# Resolution of Curve and Surface Singularities

## in Characteristic Zero

Book

Part of the Algebras and Applications book series (AA, volume 4)

1. Front Matter
Pages i-xxi
2. K. Kiyek, J. L. Vicente
Pages 1-66
3. K. Kiyek, J. L. Vicente
Pages 67-100
4. K. Kiyek, J. L. Vicente
Pages 101-142
5. K. Kiyek, J. L. Vicente
Pages 143-168
6. K. Kiyek, J. L. Vicente
Pages 169-204
7. K. Kiyek, J. L. Vicente
Pages 205-246
8. K. Kiyek, J. L. Vicente
Pages 247-302
9. K. Kiyek, J. L. Vicente
Pages 303-344
10. Back Matter
Pages 345-485

### Introduction

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}.

### Keywords

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

#### Authors and affiliations

2. 2.Departamento de AlgebraUniversidad de SevillaSevillaSpain

### Bibliographic information

• Book Title Resolution of Curve and Surface Singularities
• Book Subtitle in Characteristic Zero
• Authors K. Kiyek
J.L. Vicente
• Series Title Algebras and Applications
• DOI https://doi.org/10.1007/978-1-4020-2029-2
• Publisher Name Springer, Dordrecht
• eBook Packages
• Hardcover ISBN 978-1-4020-2028-5
• Softcover ISBN 978-90-481-6573-5
• eBook ISBN 978-1-4020-2029-2
• Series ISSN 1572-5553
• Series E-ISSN 2192-2950
• Edition Number 1
• Number of Pages XXII, 486
• Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
• Topics
• Buy this book on publisher's site
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## Reviews

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)