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Moduli of curves : CIMAT Guanajuato, Mexico 2016

Author: L Brambila; C Ciliberto; Eduardo Esteves; Margarida Melo, (Mathematician); Claire Voisin
Publisher: Cham : Springer, 2017.
Series: Lecture notes of the Unione Matematica Italiana, 21.
Edition/Format:   eBook : Document : Conference publication : EnglishView all editions and formats
Summary:
Providing a timely description of the present state of the art of moduli spaces of curves and their geometry, this volume is written in a way which will make it extremely useful both for young people who want to approach this important field, and also for established researchers, who will find references, problems, original expositions, new viewpoints, etc. The book collects the lecture notes of a number of leading  Read more...
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Genre/Form: Electronic books
Additional Physical Format: Print version:
Moduli of curves.
Cham : Springer, 2017
(OCoLC)985081654
Material Type: Conference publication, Document, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: L Brambila; C Ciliberto; Eduardo Esteves; Margarida Melo, (Mathematician); Claire Voisin
ISBN: 9783319594866 3319594869
OCLC Number: 1005564739
Description: 1 online resource
Contents: Preface --
Contents --
Lectures on Tropical Curves and Their Moduli Spaces --
1 Introduction --
2 From the Beginning: Tropical Plane Curves --
2.1 Embedded Tropicalizations --
2.2 A Very, Very Short Treatment of Berkovich Analytifications --
3 Abstract Algebraic and Tropical Curves --
3.1 Stable Curves --
3.2 Stable Models --
3.3 Dual Graphs of Stable Curves --
3.4 Abstract Tropical Curves --
3.5 From Algebraic to Tropical Curves: Abstract Tropicalization --
4 Definition of the Moduli Space of Tropical Curves --
5 Boundary Complexes of Toroidal Embeddings 5.1 Ã#x89;tale Morphisms5.2 Toroidal Embeddings --
5.3 Boundary Complexes of Toroidal Embeddings --
6 Toroidal Deligne-Mumford Stacks --
6.1 Categories Fibered in Groupoids --
6.2 Fast Forward: Deligne-Mumford Stacks, and Toroidal Embeddings --
7 Compactification of Mg, n by Stable Curves --
7.1 The Boundary Strata MG of Mg, n --
7.2 Applications to the Topology of Mg, n --
References --
Higher Dimensional Varieties and their Moduli Spaces --
1 Introduction --
1.1 Projective Surfaces --
1.2 Higher Dimensional Projective Varieties 2 Birational Geometry of Log Pairs3 Termination of Threefold Flips --
4 Some Results --
5 Minimal Model Program with Scaling --
6 Flops and Sarkisov Program --
References --
Higher-Dimensional Varieties --
1 Introduction --
2 Divisors --
2.1 Weil and Cartier Divisors --
2.2 Invertible Sheaves --
2.3 Intersection of Curves and Divisors --
2.4 Line Bundles --
2.5 Linear Systems and Morphisms to Projective Spaces --
2.6 Globally Generated Sheaves --
2.7 Ample Divisors --
2.8 Ample Divisors on Curves --
2.9 Nef Divisors --
2.10 Cones of Divisors 2.11 The Canonical Divisor3 Riemannâ#x80;#x93;Roch Theorems --
3.1 Intersecting Two Curves on a Surface --
3.2 General Intersection Numbers --
3.3 Intersection of Divisors Over the Complex Numbers --
3.4 Asymptotic Numbers of Sections --
3.5 Kodaira Dimension and Iitaka Fibrations --
4 Rational Curves on Varieties --
4.1 Parametrizing Curves --
4.2 Free Rational Curves and Uniruled Varieties --
4.3 Bend-and-Break Lemmas --
4.4 Rational Curves on Fano Varieties --
4.5 Rational Curves on Varieties Whose Canonical Divisor is not nef --
5 The Cone Theorem 5.1 Cone of Curves5.2 Mori's Cone Theorem --
5.3 Contractions of KX-Negative Extremal Rays --
5.4 Various Types of Contractions --
5.5 Fiber-Type Contractions --
5.6 Divisorial Contractions --
5.7 Small Contractions and Flips --
5.8 The Minimal Model Program --
References --
Progress on Syzygies of Algebraic Curves --
1 Introduction --
2 Syzygies of Graded Modules Over Polynomial Algebras --
3 Syzygies in Algebraic Geometry --
4 Green's Conjecture --
4.1 The Resolution of a General Canonical Curve --
4.2 The Resolution of Canonical Curves of Small Genus
Series Title: Lecture notes of the Unione Matematica Italiana, 21.
Responsibility: Leticia Brambila Paz [and others], editors.

Abstract:

Providing a timely description of the present state of the art of moduli spaces of curves and their geometry, this volume is written in a way which will make it extremely useful both for young people who want to approach this important field, and also for established researchers, who will find references, problems, original expositions, new viewpoints, etc. The book collects the lecture notes of a number of leading algebraic geometers and in particular specialists in the field of moduli spaces of curves and their geometry. This is an important subject in algebraic geometry and complex analysis which has seen spectacular developments in recent decades, with important applications to other parts of mathematics such as birational geometry and enumerative geometry, and to other sciences, including physics. The themes treated are classical but with a constant look to modern developments (see Cascini, Debarre, Farkas, and Sernesi's contributions), and include very new material, such as Bridgeland stability (see Macri's lecture notes) and tropical geometry (see Chan's lecture notes).--

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