Book contents
- Frontmatter
- Contents
- Preface
- Invited Lectures
- List of Participants
- Aspects of Gert-Martin Greuel's Mathematical Work
- Exterior Algebra Methods for the Construction of Rational Surfaces in the Projective Fourspace
- Superisolated Surface Singularities
- Linear Free Divisors and Quiver Representations
- Derived Categories of Modules and Coherent Sheaves
- Monodromy
- Algorithmic Resolution of Singularities
- Newton Polyhedra of Discriminants: A Computation
- Depth and Differential Forms
- The Geometry of the Versal Deformation
- 21 Years of SINGULAR Experiments in Mathematics
- The Patchworking Construction in Tropical Enumerative Geometry
- Adjunction Conditions for One-Forms on Surfaces in Projective Three-Space
- Sextic Surfaces with Ten Triple Points
- Sextic Surfaces with 10 Triple Points
- Topology, Geometry, and Equations of Normal Surface Singularities
Algorithmic Resolution of Singularities
Published online by Cambridge University Press: 11 November 2009
- Frontmatter
- Contents
- Preface
- Invited Lectures
- List of Participants
- Aspects of Gert-Martin Greuel's Mathematical Work
- Exterior Algebra Methods for the Construction of Rational Surfaces in the Projective Fourspace
- Superisolated Surface Singularities
- Linear Free Divisors and Quiver Representations
- Derived Categories of Modules and Coherent Sheaves
- Monodromy
- Algorithmic Resolution of Singularities
- Newton Polyhedra of Discriminants: A Computation
- Depth and Differential Forms
- The Geometry of the Versal Deformation
- 21 Years of SINGULAR Experiments in Mathematics
- The Patchworking Construction in Tropical Enumerative Geometry
- Adjunction Conditions for One-Forms on Surfaces in Projective Three-Space
- Sextic Surfaces with Ten Triple Points
- Sextic Surfaces with 10 Triple Points
- Topology, Geometry, and Equations of Normal Surface Singularities
Summary
Abstract
Although the problem of the existence of a resolution of singularities in characteristic zero was already proved by Hironaka in the 1960s and although algorithmic proofs of it have been given independently by the groups of Bierstone and Milman and of Encinas and Villamayor in the early 1990s, the explicit construction of a resolution of singularities of a given variety is still a very complicated computational task. In this article, we would like to outline the algorithmic approach of Encinas and Villamayor and simultaneously discuss the practical problems connected to the task of implementing the algorithm.
Introduction
The problem of existence and construction of a resolution of singularities is one of the central tasks in algebraic geometry. In its shortest formulation it can be stated as: Given a variety X over a field K, a resolution of singularities of X is a proper birational morphism π : Y → X such that Y is a non-singular variety.
Historically, a question of this type has first been considered in the second half of the 19th century – in the context of curves over the field of complex numbers. It was already a very active area of research at that time with a large number of contributions (of varying extent of rigor) and eventually lead to a proof of existence of resolution of singularities in this special situation at the end of the century.
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- Singularities and Computer Algebra , pp. 157 - 184Publisher: Cambridge University PressPrint publication year: 2006
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