The book offers an extensive study on the convoluted history of the research of algebraic surfaces, focusing for the first time on one of its characterizing curves: the branch curve. Starting with separate beginnings during the 19th century with descriptive geometry as well as knot theory, the book focuses on the 20th century, covering the rise of the Italian school of algebraic geometry between the 1900s till the 1930s (with Federigo Enriques, Oscar Zariski and Beniamino Segre, among others), the decline of its classical approach during the 1940s and the 1950s (with Oscar Chisini and his students), and the emergence of new approaches with Boris Moishezon’s program of braid monodromy factorization.
By focusing on how the research on one specific curve changed during the 20th century, the author provides insights concerning the dynamics of epistemic objects and configurations of mathematical research. It is in this sense that the book offers to take the branch curve as a cross-section through the history of algebraic geometry of the 20th century, considering this curve as an intersection of several research approaches and methods.Researchers in the history of science and of mathematics as well as mathematicians will certainly find this book interesting and appealing, contributing to the growing research on the history of algebraic geometry and its changing images.
Author(s): Michael Friedman
Series: Frontiers in the History of Science
Publisher: Birkhäuser
Year: 2022
Language: English
Pages: 257
City: Cham
Acknowledgements
Contents
1: Introduction
1.1 On Branch Points and Branch Curves
1.2 Dynamics of a Mathematical Object
1.2.1 Ephemeral Epistemic Configurations and the Identity of the Mathematical Objects
1.2.2 On Branch Points, Again: on Riemann´s Terminology and How (Not) to Transfer Results
1.2.3 On Branch Curves, Again: Plurality of Notations
1.2.4 Transformations Between Epistemic Configurations
1.3 An Overview: Historical Literature, Structure and Argument
1.3.1 Omitted Traditions
1.3.2 Structure of the Book: The Twentieth Century
2: Prologue: Separate Beginnings During the Nineteenth Century
2.1 The Beginning of the Nineteenth Century: Monge and the ``Contour Apparent´´
2.2 1820s-1860s: Étienne Bobillier and George Salmon
2.3 1890s-1900s: Wirtinger´s and Heegaard´s Turn Towards Knot Theory
2.4 The End of the Nineteenth Century: A Regression Toward the Local
3: 1900s-1930s: Branch Curves and the Italian School of Algebraic Geometry
3.1 Enriques: A Plurality of Methods to Investigate the Branch Curve
3.1.1 Enriques on Intuition and Visualization
3.1.2 The Turn of the nineteenth Century: First Attempts of Classification of Surfaces
3.1.2.1 On Double Covers and Branch Curves
3.1.2.2 End of the 1890s: Enriques´s Initial Configurations
3.1.3 Two Papers from 1912 and the Culmination of the Classification Project
3.1.4 1923: After the Classification Project
3.2 Zariski and Segre: Novel Approaches
3.2.1 The Late 1920s: Zariski on Existence Theorems and the Beginning of a Group-Theoretic Approach
3.2.2 1930: Segre and Special Position of the Singular Points
3.2.3 1930-1937: Before and After Zariski´s Algebraic Surfaces
3.2.3.1 1935: Zariski´s Algebraic Surfaces
3.2.3.2 After Algebraic Surfaces
3.3 Reflections on Rigor: Reassessment and New Definitions in the 1950s
3.4 Appendix to Chap. 3: Birational Maps and Genera of Curves and Surfaces
4: 1930s-1950s: Chisini´s Branch Curves: The Decline of the Classical Approach
4.1 The 1930s and Chisini´s First Conjecture
4.1.1 The ``Characteristic Bundle´´
4.1.2 On Braids, Branch Curves and Degenerations
4.1.2.1 Bernard d´Orgeval in Oflag X B
4.1.2.2 Guido Zappa´s degenerations
4.1.3 Detour. 1944: Chisini´s First `Conjecture´
4.2 Chisini´s Students: Isolation and Abandonment
4.2.1 Dedò and the New Notation of Braids
4.2.2 Tibiletti and the Second `Theorem´ of Chisini
4.3 Conclusion: Seclusion, Ignorance and Abandonment
4.4 Appendix to Chap. 4: A Short Introduction to the Braid Group
5: From the 1970s Onward: The Rise of Braid Monodromy Factorization
5.1 The 1960s: Generalization and Stagnation or the ``Rising Sea´´ and the Sunken Branch Curves
5.1.1 Detour: End of the 1950s: Abhyankar´s Conjecture
5.1.2 1971: The New Edition of Zariski´s Algebraic Surfaces
5.2 The 1970s: Livne and Moishezon on Equivalence of Factorizations
5.2.1 Livne´s MA Thesis from 1975
5.2.2 Separations of Configurations and Shifts of Contexts
5.2.3 On Surfaces with and Livne´s 1981 PhD Thesis
5.3 Moishezon´s Program
5.3.1 From the USSR to Israel and to the USA
5.3.1.1 Moishezon's Emigration and Jewish Mathematicians in the USSR
5.3.2 Before Braid Monodromy: The Shafarevich School, Moishezon and the Decomposition of Algebraic Surfaces
5.3.3 From 1981 to 1985: (Re)introducing Braid Monodromy
5.3.3.1 1981: The Search for Normal Forms
5.3.3.2 1983/1985: The Arithmetic of Braids and the Language of Factorizations
5.3.3.3 Conclusion: Moishezon and Chisini
5.4 Moishezon and Teicher Cross the Watershed
5.4.1 Coda: The Group-Theoretical Approach of the 1990s
6: Epilogue: On Ramified and Ignored Spaces
Bibliography
Index
