Geometry of Algebraic Curves: Volume II with a contribution by Joseph Daniel Harris

ISBN-13: 9783540426882 / Angielski / Twarda / 2011 / 963 str.

Enrico Arbarello;Maurizio Cornalba;Phillip Griffiths

Opis

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 Teichmuller 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.

Recenzje

From the reviews:

"This second volume will become the standard reference for researchers and students working on the algebraic geometry of curves. With almost 700 items in the rich 42-page bibliography, bibliographical notes at the end of every chapter to guide the reader and sets of (guided) exercises as in the first volume, this second volume is an interactive resource for everyone seriously interested on this beautiful part of algebraic geometry. We owe the authors a heartfelt thank you for writing such a rich, beautiful and full treatise." (Felipe Zaldivar, The Mathematical Association of America, July, 2011)

"Here, after a quarter of a century, is finally the sequel to Volume I ... . That volume, essentially devoted to properties of a single curve ... . The present volume has its focus on their moduli ... . The book under review is very helpful for reference and for learning the details ... . Summing up: every algebraic geometer should have a copy, while a Teichmüller person and a topologist should seriously consider getting one." (E. Looijenga, Mathematical Reviews, Issue 2012 e)

"Provide comprehensive and detailed foundations for the theory of moduli of complex algebraic curves, and that from multiple perspectives and various points of view. ... The bibliography at the end of the book is extremely rich and very up-to-date. ... The current book is an excellent research monograph and reference book in the theory of complex algebraic curves and their moduli, which is very likely to become an indispensable source for researchers and graduate students in both complex geometry and mathematical physics." (Werner Kleinert, Zentralblatt MATH, Vol. 1235, 2012)

Spis treści

Preface.- Guide to the Reader.- Chapter IX. The Hilbert Scheme.- Chapter X. Nodal curves.- Chapter XI. Elementary deformation theory and some applications.- Chapter XII. The moduli space of stable curves.- Chapter XIII. Line bundles on moduli.- Chapter XIV. The projectivity of the moduli space of stable curves.- Chapter XV. The Teichmüller point of view.- Chapter XVI. Smooth Galois covers of moduli spaces.- Chapter XVII. Cycles on the moduli spaces of stable curves.- Chapter XVIII. Cellular decomposition of moduli spaces.- Chapter XIX. First consequences of the cellular decomposition .- Chapter XX. Intersection theory of tautological classes.- Chapter XXI. Brill-Noether theory on a moving curve.- Bibliography.- Index.

Blurb

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 will be of value to students who wish to learn the subject and to experts as a reference source.

The first volume appeared 1985 as volume 267 of the same series.