Shreeram S. Abhyankar

Resolution of Singularities of Embedded Algebraic Surfaces

• Broschiertes Buch

Produktbeschreibung

The common solutions of a finite number of polynomial equations in a finite number of variables constitute an algebraic variety. The degrees of freedom of a moving point on the variety is the dimension of the variety. A one-dimensional variety is a curve and a two-dimensional variety is a surface. A three-dimensional variety may be called asolid. Most points of a variety are simple points. Singularities are special points, or points of multiplicity greater than one. Points of multiplicity two are double points, points of multiplicity three are tripie points, and so on. A nodal point of a curve is a double point where the curve crosses itself, such as the alpha curve. A cusp is a double point where the curve has a beak. The vertex of a cone provides an example of a surface singularity. A reversible change of variables gives abirational transformation of a variety. Singularities of a variety may be resolved by birational transformations.

Produktdetails

• Springer Monographs in Mathematics
• Verlag: Springer / Springer Berlin Heidelberg / Springer, Berlin
• Artikelnr. des Verlages: 978-3-642-08351-8
• 2. Aufl.
• Seitenzahl: 324
• Erscheinungstermin: 4. Dezember 2010
• Englisch
• Abmessung: 234mm x 156mm x 18mm
• Gewicht: 493g
• ISBN-13: 9783642083518
• ISBN-10: 364208351X
• Artikelnr.: 32078737

Autorenporträt

Besides the description of the geometric part of the author's 1965 proof of desingularization of algebraic surfaces and solids in nonzero characteristic, this new edition provides a self-contained introduction to birational algebraic geometry, based only on commutative algebra. It also gives a short proof of analytic desingularization in characteristic zero for any dimension found in 1996.

Inhaltsangabe

0 Introduction.- 1. Local Theory.- 1 Terminology and preliminaries.- 2 Resolvers and principalizers.- 3 Dominant character of a normal sequence.- 4 Unramified local extensions.- 5 Main results.- 2. Global Theory.- 6 Terminology and preliminaries.- 7 Global resolvers.- 8 Global principalizers.- 9 Main results.- 3. Some Cases of Three-Dimensional Birational Resolution.- 10 Uniformization of points of small multiplicity.- 11 Three-dimensional birational resolution over a ground field of characteristic zero.- 12 Existence of projective models having only points of small multiplicity.- 13 Three-dimensional birational resolution over an algebraically closed ground field of charateristic ? 2, 3, 5.- Appendix on Analytic Desingularization in Characteristic Zero.- Additional Bibliography.- Index of Notation.- Index of Definitions.- List of Corrections.