Algebraic Curves, the Brill and Noether Way
Algebraic Curves, the Brill and Noether Way

Author: Eduardo Casas-Alvero

Publisher: Springer Nature

Published: 2019-11-30

Total Pages: 224

ISBN-13: 3030290166

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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.

A Treatise on Algebraic Plane Curves
A Treatise on Algebraic Plane Curves

Author: Julian Lowell Coolidge

Publisher: Courier Corporation

Published: 2004-01-01

Total Pages: 554

ISBN-13: 9780486495767

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A thorough introduction to the theory of algebraic plane curves and their relations to various fields of geometry and analysis. Almost entirely confined to the properties of the general curve, and chiefly employs algebraic procedure. Geometric methods are much employed, however, especially those involving the projective geometry of hyperspace. 1931 edition. 17 illustrations.

Brill--Noether Theory Over the Hurwitz Space
Brill--Noether Theory Over the Hurwitz Space

Author: Hannah Kerner Larson

Publisher:

Published: 2022

Total Pages:

ISBN-13:

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Historically, algebraic curves were defined as the solutions to a collection of polynomial equations inside of some ambient space. In the 19th century, however, mathematicians defined the notion of an abstract curve. With this perspective, the same abstract curve may sit in an ambient (projective) space in more than one way. The foundational Brill--Noether theorem, proved in the 1970s and 1980s, bridges these two perspectives by describing the maps of "most" abstract curves to projective spaces. However, the theorem does not hold for all curves. In nature, we often encounter curves already in (or mapping to) a projective space, and the presence of such a map may force the curve to have unexpected maps to other projective spaces! The first case of this is a curve that already has a map to the projective line. From the 1990s through the late 2010s, several mathematicians investigated this first case. They found that the space of maps of such a curve to other projective spaces can have multiple components of varying dimensions and eventually determined the dimension of the largest component. In this thesis, I develop analogues of all the main theorems of Brill--Noether theory for curves that already have a map to the projective line. The moduli space of curves together with a map to the line is called the Hurwitz space, so we call this work Brill--Noether theory over the Hurwitz space.

Lectures on Curves on an Algebraic Surface. (AM-59), Volume 59
Lectures on Curves on an Algebraic Surface. (AM-59), Volume 59

Author: David Mumford

Publisher: Princeton University Press

Published: 2016-03-02

Total Pages: 212

ISBN-13: 1400882060

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These lectures, delivered by Professor Mumford at Harvard in 1963-1964, are devoted to a study of properties of families of algebraic curves, on a non-singular projective algebraic curve defined over an algebraically closed field of arbitrary characteristic. The methods and techniques of Grothendieck, which have so changed the character of algebraic geometry in recent years, are used systematically throughout. Thus the classical material is presented from a new viewpoint.

Algebraic Curves
Algebraic Curves

Author: William Fulton

Publisher:

Published: 2008

Total Pages: 120

ISBN-13:

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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.

Computational Aspects Of Algebraic Curves
Computational Aspects Of Algebraic Curves

Author: Tanush Shaska

Publisher: World Scientific

Published: 2005-08-24

Total Pages: 286

ISBN-13: 9814479578

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The development of new computational techniques and better computing power has made it possible to attack some classical problems of algebraic geometry. The main goal of this book is to highlight such computational techniques related to algebraic curves. The area of research in algebraic curves is receiving more interest not only from the mathematics community, but also from engineers and computer scientists, because of the importance of algebraic curves in applications including cryptography, coding theory, error-correcting codes, digital imaging, computer vision, and many more.This book covers a wide variety of topics in the area, including elliptic curve cryptography, hyperelliptic curves, representations on some Riemann-Roch spaces of modular curves, computation of Hurwitz spectra, generating systems of finite groups, Galois groups of polynomials, among other topics.

Geometry of Algebraic Curves
Geometry of Algebraic Curves

Author: Enrico Arbarello

Publisher: Springer

Published: 2013-08-30

Total Pages: 387

ISBN-13: 9781475753240

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In recent years there has been enormous activity in the theory of algebraic curves. Many long-standing problems have been solved using the general techniques developed in algebraic geometry during the 1950's and 1960's. Additionally, unexpected and deep connections between algebraic curves and differential equations have been uncovered, and these in turn shed light on other classical problems in curve theory. It seems fair to say that the theory of algebraic curves looks completely different now from how it appeared 15 years ago; in particular, our current state of knowledge repre sents a significant advance beyond the legacy left by the classical geometers such as Noether, Castelnuovo, Enriques, and Severi. These books give a presentation of one of the central areas of this recent activity; namely, the study of linear series on both a fixed curve (Volume I) and on a variable curve (Volume II). Our goal is to give a comprehensive and self-contained account of the extrinsic geometry of algebraic curves, which in our opinion constitutes the main geometric core of the recent advances in curve theory. Along the way we shall, of course, discuss appli cations of the theory of linear series to a number of classical topics (e.g., the geometry of the Riemann theta divisor) as well as to some of the current research (e.g., the Kodaira dimension of the moduli space of curves).