Lectures on Curves on an Algebraic Surface
Lectures on Curves on an Algebraic Surface

Author: David Mumford

Publisher: Princeton University Press

Published: 2016-03-02

Total Pages: 219

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.

Singularities of Plane Curves
Singularities of Plane Curves

Author: Eduardo Casas-Alvero

Publisher: Cambridge University Press

Published: 2000-08-31

Total Pages: 363

ISBN-13: 0521789591

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Comprehensive and self-contained exposition of singularities of plane curves, including new, previously unpublished results.

Introduction to the Theory of Schemes
Introduction to the Theory of Schemes

Author: Yuri I. Manin

Publisher: Springer

Published: 2018-05-15

Total Pages: 217

ISBN-13: 3319743163

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This English edition of Yuri I. Manin's well-received lecture notes provides a concise but extremely lucid exposition of the basics of algebraic geometry and sheaf theory. The lectures were originally held in Moscow in the late 1960s, and the corresponding preprints were widely circulated among Russian mathematicians. This book will be of interest to students majoring in algebraic geometry and theoretical physics (high energy physics, solid body, astrophysics) as well as to researchers and scholars in these areas. "This is an excellent introduction to the basics of Grothendieck's theory of schemes; the very best first reading about the subject that I am aware of. I would heartily recommend every grad student who wants to study algebraic geometry to read it prior to reading more advanced textbooks."- Alexander Beilinson

Geometric Invariant Theory for Polarized Curves
Geometric Invariant Theory for Polarized Curves

Author: Gilberto Bini

Publisher: Springer

Published: 2014-11-07

Total Pages: 217

ISBN-13: 3319113372

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We investigate GIT quotients of polarized curves. More specifically, we study the GIT problem for the Hilbert and Chow schemes of curves of degree d and genus g in a projective space of dimension d-g, as d decreases with respect to g. We prove that the first three values of d at which the GIT quotients change are given by d=a(2g-2) where a=2, 3.5, 4. We show that, for a>4, L. Caporaso's results hold true for both Hilbert and Chow semistability. If 3.5a

Foundational Essays on Topological Manifolds, Smoothings, and Triangulations. (AM-88), Volume 88
Foundational Essays on Topological Manifolds, Smoothings, and Triangulations. (AM-88), Volume 88

Author: Robion C. Kirby

Publisher: Princeton University Press

Published: 2016-03-02

Total Pages: 368

ISBN-13: 1400881501

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Since Poincaré's time, topologists have been most concerned with three species of manifold. The most primitive of these--the TOP manifolds--remained rather mysterious until 1968, when Kirby discovered his now famous torus unfurling device. A period of rapid progress with TOP manifolds ensued, including, in 1969, Siebenmann's refutation of the Hauptvermutung and the Triangulation Conjecture. Here is the first connected account of Kirby's and Siebenmann's basic research in this area. The five sections of this book are introduced by three articles by the authors that initially appeared between 1968 and 1970. Appendices provide a full discussion of the classification of homotopy tori, including Casson's unpublished work and a consideration of periodicity in topological surgery.

Moduli of Curves
Moduli of Curves

Author: Leticia Brambila Paz

Publisher: Springer

Published: 2017-10-03

Total Pages: 248

ISBN-13: 3319594869

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Providing a timely description of the present state of the art of moduli spaces of curves and their geometry, this volume is written in a way which will make it extremely useful both for young people who want to approach this important field, and also for established researchers, who will find references, problems, original expositions, new viewpoints, etc. The book collects the lecture notes of a number of leading algebraic geometers and in particular specialists in the field of moduli spaces of curves and their geometry. This is an important subject in algebraic geometry and complex analysis which has seen spectacular developments in recent decades, with important applications to other parts of mathematics such as birational geometry and enumerative geometry, and to other sciences, including physics. The themes treated are classical but with a constant look to modern developments (see Cascini, Debarre, Farkas, and Sernesi's contributions), and include very new material, such as Bridgeland stability (see Macri's lecture notes) and tropical geometry (see Chan's lecture notes).

Geometry of Algebraic Curves
Geometry of Algebraic Curves

Author: Enrico Arbarello

Publisher: Springer Science & Business Media

Published: 2011-03-10

Total Pages: 983

ISBN-13: 3540693920

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The second volume of the Geometry of Algebraic Curves is devoted to the foundations of the theory of moduli of algebraic curves. Its authors are research mathematicians who have actively participated in the development of the Geometry of Algebraic Curves. The subject is an extremely fertile and active one, both within the mathematical community and at the interface with the theoretical physics community. The approach is unique in its blending of algebro-geometric, complex analytic and topological/combinatorial methods. It treats important topics such as Teichmüller theory, the cellular decomposition of moduli and its consequences and the Witten conjecture. The careful and comprehensive presentation of the material is of value to students who wish to learn the subject and to experts as a reference source. The first volume appeared 1985 as vol. 267 of the same series.

Arithmetic Moduli of Elliptic Curves. (AM-108), Volume 108
Arithmetic Moduli of Elliptic Curves. (AM-108), Volume 108

Author: Nicholas M. Katz

Publisher: Princeton University Press

Published: 2016-03-02

Total Pages: 528

ISBN-13: 1400881714

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This work is a comprehensive treatment of recent developments in the study of elliptic curves and their moduli spaces. The arithmetic study of the moduli spaces began with Jacobi's "Fundamenta Nova" in 1829, and the modern theory was erected by Eichler-Shimura, Igusa, and Deligne-Rapoport. In the past decade mathematicians have made further substantial progress in the field. This book gives a complete account of that progress, including not only the work of the authors, but also that of Deligne and Drinfeld.