Τhe classification of complex algebraic surfaces is a very classical subject which goes back to the old Italian school of algebraic geometry with Enriques and Castelnuovo. However, the exposition in the present book is modern and follows Mori's approach to the classification of algebraic varieties. The text includes the P12
theorem, the Sarkisov programme in the surface case and the Noether–Castelnuovo theorem in its classical version.
This book serves as a relatively quick and handy introduction to the theory of algebraic surfaces and is intended for readers with a good knowledge of basic algebraic geometry. Although an acquaintance with the basic parts of books like Principles of Algebraic Geometry by Griffiths and Harris or Algebraic Geometry by Hartshorne should be sufficient, the author strove to make the text as self-contained as possible and, for this reason, a first chapter is devoted to a quick exposition of some preliminaries.
Keywords: Algebraic surfaces, classification
Author(s): Ciro Ciliberto
Series: EMS Series of Lectures in Mathematics
Publisher: European Mathematical Society
Year: 2020
Language: English
Pages: 145
Preface
Some preliminaries
Projective morphisms
Basic invariants of surfaces
The ramification formula
Basic formulas
Ample line bundles
The Hodge index theorem
Blow-up
Rational and birational maps
The relative canonical sheaf
Cones
Complete intersections
Stein factorization
Abelian varieties
The Albanese variety
Double covers
The Riemann existence theorem
Relative duality
Characterization of the complex projective plane
Minimal models
Ruled surfaces
Surfaces with non-nef canonical bundle
Proof of the rationality theorem
Zariski's lemma
Proof of the base point freeness theorem
Boundedness of denominators
Proof of the extremal contraction theorem
The cone theorem
Step 1 of the proof
Step 2 of the proof
Step 3 of the proof
Step 4 of the proof: the contraction theorem
Step 5 of the proof
The minimal model programme
The Castelnuovo rationality criterion
The fundamental theorem of the classification
The Castelnuovo–De Franchis theorem
The canonical bundle formula for elliptic fibrations
Basic lemmas
Proof of Theorem 9.2
Classification and the abundance theorem
Surfaces with kappa=-infinity
The abundance theorem: statement
Surfaces with kappa=2
Surfaces with kappa=1
Surfaces with kappa=0
Surfaces of general type
Some vanishing theorems
Connectedness of pluricanonical divisors
Base point freeness
Birationality
The Bagnera–De Franchis classification of bielliptic surfaces
The P_{12}-theorem
The Sarkisov programme
Sarkisov links
The Noether–Castelnuovo theorem: statement
Sarkisov degree
The Noether–Fano–Iskovskih theorem
Step 1: µ=µ'
Step 2: invariance of the adjoints
Step 3: conclusion
Sarkisov algorithm
The classical Noether–Castelnuovo theorem
Infinitely near points
Homaloidal nets
The simplicity
Proof of the classical Noether–Castelnuovo theorem
Examples
Negative curves
The blown-up plane
Products of elliptic curves
Bibliography
Index
