Algebraic Geometry and Singularities

Inhaltsverzeichnis

• I Resolution of Singularities.
• Désingularisation en dimension 3 et caractéristique p.
• 1 Différentes notions de désingularisation.
• 2 Première réduction.
• 3 Deuxième réduction, construction d’un modèle projectif.
• 4 Troisième réduction, birationnel devient projectif.
• 5 Final: Morphisme projectif birationnel devient désingularisation.
• Sur l’espace des courbes tracées sur une singularité.
• 1 Introduction.
• 2 Structure pro-algébrique de Tespace des courbes et la fonction de M. Art in d’une singularité.
• 3 Families de courbes (selon J. Nash) et désingularisations.
• 4 Courbes sur une singularité isolée d’hypersurface.
• 5 Courbes lisses sur une singularité de surface.
• 6 Deux exemples.
• Blowing up acyclic graphs and geometrical configurations.
• 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.
• 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.
• 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.
• 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.
• 2 Preliminaries on linear systems.
• 3 The polarity map.
• 4 Plücker’s formula.
• 5 The net of polars.
• 6 Some calculus.
• On moduli spaces of semiquasihomogeneous singularities.
• 2 Versal µ-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.
• 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.
• 2 Compactification.
• 3 Globalization of deformations.
• Caractérisation géométrique de l’existence du polynôme de Bernstein relatif.
• 1 Polynôme de Bernstein relatif.
• 2 DX×T Module holonome régulier relativement cohérent.
• Le Polygone de Newton d’un DX-module.
• 2 Le cas d’une variable.
• 3 La catégorie des faisceaux pervers.
• 4 Le faisceau d’irrégularité et le cycle d’irrégularité.
• 5 La filtration du faisceau d’irrégularité.
• 6 Le poly gone de Newton d’un DX-module.
• 7 Sur l’existence d’une équation fonctionnelle régulière.
• How good are real pictures?.
• 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.
• 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].
• 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.
• 2 The concept of approximant.
• 3 Curves associated to a semigroup.
• 4 A family of examples.
• Cubic surfaces with double points in positive characteristic.
• 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.