. as an arbitrary constant, not just a "lucky" resp. "random" one. Correctness of the algorithm will be deduced from a general semicontinuity theorem due to the first two authors. The computer algebra system Singular provides already the functionality to realize the algorithm and we present several examples illustrating its power.

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arXiv2021

On Delta for parameterized Curve Singularities

We consider families of parameterizations of reduced curve singularities over a Noetherian base scheme and prove that the delta invariant is semicontinuous. In our setting, each curve singularity in the family is the image of a parameterization and not the fiber of a morphism. The problem came up in connection with the right-left classification of parameterizations of curve singularities defined over a field of positive characteristic. We prove a bound for right-left determinacy of a parameterization in terms of delta and the semicontinuity theorem provides a simultaneous bound for the determinacy in a family. The fact that the base space can be an arbitrary Noetherian scheme causes some difficulties but is (not only) of interest for computational purposes.

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arXiv2020

Plane algebraic curves with prescribed singularities

We report on the problem of the existence of complex and real algebraic curves in the plane with prescribed singularities up to analytic and topological equivalence. The question is whether, for a given positive integer $d$ and a finite number of given analytic or topological singularity types, there exist a plane (irreducible) curve of degree $d$ having singular points of the given type as its only singularities. The set of all such curves is a quasi-projective variety, which we call an equisingular family (ESF). We describe, in terms of numerical invariants of the curves and their singularities, the state of the art concerning necessary and sufficient conditions for the non-emptiness and $T$-smoothness (i.e., smooth of expected dimension) of the corresponding ESF. The considered singularities can be arbitrary, but we spend special attention to plane curves with nodes and cusps, the most studied case, where still no complete answer is known in general. An important result is, however, that the necessary and the sufficient conditions show the same asymptotics for $T$-smooth equisingular families if the degree goes to infinity.

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arXiv2020

Semicontinuity of Singularity Invariants in Families of Formal Power Series

The problem we are considering came up in connection with the classification of singularities in positive characteristic. Then it is important that certain invariants like the determinacy can be bounded simultaneously in families of formal power series parametrized by some algebraic variety. In contrast to the case of analytic or algebraic families, where such a bound is well known, the problem is rather subtle, since the modules defining the invariants are quasi-finite but not finite over the base space. In fact, in general the fibre dimension is not semicontinuous and the quasi-finite locus is not open. However, if we pass to the completed fibers in a family of modules we can prove that their fiber dimension is semicontinuous under some mild conditions. We prove this in a rather general framework by introducing and using the completed and the Henselian tensor product, the proof being more involved as one might think. Finally we apply this to the Milnor number and the Tjurina number in families of hypersurfaces and complete intersections and to the determinacy in a family of ideals.

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arXiv2019

Straight Equisingular Deformations and Punctual Hilbert Schemes

We study "straight equisingular deformations", a linear subfunctor of all equisingular deformations and describe their seminuniversal deformation by an ideal containing the fixed Tjurina ideal. Moreover, we show that the base space of the seminuniversal straight equisingular deformation appears as the fibre of a morphism from the μ-constant stratum onto a punctual Hilbert scheme parametrizing certain zero-dimensional schemes concentrated in the singular point. Although equisingular deformations of plane curve singularities are very well understood, we believe that this aspect may give a new insight in their inner structure.

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arXiv2019

Finite determinacy of matrices and ideals

The main aim of this paper is to characterize ideals I in the power series ring R=K[[x1,...,xs]] that are finitely determined up to contact equivalence by proving that this is the case if and only if I is an isolated complete intersection singularity, provided dim(R/I) > 0 and K is an infinite field (of arbitrary characteristic). Here two ideals I and J are contact equivalent if the local K-algebras R/I and R/J are isomorphic. If I is minimally generated by a1,...,am, we call I finitely contact determined if it is contact equivalent to any ideal J that can be generated by b1,...,bm with ai - bi in ^k for some integer k. We give also computable and semicontinuous determinacy bounds. The above result is proved by considering left-right equivalence on the ring M of m x n matrices A with entries in R and we show that the Fitting ideals of a finitely determined matrix in M have maximal height, a result of independent interest. The case of ideals is treated by considering 1-column matrices. Fitting ideals together with a special construction are used to prove the characterization of finite determinacy for ideals in R. Some results of this paper are known in characteristic 0, but they need new (and more sophisticated) arguments in positive characteristic partly because the tangent space to the orbit of the left-right group cannot be described in the classical way. In addition we point out several other oddities, including the concept of specialization for power series, where the classical approach (due to Krull) does not work anymore. We include some open problems and a conjecture.

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arXiv2019

Deformation and Smoothing of Singularities

We give a survey on some aspects of deformations of isolated singularities. In addition to the presentation of the general theory, we report on the question of the smoothability of a singularity and on relations between different invariants, such as the Milnor number, the Tjurina number, and the dimension of a smoothing component.

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arXiv2018

On 1-forms on isolated complete intersection curve singularities

We collect some classical results about holomorphic 1-forms of a reduced complex curve singularity. They are used to study the pull-back of holomorphic 1-forms on an isolated complete intersection curve singularity under the normalization morphism. We wonder whether the Milnor number $μ$ and the Tjurina number $τ$ of any isolated plane curve singularity satisfy the inequality $3μ<4τ$.

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arXiv2017

Life and work of Egbert Brieskorn (1936 - 2013)

Egbert Brieskorn died on July 11, 2013, a few days after his 77th birthday. He was an impressive personality who has left a lasting impression on all who knew him, whether inside or outside of mathematics. Brieskorn was a great mathematician, but his interests, his knowledge, and activities ranged far beyond mathematics. In this contribution, which is strongly influenced by many years of personal connectedness of the authors with Brieskorn, we try to give a deeper insight into the life and work of Brieskorn. We illuminate both his personal commitment to peace and the environment as well as his long-term study of the life and work of Felix Hausdorff and the publication of Hausdorff's collected works. However, the main focus of the article is on the presentation of his remarkable and influential mathematical work.

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arXiv2017

Singularities in Positive Characteristic

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 singularities, classification of hypersurface singularities and determinacy of arbitrary singularities. The section on equisingularity has its roots in two important early papers by Antonio Campillo. One emphasis is on the differences between positive and zero characteristic and on open problems.

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