Curves and Their Jacobians
Title | Curves and Their Jacobians PDF eBook |
Author | David Mumford |
Publisher | Ann Arbor : University of Michigan Press, c1975, 1976 printing. |
Pages | 120 |
Release | 1975 |
Genre | Mathematics |
ISBN |
Rigid Geometry of Curves and Their Jacobians
Title | Rigid Geometry of Curves and Their Jacobians PDF eBook |
Author | Werner Lütkebohmert |
Publisher | Springer |
Pages | 398 |
Release | 2016-01-26 |
Genre | Mathematics |
ISBN | 331927371X |
This book presents some of the most important aspects of rigid geometry, namely its applications to the study of smooth algebraic curves, of their Jacobians, and of abelian varieties - all of them defined over a complete non-archimedean valued field. The text starts with a survey of the foundation of rigid geometry, and then focuses on a detailed treatment of the applications. In the case of curves with split rational reduction there is a complete analogue to the fascinating theory of Riemann surfaces. In the case of proper smooth group varieties the uniformization and the construction of abelian varieties are treated in detail. Rigid geometry was established by John Tate and was enriched by a formal algebraic approach launched by Michel Raynaud. It has proved as a means to illustrate the geometric ideas behind the abstract methods of formal algebraic geometry as used by Mumford and Faltings. This book should be of great use to students wishing to enter this field, as well as those already working in it.
The Red Book of Varieties and Schemes
Title | The Red Book of Varieties and Schemes PDF eBook |
Author | David Mumford |
Publisher | Springer |
Pages | 316 |
Release | 2004-02-21 |
Genre | Mathematics |
ISBN | 3540460217 |
Mumford's famous "Red Book" gives a simple, readable account of the basic objects of algebraic geometry, preserving as much as possible their geometric flavor and integrating this with the tools of commutative algebra. It is aimed at graduates or mathematicians in other fields wishing to quickly learn aboutalgebraic geometry. This new edition includes an appendix that gives an overview of the theory of curves, their moduli spaces and their Jacobians -- one of the most exciting fields within algebraic geometry.
Lectures on Algebraic Geometry II
Title | Lectures on Algebraic Geometry II PDF eBook |
Author | Günter Harder |
Publisher | Springer Science & Business Media |
Pages | 376 |
Release | 2011-04-21 |
Genre | Mathematics |
ISBN | 3834881597 |
This second volume introduces the concept of shemes, reviews some commutative algebra and introduces projective schemes. The finiteness theorem for coherent sheaves is proved, here again the techniques of homological algebra and sheaf cohomology are needed. In the last two chapters, projective curves over an arbitrary ground field are discussed, the theory of Jacobians is developed, and the existence of the Picard scheme is proved. Finally, the author gives some outlook into further developments- for instance étale cohomology- and states some fundamental theorems.
Algebraic Curves and One-Dimensional Fields
Title | Algebraic Curves and One-Dimensional Fields PDF eBook |
Author | Fedor Bogomolov |
Publisher | American Mathematical Soc. |
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.
Degenerating Curves and Their Jacobians
Title | Degenerating Curves and Their Jacobians PDF eBook |
Author | Dino Jacques Lorenzini |
Publisher | |
Pages | 174 |
Release | 1988 |
Genre | |
ISBN |
Algebraic Curves and Their Applications
Title | Algebraic Curves and Their Applications PDF eBook |
Author | Lubjana Beshaj |
Publisher | American Mathematical Soc. |
Pages | 344 |
Release | 2019-02-26 |
Genre | Curves, Algebraic |
ISBN | 1470442477 |
This volume contains a collection of papers on algebraic curves and their applications. While algebraic curves traditionally have provided a path toward modern algebraic geometry, they also provide many applications in number theory, computer security and cryptography, coding theory, differential equations, and more. Papers cover topics such as the rational torsion points of elliptic curves, arithmetic statistics in the moduli space of curves, combinatorial descriptions of semistable hyperelliptic curves over local fields, heights on weighted projective spaces, automorphism groups of curves, hyperelliptic curves, dessins d'enfants, applications to Painlevé equations, descent on real algebraic varieties, quadratic residue codes based on hyperelliptic curves, and Abelian varieties and cryptography. This book will be a valuable resource for people interested in algebraic curves and their connections to other branches of mathematics.