Resolution of Singularities of Embedded Algebraic Surfaces

Shreeram S. Abhyankar (Autor)

Buch | Hardcover

XII, 312 Seiten

1998 | 2nd enlarged ed. 1998
Springer Berlin (Verlag)
978-3-540-63719-6 (ISBN)

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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.

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.

Erscheint lt. Verlag	5.3.1998
Reihe/Serie	Springer Monographs in Mathematics
Zusatzinfo	XII, 312 p.
Verlagsort	Berlin
Sprache	englisch
Maße	155 x 235 mm
Gewicht	600 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie
Themenwelt	Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
Schlagworte	Algebraische Geometrie • characteristic • desingularization • Fläche • Number Theory • Resolution • Singularitätstheorie • Singularities • solids • transformations
ISBN-10	3-540-63719-2 / 3540637192
ISBN-13	978-3-540-63719-6 / 9783540637196
Zustand	Neuware