Algebraic Curves: Towards Moduli Spaces: Moscow Lectures, cartea 2
Autor Maxim E. Kazaryan, Sergei K. Lando, Victor V. Prasolov Traducere de Natalia Tsilevichen Limba Engleză Hardback – 6 feb 2019
The book begins by studying individual smooth algebraic curves, including the most beautiful ones, before addressing families of curves. Studying families of algebraic curves often proves to be more efficient than studying individual curves: these families and their total spaces can still be smooth, even if there are singular curves among their members. A major discovery of the 20th century, attributed to P. Deligne and D. Mumford, was that curves with only mild singularities form smooth compact moduli spaces. An unexpected byproduct of this discovery was the realization that the analysis of more complex curve singularities is not a necessary step in understanding the geometry of the moduli spaces. Thebook does not use the sophisticated machinery of modern algebraic geometry, and most classical objects related to curves – such as Jacobian, space of holomorphic differentials, the Riemann-Roch theorem, and Weierstrass points – are treated at a basic level that does not require a profound command of algebraic geometry, but which is sufficient for extending them to vector bundles and other geometric objects associated to moduli spaces. Nevertheless, it offers clear information on the construction of the moduli spaces, and provides readers with tools for practical operations with this notion.
Based on several lecture courses given by the authors at the Independent University of Moscow and Higher School of Economics, the book also includes a wealth of problems, making it suitable not only for individual research, but also as a textbook for undergraduate and graduate coursework
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Specificații
ISBN-13: 9783030029425
ISBN-10: 3030029425
Pagini: 220
Ilustrații: XIV, 231 p. 37 illus.
Dimensiuni: 155 x 235 x 16 mm
Greutate: 0.54 kg
Ediția:1st ed. 2018
Editura: Springer International Publishing
Colecția Springer
Seria Moscow Lectures
Locul publicării:Cham, Switzerland
ISBN-10: 3030029425
Pagini: 220
Ilustrații: XIV, 231 p. 37 illus.
Dimensiuni: 155 x 235 x 16 mm
Greutate: 0.54 kg
Ediția:1st ed. 2018
Editura: Springer International Publishing
Colecția Springer
Seria Moscow Lectures
Locul publicării:Cham, Switzerland
Cuprins
Introduction.- 1 Preliminaries – 2 Algebraic curves.- 3 Complex structure and the topology of curves.- 4 Curves in projective spaces.- 5 Plücker formulas.- 6 Mappings of curves.- 7 Differential 1-forms on curves.- 8 Line bundles, linear systems, and divisors.- 9 Riemann–Roch formula and its applications.- 10 Proof of the Riemann–Roch formula.- 11 Weierstrass points.- 12 Abel’s theorem.- 13 Examples of moduli spaces.- 14 Approaches to constructing moduli spaces.- 15 Moduli spaces of rational curves with marked points.- 16 Stable curves.- 17 A backward look from the viewpoint of characteristic classes.- 18 Moduli spaces of stable maps.- 19 Exam problems.- References.- Index.
Recenzii
“The book under review is an accessible introduction to the study of complex algebraic curves and their moduli spaces. … The book comes with sets of exercises in each of its chapters and can be used as a textbook for a graduate course or for self-study by a motivated reader.” (Felipe Zaldivar, MAA Reviews, April 22, 2019)
Notă biografică
Maxim Kazaryan is a researcher at the Steklov Mathematical Institute RAS. He also works as a professor of mathematics at the NRU Higher School of Economics since 2008 and at the Skolkovo Institute of Science and Technology since 2016.
Sergei Lando is a professor of mathematics at the NRU Higher School of Economics since 2008 and at the Skolkovo Institute of Science and Technology since 2016. He was the first Dean of the Department of Mathematics at the NRU HSE. He also is a Vice-President of the Moscow Mathematical Society.
Victor Prasolov is a permanent teacher of mathematics at the Independent University of Moscow.
Sergei Lando is a professor of mathematics at the NRU Higher School of Economics since 2008 and at the Skolkovo Institute of Science and Technology since 2016. He was the first Dean of the Department of Mathematics at the NRU HSE. He also is a Vice-President of the Moscow Mathematical Society.
Victor Prasolov is a permanent teacher of mathematics at the Independent University of Moscow.
Textul de pe ultima copertă
This book offers a concise yet thorough introduction to the notion of moduli spaces of complex algebraic curves. Over the last few decades, this notion has become central not only in algebraic geometry, but in mathematical physics, including string theory, as well.The book begins by studying individual smooth algebraic curves, including the most beautiful ones, before addressing families of curves. Studying families of algebraic curves often proves to be more efficient than studying individual curves: these families and their total spaces can still be smooth, even if there are singular curves among their members. A major discovery of the 20th century, attributed to P. Deligne and D. Mumford, was that curves with only mild singularities form smooth compact moduli spaces. An unexpected byproduct of this discovery was the realization that the analysis of more complex curve singularities is not a necessary step in understanding the geometry of the moduli spaces.
The book does not use the sophisticated machinery of modern algebraic geometry, and most classical objects related to curves – such as Jacobian, space of holomorphic differentials, the Riemann-Roch theorem, and Weierstrass points – are treated at a basic level that does not require a profound command of algebraic geometry, but which is sufficient for extending them to vector bundles and other geometric objects associated to moduli spaces. Nevertheless, it offers clear information on the construction of the moduli spaces, and provides readers with tools for practical operations with this notion.
Based on several lecture courses given by the authors at the Independent University of Moscow and Higher School of Economics, the book also includes a wealth of problems, making it suitable not only for individual research, but also as a textbook for undergraduate and graduate coursework
Based on several lecture courses given by the authors at the Independent University of Moscow and Higher School of Economics, the book also includes a wealth of problems, making it suitable not only for individual research, but also as a textbook for undergraduate and graduate coursework
Caracteristici
Leads a reader to far advanced topics widely used in modern research, using basic tools from the first two years of university studies From the very beginning, the study of algebraic curves is aimed at the construction of their moduli spaces in the final chapters Supplied with numerous exercises and problems both making the book a convenient base for a university lecture course and allowing the reader to control his/her progress