Algebraic Geometry between Tradition and Future: An Italian Perspective

Contents
About the Editor
Introduction
Francesco Severi's Mathematical Library
1 The Relevance of Private Libraries in Historical Perspective
2 Francesco Severi and His Private Library
3 Mathematical Library of Francesco Severi
Francesco Severi and the Fascist Regime
1 Severi: The Mathematician and the Politician
2 Severi in Padua
3 Moving to Rome
4 The Oath
5 1938
6 The End of 20 Years of Fascism
7 The Purge
8 The Purge at the Lincei
Fabio Conforto (1909–1954): His Scientific and Academic Career at the University of Rome
1 Introduction
2 Early Career of Conforto
3 Fabio Conforto in Rome
3.1 Conforto at INAC
3.2 The Scientific Relationship with Francesco Severi
4 Conforto and WWII
4.1 Difficult Years: Rome, Reggio Calabria, Lecce
5 Conforto's Scientific Journeys
6 Conclusions
References
Alessandro Terracini (1889 –1968): Teaching and Research from the University Years to the Racial Laws
Abbreviations
1 Introduction
2 Training, First Works, and First Scientific Contacts
3 Participation in the Great War and First Appointments
4 At the Universities of Modena and Catania
5 Return to Turin: From Tenure to the Racial Laws
5.1 Research and Teaching
5.2 “Amputation” of the Italian Scientific Community and the Decision to Emigrate
References
Higher-Dimensional Geometry from Fano to Mori and Beyond
1 Introduction
2 Fano Varieties and Fano-Mori Contractions
3 Classifications of Fano Varieties and Fano-Mori Contractions
4 Rational Curves on Fano Varieties: Rationally Connected
5 Elephants and Base Point Freeness
6 Kähler-Einstein Metrics
References
Gino Fano (1871 –1952)
Acronyms and Abbreviations
1 Introduction
2 From Segre's School to Achievements on the International Scene
2.1 Early Research as a Student
2.2 Fano in Göttingen
2.3 Research, Epistemological Vision, and Teaching
2.3.1 The Epistemological Vision
2.3.2 “Fighting Prejudices Against the Supposed Mysteries of Mathematics.” Fano's Commitment to Education
2.3.3 Scientific Dissemination
3 The Late Fano
3.1 Discrimination and the Collapse of the Three Pillars of Life: Family, Country, and Profession
3.2 “So the Time Came to Flee Turin”: Emigration
3.3 The Latest Studies on Varieties
3.4 The Return to Teaching in the Italian University Camp in Lausanne
3.4.1 Keynote Lectures at the Cercle Mathématique
3.5 Between Turin and the United States
4 On Fano's Material and Immaterial Heritage
4.1 Fano's Work Within the Cultural Heritage of the Italian Geometric Tradition
4.2 A Specific Mathematical Heritage: Methods and Results
4.3 Fano's Legacy in the Short and Long Terms
5 Conclusive Remarks
References
From Enriques Surface to Artin-Mumford Counterexample
1 Perspectives on Enriques Surface and Rationality
2 Webs of Quadrics and Enriques Surfaces
2.1 Webs of Quadrics
2.2 Reye Congruences of Lines
2.3 Further Notation
3 Reye Congruences and Symmetroids
3.1 The Classical Construction
3.2 Order and Class of S
3.3 The Quartic Symmetroid
3.4 Quartic Double Solids
4 The Artin-Mumford Counterexample Revisited
4.1 Artin-Mumford Double Solids
4.2 The Congruence of Bitangent Lines F(S̃+)
4.3 The Rational Map ψ: G "044BW
4.4 S and the Fano Surface F(W̃')
4.5 The Counterexample of Artin-Mumford
References
The Theorem of Completeness of the Characteristic Series: Enriques' Contribution
1 Introduction
2 The Theorem of Completeness of the Characteristic Series
3 Enriques–Poincaré Theorem
4 The Fundamental Theorem of Irregular Surfaces
5 Enriques' Attempts for the Proof of Theorem 9
References
Severi, Zappa, and the Characteristic System
1 Introduction
2 The Fundamental Problem
3 The Examples
References
Two Letters by Guido Castelnuovo
1 Introduction
2 The Letters
2.1 Guido Castelnuovo to Francesco Severi
2.2 Guido Castelnuovo to Beniamino Segre
3 Regular 1-Forms on a Surface
3.1 The General Set Up
3.2 The Expression of 1-Forms on a Surface
3.3 Closedness of 1-Forms
3.4 Homogeneous Form of Picard's Relation
4 Comments on Castelnuovo's Letters
5 Algebraic Proofs via the Hodge–Frölicher Spectral Sequence
5.1 Global Regular 1-Forms Are Closed in Characteristic Zero
5.2 Analytic Irregularity and Arithmetic Irregularity Coincide
References
Guido Castelnuovo and His Heritage: Geometry, Combinatorics, and Teaching
1 Introduction
2 A Trilogy of Papers
3 A Few Remarks
3.1 Guido Castelnuovo and Corrado Segre
3.2 Geometry and Probability
3.3 Guido ed Emma Castelnuovo
References
Guido Castelnuovo and His Family
1 Guido Castelnuovo, His Times, and His Family
1.1 Family Words
2 Enrico Castelnuovo (1839–1915): Passion and Duty
3 Luigi Luzzatti (1841–1927): Solidarity and Institutions
3.1 The Higher School of Commerce of Venice
3.2 Literary Interlude
3.3 Freedom of Conscience and Science
3.4 Luigi Luzzatti's Judaism
3.5 Solidarity and Social Security
3.6 A Letter from Paolo Medolaghi to Friedrich Engel
3.7 Guido Castelnuovo's Commitment to Promoting Statistics and Probability Theory
4 Adele Levi Della Vida (1822–1915): Education and Emancipation
4.1 Children's Education
4.2 Fröbel's Gifts
5 Castelnuovo and the Problem of Higher Education
5.1 The Attitudes to be Developed in Teaching, and How to Develop Them
5.2 The Importance of Linking Disciplines in Teaching
References
The Genesis of the Italian School of Algebraic Geometry Through the Correspondence Between Luigi Cremona and Some of His Students
1 Introduction
2 The First Student of Cremona: Eugenio Bertini
3 Ettore Caporali in His Correspondence with Cremona
4 Riccardo De Paolis Through the Memory of Corrado Segre
5 Conclusions
References
Veronese, Cremona, and the Mystical Hexagram
1 Prologue
2 Before Veronese (1639–1877)
3 Giuseppe Veronese's First Work: 1877
3.1 Steiner Points
3.2 Kirkman Points
3.3 Veronese's Six Configurations
4 Veronese's Multi-Mystic
4.1 The First Step of the Multi-Mystic
4.2 The Second Step
5 Cremona's Interpretation of the Mystical Hexagram in the Theory of Cubic Surfaces
5.1 Cremona's Study on a Singular Cubic Surface
5.2 Cremona and the Multi-Mystic
6 Conclusions
References