Algebraic Geometry and Singularities

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.- 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 Plücker’s formula.- 5 The net of polars.- 6 Some calculus.- On moduli spaces of semiquasihomogeneous singularities.- 1 Introduction.- 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.- 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.- 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.- 1 Introduction.- 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?.- 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.