Birational geometry of algebraic surfaces (Master thesis)

Ζήκας, Σωκράτης

The topic of this expository M.Sc. thesis is the study of the birational geometry of algebraic surfaces in view of the Minimal Model Program (MMP). The main reference used is [1]. The first chapter is introductory, mainly presenting historical aspects related to the study of surfaces and to the development of the MMP. In the second chapter we briefly discuss the main tools which enable the interplay between the algebraic and the complex analytic categories. We also introduce some fundamental results centered around complex manifolds and sheaf cohomology. The third chapter is dedicated to the detailed view of the main theorems and ideas behind the MMP in dimension 2. We briefly touch on some foundational material, such as Intersection Theory and Blowups, and subsequently we present the main tools such as the Castelnuovo blow down theorem and the existence of Extremal contractions. Finally we translate the aforementioned results in terms of the Kleiman-Mori cone of effective 1-cycles, giving a more modern approach to the subject. In chapter four we study the end results of the MMP in dimension 2, namely the Mori fibre spaces and the Minimal models. In the first case, we present a theorem detailing their structure. The rest of the chapter is devoted to the detailed construction of the canonical model for a minimal model with Kodaira dimension, κ(S) = 2 as well as a brief discussion of the Hard Dichotomy and Abundance theorems. The fifth chapter contains the Enriques classification of algebraic surfaces up to birational equivalence as well as a sketch of the proof. In the last part of the chapter we introduce some examples for each birational class in the above classification and study some of their basic properties. In the sixth and final chapter we formulate and prove the Sarkisov program in dimension 2 and apply it to the study of birational relations between Mori fibre spaces.
Institution and School/Department of submitter: Πανεπιστήμιο Ιωαννίνων. Σχολή Θετικών Επιστημών. Τμήμα Μαθηματικών
Subject classification: Γεωμετρία, Αλγεβρική
Keywords: Αλγεβρική επιφάνεια,Birational γεωμετρία,Minimal Model Program (MMP)
Appears in Collections:Διατριβές Μεταπτυχιακής Έρευνας (Masters)

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