# Algebraic Geometry and Singularities (Progress in Mathematics v. 134)

By: Luis Narvaez Macarro (editor), Antonio Campillo Lopez (editor)Hardback

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### Description

The volume contains both general and research papers. Among the first ones are papers showing recent and original developments or methods in subjects such as resolution of singularities, D-module theory, singularities of maps and geometry of curves. The research papers deal on topics related to, or close to, those listed_above. The contributions are organized in three parts according to their contents. Part I presents a set of papers on resolution of singularities, a topic of renewed activity. It deals with important topics of current interest, such as canonical, algorithmic, combinatorial and graphical procedures (Villamayor, Oka, Marijmin), as well as special results on desingularization in characteristic p (Cossart, Moh), and connections between resolution and structure of the space of arcs through a singularity (Gonz81ez-Sprinberg-Lejeune-Jalabert). Part II contains a series of papers on the study~of singularities and its connections with differential systems and deformation or perturbation theo- ries. Two expository papers (Maisonobe-Briam;on, :'vlebkhout) describe, in an algebro-geometric way, the interaction between singularities and D-module t.he- ory including recent progress on Bernstein polynomials and Newton polygon techniques. Geometry of foliations (Henaut, Garcfa-Reguera), polar varieties and stratifications (Hajto) are also topics treated here. Two other papers (Wall, Greuel-Pfister) deal with quasihomogeneous singularities in the contexts of per- turbations and moduli spaces. Globalization of deformations of singularities (de Jong) and determination of complex topology from the real one (~10nd) com- plete this series of papers. Part III consists of papers on algebraic geometry of curves and surfaces.

### Contents

I Resolution of Singularities.- Desingularisation en dimension 3 et caracteristique p.- 1 Differentes notions de desingularisation.- 2 Premiere reduction.- 3 Deuxieme reduction, construction d'un modele projectif.- 4 Troisieme reduction, birationnel devient projectif.- 5 Final: Morphisme projectif birationnel devient desingularisation.- Sur l'espace des courbes tracees sur une singularite.- 1 Introduction.- 2 Structure pro-algebrique de Tespace des courbes et la fonction de M. Art in d'une singularite.- 3 Families de courbes (selon J. Nash) et desingularisations.- 4 Courbes sur une singularite isolee d'hypersurface.- 5 Courbes lisses sur une singularite de surface.- 6 Deux exemples.- Blowing up acyclic graphs and geometrical configurations.- 1 Introduction.- 2 Basic concepts and notations.- 3 Blowing up acyclic graphs.- 4 Graphic representation of the blowing up for a geometric configuration.- 5 Geometric modification for acyclic graphs.- On a Newton polygon approach to the uniformization of singularities of characteristic p.- 1 Introduction.- 2 Newton polygon and uniformization for ?1 ? n ? 1.- 3 Jumping lemma and Uniformization for ?1 = n ? 2.- 4 The classification of 3-dimensional singularities and uniformization for ?2 ? 3 or ?2 = $${\pi {\mathop 2\limits^ * }} \geqslant 2$$.- 5 Uniformization for ?2 = 2 and $${\pi {\mathop 2\limits^ * }}$$ = 1.- 6 Uniformization for ?2 = 1.- Geometry of plane curves via toroidal resolution.- 1 Introduction.- 2 Toric blowing-up and a tower of toric blowing-ups.- 3 Dual Newton diagram and an admissible toric blowing-up.- 4 Resolution complexity.- 5 Characteristic power and Puiseux Pairs.- 6 The Puiseux pairs of normal slice curves.- 7 Geometry of plane curves via a toroidal resolution.- 8 Iterated generic hyperplane section curves.- to the algorithm of resolution.- 1 Introduction.- 2 Stating the problem of resolution of singularities.- 3 Auxiliary result: Idealistic pairs.- 4 Constructive resolutions.- 5 The language of groves and the problem of patching.- 6 Examples.- II Complex Singularities and Differential Systems.- Polarity with respect to a foliation.- 1 Introduction.- 2 Preliminaries on linear systems.- 3 The polarity map.- 4 Plucker's formula.- 5 The net of polars.- 6 Some calculus.- On moduli spaces of semiquasihomogeneous singularities.- 1 Introduction.- 2 Versal -constant deformations and kernel of Kodaira-Spencer map.- 3 Existence of a geometric quotient for fixed Hilbert function of the Tjurina algebra.- 4 The automorphism group of semi Brieskorn singularities.- 5 Problems.- Stratification Properties of Constructible Sets.- 1 Introduction.- 2 Grassmann blowing-up.- 3 Analytically constructible sets.- 4 An application: the Henry-Merle Proposition.- 5 Canonical stratification.- On the linearization problem and some questions for webs in ?2.- 1 Introduction in the form of a survey.- 2 Linearization of webs in (?2,0).- 3 Geometry of the abelian relation space and the linearization problem in the maximum rank case.- 4 Some questions on wrebs in ?2.- Globalization of Admissible Deformations.- 1 Introduction.- 2 Compactification.- 3 Globalization of deformations.- Caracterisation geometrique de l'existence du polynome de Bernstein relatif.- 1 Polynome de Bernstein relatif.- 2 DXxT Module holonome regulier relativement coherent.- Le Polygone de Newton d'un DX-module.- 1 Introduction.- 2 Le cas d'une variable.- 3 La categorie des faisceaux pervers.- 4 Le faisceau d'irregularite et le cycle d'irregularite.- 5 La filtration du faisceau d'irregularite.- 6 Le poly gone de Newton d'un DX-module.- 7 Sur l'existence d'une equation fonctionnelle reguliere.- How good are real pictures?.- 1 Introduction.- 2 Comparison of real and complex discriminants and images.- 3 Codimension 1 germs.- 4 Good real forms and their perturbations.- 5 Bad real pictures.- Weighted homogeneous complete intersections.- 1 Introduction.- 2 Notation.- 3 Ideals and C-equivalence.- 4 Submodules.- 5 K-equivalence.- 6 Combinatorial arguments.- 7 A-equivalence.- 8 Other ground fields.- III Curves and Surfaces.- Degree 8 and genus 5 curves in ?3 and the Horrocks-Mumford bundle.- 1 Construction of curves of degree 8 and genus 5 on a Kummer surface S ? ?3.- 2 Barth's Construction.- 3 A generic curve of degree 8 and genus 5 in ?3.- Irreducible Polynomials of k((X))[Y].- 1 Introduction.- 2 Reduction of the Problem.- 3 Some Maximal Ideals of k?X?[Y].- 4 Irreducibility Criterion for Monic Polynomials of k?X?[Y].- 5 Some Ideas to Compute V[n/2](P).- Examples of Abelian Surfaces with Polarization type (1,3).- 1 Abstract.- 2 Introduction.- 3 Preliminaries.- 4 First examples: products of elliptic curves.- 5 The two-dimensional families of T-invariant quartic surfaces.- 6 The Family FAE.- 7 The Family t?1(L0, 1, 2).- 8 The Family FAB ? TAE.- Semigroups and Clusters at Infinity.- 1 Introduction.- 2 The concept of approximant.- 3 Curves associated to a semigroup.- 4 A family of examples.- Cubic surfaces with double points in positive characteristic.- 1 Introduction.- 2 Two characterizations of rational double points.- 3 Singularities and normal forms.- On the classification of reducible curve singularities.- 1 Reducible curve singularities.- 2 Decomposable curves.- 3 Classification.- 4 Deformations and smoothings.

### Product Details

• publication date: 26/01/1995
• ISBN13: 9783764353346
• Format: Hardback
• Number Of Pages: 407
• ID: 9783764353346
• weight: 835
• ISBN10: 3764353341

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