Resolution of Surface Singularities

The starting point for a new development in the resolution of surfaces, after Hironaka's proof in 1964, was Zariski's ... By using suitable projections as for equisingularity, he introduced the notion of quasi-ordinary singularities.

Author: Vincent Cossart

Publisher: Springer

ISBN: 9783540391258

Category: Mathematics

Page: 134

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Lectures on Resolution of Singularities AM 166

Surface singularities are much more complicated than curves, and re- solving them is a rather subtle problem. It seems that surfaces are still special enough to connect resolutions directly to various geometric properties but not ...

Author: János Kollár

Publisher: Princeton University Press

ISBN: 9781400827800

Category: Mathematics

Page: 208

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Resolution of singularities is a powerful and frequently used tool in algebraic geometry. In this book, János Kollár provides a comprehensive treatment of the characteristic 0 case. He describes more than a dozen proofs for curves, many based on the original papers of Newton, Riemann, and Noether. Kollár goes back to the original sources and presents them in a modern context. He addresses three methods for surfaces, and gives a self-contained and entirely elementary proof of a strong and functorial resolution in all dimensions. Based on a series of lectures at Princeton University and written in an informal yet lucid style, this book is aimed at readers who are interested in both the historical roots of the modern methods and in a simple and transparent proof of this important theorem.
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Resolution of Curve and Surface Singularities in Characteristic Zero

The complete neighborhood of a singularity of a complex curve can be analytically parametrized. ... The Surfaces Jung Methods When we dealt before with resolution of singularities of curves, we explained how to take advantage of the two ...

Author: K. Kiyek

Publisher: Springer Science & Business Media

ISBN: 9781402020292

Category: Mathematics

Page: 486

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The Curves The Point of View of Max Noether Probably the oldest references to the problem of resolution of singularities are found in Max Noether's works on plane curves [cf. [148], [149]]. And probably the origin of the problem was to have a formula to compute the genus of a plane curve. The genus is the most useful birational invariant of a curve in classical projective geometry. It was long known that, for a plane curve of degree n having l m ordinary singular points with respective multiplicities ri, i E {1, . . . , m}, the genus p of the curve is given by the formula = (n - l)(n - 2) _ ~ "r. (r. _ 1) P 2 2 L. . ,. •• . Of course, the problem now arises: how to compute the genus of a plane curve having some non-ordinary singularities. This leads to the natural question: can we birationally transform any (singular) plane curve into another one having only ordinary singularities? The answer is positive. Let us give a flavor (without proofs) 2 on how Noether did it • To solve the problem, it is enough to consider a special kind of Cremona trans formations, namely quadratic transformations of the projective plane. Let ~ be a linear system of conics with three non-collinear base points r = {Ao, AI, A }, 2 and take a projective frame of the type {Ao, AI, A ; U}.
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Resolution of Singularities

He was able to patch together local solutions to prove the existence of a resolution of singularities for algebraic surfaces over an algebraically closed field of characteristic zero. He later was able to prove local uniformization for ...

Author: Steven Dale Cutkosky

Publisher: American Mathematical Soc.

ISBN: 9780821835555

Category: Mathematics

Page: 186

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The notion of singularity is basic to mathematics. In algebraic geometry, the resolution of singularities by simple algebraic mappings is truly a fundamental problem. It has a complete solution in characteristic zero and partial solutions in arbitrary characteristic. The resolution of singularities in characteristic zero is a key result used in many subjects besides algebraic geometry, such as differential equations, dynamical systems, number theory, the theory of $\mathcal{D}$-modules, topology, and mathematical physics. This book is a rigorous, but instructional, look at resolutions. A simplified proof, based on canonical resolutions, is given for characteristic zero. There are several proofs given for resolution of curves and surfaces in characteristic zero and arbitrary characteristic. Besides explaining the tools needed for understanding resolutions, Cutkosky explains the history and ideas, providing valuable insight and intuition for the novice (or expert). There are many examples and exercises throughout the text. The book is suitable for a second course on an exciting topic in algebraic geometry. A core course on resolutions is contained in Chapters 2 through 6. Additional topics are covered in the final chapters. The prerequisite is a course covering the basic notions of schemes and sheaves.
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Resolution of Singularities

Giraud, J.: Desingularization in low dimension. In: Resolution of surface singularities. Lecture Notes in Math. vol. 1101, Springer 1984, 51–78. van Geemen, B., Oort, F.: A compactification of a fine moduli spaces of curves.

Author: Herwig Hauser

Publisher: Birkhäuser

ISBN: 9783034883993

Category: Mathematics

Page: 598

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In September 1997, the Working Week on Resolution of Singularities was held at Obergurgl in the Tyrolean Alps. Its objective was to manifest the state of the art in the field and to formulate major questions for future research. The four courses given during this week were written up by the speakers and make up part I of this volume. They are complemented in part II by fifteen selected contributions on specific topics and resolution theories. The volume is intended to provide a broad and accessible introduction to resolution of singularities leading the reader directly to concrete research problems.
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Resolution of Singularities of Embedded Algebraic Surfaces

A one-dimensional variety is a curve and a two-dimensional variety is a surface. A three-dimensional variety may be called a solid. Most points of a variety are simple points. Singularities are special points, or points of multiplicity ...

Author: Shreeram S. Abhyankar

Publisher: Springer Science & Business Media

ISBN: 9783662035801

Category: Mathematics

Page: 312

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The common solutions of a finite number of polynomial equations in a finite number of variables constitute an algebraic variety. The degrees of freedom of a moving point on the variety is the dimension of the variety. A one-dimensional variety is a curve and a two-dimensional variety is a surface. A three-dimensional variety may be called asolid. Most points of a variety are simple points. Singularities are special points, or points of multiplicity greater than one. Points of multiplicity two are double points, points of multiplicity three are tripie points, and so on. A nodal point of a curve is a double point where the curve crosses itself, such as the alpha curve. A cusp is a double point where the curve has a beak. The vertex of a cone provides an example of a surface singularity. A reversible change of variables gives abirational transformation of a variety. Singularities of a variety may be resolved by birational transformations.
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Local Dynamics of Non Invertible Maps Near Normal Surface Singularities

surface. singularities,. resolutions,. and. intersection. theory. In this section we recall some ideas and constructions from the theory of surface singularities and, in particular, the resolution of such singularities.

Author: William Gignac

Publisher: American Mathematical Society

ISBN: 9781470449582

Category: Mathematics

Page: 100

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View the abstract.
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Deformations of Surface Singularities

Kollár, J. and Shepherd-Barron, N. I., Threefolds and deformations of surface singularities, Invent. ... Laufer, H. B., Weak simultaneous resolution for deformations of Gorenstein surface singularities, Proc. of Symp. in Pure Math., 40, ...

Author: Andras Némethi

Publisher: Springer Science & Business Media

ISBN: 9783642391316

Category: Mathematics

Page: 280

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The present publication contains a special collection of research and review articles on deformations of surface singularities, that put together serve as an introductory survey of results and methods of the theory, as well as open problems and examples. The aim is to collect material that will help mathematicians already working or wishing to work in this area to deepen their insight and eliminate the technical barriers in this learning process. Additionally, we introduce some material which emphasizes the newly found relationship with the theory of Stein fillings and symplectic geometry. This links two main theories of mathematics: low dimensional topology and algebraic geometry.​ The theory of normal surface singularities is a distinguished part of analytic or algebraic geometry with several important results, its own technical machinery, and several open problems. Recently several connections were established with low dimensional topology, symplectic geometry and theory of Stein fillings. This created an intense mathematical activity with spectacular bridges between the two areas. The theory of deformation of singularities is the key object in these connections.
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Introduction to Lipschitz Geometry of Singularities

We have already seen that every analytic space admits a resolution of singularities. More specific results are achieved for surfaces. Theorem 2.4.4 (O. Zariski [Zar39]) A normal surface singularity can be resolved by the iteration of a ...

Author: Walter Neumann

Publisher: Springer Nature

ISBN: 9783030618070

Category: Mathematics

Page: 346

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This book presents a broad overview of the important recent progress which led to the emergence of new ideas in Lipschitz geometry and singularities, and started to build bridges to several major areas of singularity theory. Providing all the necessary background in a series of introductory lectures, it also contains Pham and Teissier's previously unpublished pioneering work on the Lipschitz classification of germs of plane complex algebraic curves. While a real or complex algebraic variety is topologically locally conical, it is in general not metrically conical; there are parts of its link with non-trivial topology which shrink faster than linearly when approaching the special point. The essence of the Lipschitz geometry of singularities is captured by the problem of building classifications of the germs up to local bi-Lipschitz homeomorphism. The Lipschitz geometry of a singular space germ is then its equivalence class in this category. The book is aimed at graduate students and researchers from other fields of geometry who are interested in studying the multiple open questions offered by this new subject.
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Milnor Fiber Boundary of a Non isolated Surface Singularity

We did not determine here the resolution and the real analytic type of the singularity situating above a double point of type 2. Nevertheless, these singularities are also identified by Lemma 11.6.12, up to an orientation reversing ...

Author: András Némethi

Publisher: Springer

ISBN: 9783642236471

Category: Mathematics

Page: 240

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In the study of algebraic/analytic varieties a key aspect is the description of the invariants of their singularities. This book targets the challenging non-isolated case. Let f be a complex analytic hypersurface germ in three variables whose zero set has a 1-dimensional singular locus. We develop an explicit procedure and algorithm that describe the boundary M of the Milnor fiber of f as an oriented plumbed 3-manifold. This method also provides the characteristic polynomial of the algebraic monodromy. We then determine the multiplicity system of the open book decomposition of M cut out by the argument of g for any complex analytic germ g such that the pair (f,g) is an ICIS. Moreover, the horizontal and vertical monodromies of the transversal type singularities associated with the singular locus of f and of the ICIS (f,g) are also described. The theory is supported by a substantial amount of examples, including homogeneous and composed singularities and suspensions. The properties peculiar to M are also emphasized.
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