Rational Algebraic Curves
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| Author | : J. Rafael Sendra |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 273 |
| Release | : 2007-12-10 |
| Genre | : Mathematics |
| ISBN | : 3540737251 |
The central problem considered in this introduction for graduate students is the determination of rational parametrizability of an algebraic curve and, in the positive case, the computation of a good rational parametrization. This amounts to determining the genus of a curve: its complete singularity structure, computing regular points of the curve in small coordinate fields, and constructing linear systems of curves with prescribed intersection multiplicities. The book discusses various optimality criteria for rational parametrizations of algebraic curves.
| Author | : Janos Kollar |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 330 |
| Release | : 2013-04-09 |
| Genre | : Mathematics |
| ISBN | : 3662032767 |
The aim of this book is to provide an introduction to the structure theory of higher dimensional algebraic varieties by studying the geometry of curves, especially rational curves, on varieties. The main applications are in the study of Fano varieties and of related varieties with lots of rational curves on them. This Ergebnisse volume provides the first systematic introduction to this field of study. The book contains a large number of examples and exercises which serve to illustrate the range of the methods and also lead to many open questions of current research.
| Author | : J. Rafael Sendra |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 273 |
| Release | : 2007-10-19 |
| Genre | : Mathematics |
| ISBN | : 3540737243 |
The central problem considered in this introduction for graduate students is the determination of rational parametrizability of an algebraic curve and, in the positive case, the computation of a good rational parametrization. This amounts to determining the genus of a curve: its complete singularity structure, computing regular points of the curve in small coordinate fields, and constructing linear systems of curves with prescribed intersection multiplicities. The book discusses various optimality criteria for rational parametrizations of algebraic curves.
| Author | : Serguei A. Stepanov |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 444 |
| Release | : 1994-12-31 |
| Genre | : Mathematics |
| ISBN | : 9780306110368 |
Author S.A. Stepanov thoroughly investigates the current state of the theory of Diophantine equations and its related methods. Discussions focus on arithmetic, algebraic-geometric, and logical aspects of the problem. Designed for students as well as researchers, the book includes over 250 excercises accompanied by hints, instructions, and references. Written in a clear manner, this text does not require readers to have special knowledge of modern methods of algebraic geometry.
| Author | : Fedor Bogomolov |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 229 |
| Release | : 2002 |
| Genre | : Mathematics |
| ISBN | : 0821828622 |
This text covers the essential topics in the geometry of algebraic curves, such as line and vector bundles, the Riemann-Roch Theorem, divisors, coherent sheaves, and zeroth and first cohomology groups. It demonstrates how curves can act as a natural introduction to algebraic geometry.
| Author | : Frances Clare Kirwan |
| Publisher | : Cambridge University Press |
| Total Pages | : 278 |
| Release | : 1992-02-20 |
| Genre | : Mathematics |
| ISBN | : 9780521423533 |
This development of the theory of complex algebraic curves was one of the peaks of nineteenth century mathematics. They have many fascinating properties and arise in various areas of mathematics, from number theory to theoretical physics, and are the subject of much research. By using only the basic techniques acquired in most undergraduate courses in mathematics, Dr. Kirwan introduces the theory, observes the algebraic and topological properties of complex algebraic curves, and shows how they are related to complex analysis.
| Author | : William Fulton |
| Publisher | : |
| Total Pages | : 120 |
| Release | : 2008 |
| Genre | : Mathematics |
| ISBN | : |
The aim of these notes is to develop the theory of algebraic curves from the viewpoint of modern algebraic geometry, but without excessive prerequisites. We have assumed that the reader is familiar with some basic properties of rings, ideals and polynomials, such as is often covered in a one-semester course in modern algebra; additional commutative algebra is developed in later sections.
| Author | : Eduardo Casas-Alvero |
| Publisher | : Springer Nature |
| Total Pages | : 224 |
| Release | : 2019-11-30 |
| Genre | : Mathematics |
| ISBN | : 3030290166 |
The book presents the central facts of the local, projective and intrinsic theories of complex algebraic plane curves, with complete proofs and starting from low-level prerequisites. It includes Puiseux series, branches, intersection multiplicity, Bézout theorem, rational functions, Riemann-Roch theorem and rational maps. It is aimed at graduate and advanced undergraduate students, and also at anyone interested in algebraic curves or in an introduction to algebraic geometry via curves.
| Author | : Joseph H. Silverman |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 292 |
| Release | : 2013-04-17 |
| Genre | : Mathematics |
| ISBN | : 1475742525 |
The theory of elliptic curves involves a blend of algebra, geometry, analysis, and number theory. This book stresses this interplay as it develops the basic theory, providing an opportunity for readers to appreciate the unity of modern mathematics. The book’s accessibility, the informal writing style, and a wealth of exercises make it an ideal introduction for those interested in learning about Diophantine equations and arithmetic geometry.
| Author | : Ernst Kunz |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 285 |
| Release | : 2007-06-10 |
| Genre | : Mathematics |
| ISBN | : 0817644431 |
* Employs proven conception of teaching topics in commutative algebra through a focus on their applications to algebraic geometry, a significant departure from other works on plane algebraic curves in which the topological-analytic aspects are stressed *Requires only a basic knowledge of algebra, with all necessary algebraic facts collected into several appendices * Studies algebraic curves over an algebraically closed field K and those of prime characteristic, which can be applied to coding theory and cryptography * Covers filtered algebras, the associated graded rings and Rees rings to deduce basic facts about intersection theory of plane curves, applications of which are standard tools of computer algebra * Examples, exercises, figures and suggestions for further study round out this fairly self-contained textbook
