Resolution of Curve and Surface Singularities in Characteristic Zero

1. Front Matter

   Pages i-xxi

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2. Valuation Theory

   • K. Kiyek, J. L. Vicente

   Pages 1-66

3. One-Dimensional Semilocal Cohen-Macaulay Rings

   • K. Kiyek, J. L. Vicente

   Pages 67-100

4. Differential Modules and Ramification

   • K. Kiyek, J. L. Vicente

   Pages 101-142

5. Formal and Convergent Power Series Rings

   • K. Kiyek, J. L. Vicente

   Pages 143-168

6. Quasiordinary Singularities

   • K. Kiyek, J. L. Vicente

   Pages 169-204

7. The Singularity Z^q = XY^p

   • K. Kiyek, J. L. Vicente

   Pages 205-246

8. Two-Dimensional Regular Local Rings

   • K. Kiyek, J. L. Vicente

   Pages 247-302

9. Resolution of Singularities

   • K. Kiyek, J. L. Vicente

   Pages 303-344

10. Back Matter

    Pages 345-485

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The Curves The Point of View of Max Noether Probably the oldest references to the problem of resolution of singularities are found in Max Noether's works on plane curves [cf. [148], [149]]. And probably the origin of the problem was to have a formula to compute the genus of a plane curve. The genus is the most useful birational invariant of a curve in classical projective geometry. It was long known that, for a plane curve of degree n having l m ordinary singular points with respective multiplicities ri, i E {1, . . . , m}, the genus p of the curve is given by the formula = (n - l)(n - 2) _ ~ "r. (r. _ 1) P 2 2 L. . ,. •• . Of course, the problem now arises: how to compute the genus of a plane curve having some non-ordinary singularities. This leads to the natural question: can we birationally transform any (singular) plane curve into another one having only ordinary singularities? The answer is positive. Let us give a flavor (without proofs) 2 on how Noether did it • To solve the problem, it is enough to consider a special kind of Cremona trans­ formations, namely quadratic transformations of the projective plane. Let ~ be a linear system of conics with three non-collinear base points r = {Ao, AI, A }, 2 and take a projective frame of the type {Ao, AI, A ; U}.

• Abelian group
• Blowing up
• Dimension
• Divisor
• Grad
• algebraic geometry
• brandonwiskunde
• commutative algebra

From the reviews:

"As indicated in the title … describes different methods of resolution of singularities of curves and surfaces … . The first seven chapters are dedicated to developing the material … . The two appendixes, on algebraic geometry and commutative algebra, contain generalities and classical results needed in the previous chapters. This completes one of the aims of the authors: To write a book as self-contained as possible. ... In conclusion, the book is an interesting exposition of resolution of singularities in low dimensions … ." (Ana Bravo, Mathematical Reviews, 2005e)

"The monograph presents a modern theory of resolution of isolated singularities of algebraic curves and surfaces over algebraically closed fields of characteristic zero. … The exposition is self-contained and is supplied by an appendix, covering some classical algebraic geometry and commutative algebra." (Eugenii I. Shustin, Zentralblatt MATH, Vol. 1069 (20), 2005)

• Department of Mathematics, University of Paderborn, Paderborn, Germany

  K. Kiyek

• Departamento de Algebra, Universidad de Sevilla, Sevilla, Spain

  J. L. Vicente

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