THREE KEY THEOREMS ON INFINITELY NEAR SINGULARITIES
Heisuke Hironaka · 2005
The notion of infinitely near singular points is classical and well under- stood for plane curves. We generalize the notion to higher dimensions and to develop a general theory, in terms of idealistic exponents and certain graded algebras associ- ated with them. We then gain a refined generalization of the classical notion of first characteristic exponents. On the level of technical base in the higher dimensional theory, there are some powerful tools, referred to as Three Key Theorems, which are namely Differentiation Theorem, Numerical Exponent Theorem and Ambient Reduc- tion Theorem. Resume (Trois theoremes-clefs sur les singularites infiniment pro ches). — La notion de points singuliers infiniment proches est classique et bien comprise pour les courbes planes. On generalise cette notion aux plus grandes dimensions et on developpe une theorie generale, en termes de d'exposants idealistes et certaines algebres graduees as- sociees. Ainsi on obtient une generalisation raffinee dela notion classique des premiers exposants caracteristiques. Au niveau technique de base dans la theorie de dimension plus grande, on a des outils puissants, appeles les Trois theoremes-clefs. Ce sont le Theoreme de differenciation, le Theoreme de l'exposant numerique et le Theoreme de reduction de l'espace ambiant.