Birational Geometry and Moduli Spaces (Springer INdAM Series)

Preface
Contents
About the Editors
Negative Rational Curves and Their Deformations on Hyperkähler Manifolds
	1 Hyperkähler Manifolds
	2 Rational Curves
		2.1 Lower Bound for the Dimension of a Family
		2.2 Minimality
	3 Deformation Spaces
		3.1 Teichmüller and Torelli
		3.2 Kähler Chambers and MBM Classes
	4 Main Result
	5 Ergodicity
	6 Proof of the Main Result
	References
Moduli Spaces of Cubic Threefolds and of Irreducible Holomorphic Symplectic Manifolds
	1 Introduction
	2 An Isomorphism of Moduli Spaces
		2.1 Fano Varieties of Cyclic Cubic Fourfolds
		2.2 An ``Occult Period Map\'\' for Cubic Threefolds
		2.3 Moduli Spaces of (ρ,T)-Polarized IHS Fourfolds
		2.4 The Isomorphism
	3 Degenerations
		3.1 The Nodal Locus
		3.2 The Chordal Locus
	4 Further Developments and Open Questions
		4.1 Pfaffian Cyclic Cubic Fourfolds
		4.2 Higher Dimensional IHS Manifolds
		4.3 Another Case
	References
A Note on Severi Varieties of Nodal Curves on Enriques Surfaces
	1 Introduction
	2 Regularity of Severi Varieties on Enriques Surfaces
	References
A Travel Guide to the Canonical Bundle Formula
	1 Introduction
	2 Lc-Trivial Fibrations
		2.1 Definition and First Examples
		2.2 Base Change Property
		2.3 Inversion of Adjunction
		2.4 Coefficients of the Moduli Divisor
		2.5 Goodness of Moduli Divisors
	3 B-Semiampleness Conjectures
		3.1 Restrictions to Divisors
		3.2 Reduction Result
		3.3 Generalisation
	4 Parabolic Fibrations
	References
Some Examples of Calabi–Yau Pairs with Maximal Intersection and No Toric Model
	1 Introduction and Motivation
	2 Auxiliary Results on Threefold CY Pairs
		2.1 Birational Rigidity
		2.2 Singularities of the Boundary
	3 Examples of Rigid Maximal Intersection Threefold CY Pairs
		3.1 Examples with Normal Boundary
		3.2 Examples with Non-normal Boundary
	4 Further Results on Quartic Threefold CY Pairs: Beyond Maximal Intersection
	References
On Deformations of Diagrams of Commutative Algebras
	1 Introduction
		1.1 Notation and Setup
	2 Commutative Differential Graded Algebras
	3 A Very Short Introduction to Model Structures
		3.1 Pre-model Structures
	4 Model Structure on DG-Algebras
	5 Modules and Derivations
	6 Deformations of Diagrams via Projective Cofibrant Resolutions
	7 Simplex Categories
	8 Reedy Model Structures
	References
The Lefschetz Principle in Birational Geometry: Birational Twin Varieties
	1 Introduction
	2 Lefschetz Divisorially Equivalent Varieties and Birational Twins
	3 Birational Twin Varieties: Complete Collineations and Quadrics
		3.1 Birational Geometry of the Intermediate Spaces
			3.1.1 Generators of Cox(X(n,m)i)
		3.2 The Mori Chamber Decomposition of the Intermediate Space of Picard Rank Three
	4 The Flip of the Spaces of Complete Quadric Surfaces
	References
What is the Monodromy Property for Degenerations of Calabi-Yau Varieties?
	1 Introduction
	2 Models for Calabi-Yau Varieties over K
	3 Monodromy Property for Calabi-Yau Varieties
		3.1 The Monodromy Action
		3.2 The Motivic Zeta Function
	4 Some Well-Understood Cases
	5 Triple-Points-Free Models of K3 and Why they are Interesting
	6 What\'s Next
	References
Examples of Irreducible Symplectic Varieties
	1 Introduction
	2 Irreducible Symplectic Varieties
		2.1 Orbifolds
			2.1.1 The Decomposition Theorem for Orbifolds
			2.1.2 Related Notions and Examples
		2.2 Varieties with Canonical Singularities
			2.2.1 The Decomposition for Singular Projective Varieties
			2.2.2 Relation Between the Previous Notions
			2.2.3 Related Notions
			2.2.4 Examples
	References
An Example of Mirror Symmetry for Fano Threefolds
	1 Introduction
		1.1 Aim
		1.2 The Example
		1.3 The General Picture
		1.4 Notation and Conventions
	2 The Geometry of X
		2.1 Toric Geometry
		2.2 Equations
		2.3 Minkowski Sums and Deformations of U
		2.4 The Two Smoothings of X
	3 Mirror Symmetry
		3.1 Gromov–Witten Invariants and Quantum Periods
		3.2 Laurent Polynomials
		3.3 Classical Periods
		3.4 Fano/Landau–Ginzburg Correspondence
	References
Chern Numbers of Uniruled Threefolds
	1 Introduction
		1.1 Conventions
	2 Mori Fibre Spaces Over Surfaces
	3 Uniruled Threefolds
		3.1 Invariant Triples
		3.2 Terminal Singularities
		3.3 Proof of Theorem 3
	References