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Resolution of Singularities

2004 Mathematics
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Release : 2004
Genre : Mathematics
Kind : eBook
Book Rating : 556/5 ( reviews)

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Book Synopsis Resolution of Singularities by : Steven Dale Cutkosky

Download or read book Resolution of Singularities written by Steven Dale Cutkosky. This book was released on 2004. Available in PDF, EPUB and Kindle. Book excerpt: The notion of singularity is basic to mathematics. In algebraic geometry, the resolution of singularities by simple algebraic mappings is truly a fundamental problem. It has a complete solution in characteristic zero and partial solutions in arbitrary characteristic. The resolution of singularities in characteristic zero is a key result used in many subjects besides algebraic geometry, such as differential equations, dynamical systems, number theory, the theory of $\mathcal{D}$-modules, topology, and mathematical physics. This book is a rigorous, but instructional, look at resolutions. A simplified proof, based on canonical resolutions, is given for characteristic zero. There are several proofs given for resolution of curves and surfaces in characteristic zero and arbitrary characteristic. Besides explaining the tools needed for understanding resolutions, Cutkosky explains the history and ideas, providing valuable insight and intuition for the novice (or expert). There are many examples and exercises throughout the text. The book is suitable for a second course on an exciting topic in algebraic geometry. A core course on resolutions is contained in Chapters 2 through 6. Additional topics are covered in the final chapters. The prerequisite is a course covering the basic notions of schemes and sheaves.

Resolution of Curve and Surface Singularities in Characteristic Zero

2012-09-11 Mathematics
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Release : 2012-09-11
Genre : Mathematics
Kind : eBook
Book Rating : 295/5 ( reviews)

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Book Synopsis Resolution of Curve and Surface Singularities in Characteristic Zero by : K. Kiyek

Download or read book Resolution of Curve and Surface Singularities in Characteristic Zero written by K. Kiyek. This book was released on 2012-09-11. Available in PDF, EPUB and Kindle. Book excerpt: 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}.

Resolution of Surface Singularities

2006-11-14 Mathematics
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Release : 2006-11-14
Genre : Mathematics
Kind : eBook
Book Rating : 258/5 ( reviews)

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Book Synopsis Resolution of Surface Singularities by : Vincent Cossart

Download or read book Resolution of Surface Singularities written by Vincent Cossart. This book was released on 2006-11-14. Available in PDF, EPUB and Kindle. Book excerpt:

Lectures on Resolution of Singularities (AM-166)

2009-01-10 Mathematics
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Release : 2009-01-10
Genre : Mathematics
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Book Rating : 809/5 ( reviews)

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Book Synopsis Lectures on Resolution of Singularities (AM-166) by : János Kollár

Download or read book Lectures on Resolution of Singularities (AM-166) written by János Kollár. This book was released on 2009-01-10. Available in PDF, EPUB and Kindle. Book excerpt: Resolution of singularities is a powerful and frequently used tool in algebraic geometry. In this book, János Kollár provides a comprehensive treatment of the characteristic 0 case. He describes more than a dozen proofs for curves, many based on the original papers of Newton, Riemann, and Noether. Kollár goes back to the original sources and presents them in a modern context. He addresses three methods for surfaces, and gives a self-contained and entirely elementary proof of a strong and functorial resolution in all dimensions. Based on a series of lectures at Princeton University and written in an informal yet lucid style, this book is aimed at readers who are interested in both the historical roots of the modern methods and in a simple and transparent proof of this important theorem.

Resolution of Surface Singularities

1984 Asymptotic expansions
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Release : 1984
Genre : Asymptotic expansions
Kind : eBook
Book Rating : 043/5 ( reviews)

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Book Synopsis Resolution of Surface Singularities by : Andrei Verona

Download or read book Resolution of Surface Singularities written by Andrei Verona. This book was released on 1984. Available in PDF, EPUB and Kindle. Book excerpt:

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