Intro
Preface
Contents
Classification of Exceptional Complements: Elliptic Curve Case
1 Introduction
1.1 The Dual Graph of a Configuration
1.2 Types of Singularities on C
2 Elliptic Curve Case
2.1 Reduction to mathbbF2
2.2 The Search for Exceptions
References
Cylinders in Del Pezzo Surfaces of Degree Two
1 Introduction
2 Del Pezzo Surfaces with Singularities of Type A1
3 Smooth Del Pezzo Surfaces
3.1 P is an Intersection Point of Two (-1)-Curves
3.2 P is Not an Intersection Point of Two (-1)-Curves
References

Finiteness of Real Structures on KLT Calabi-Yau Regular Smooth Pairs of Dimension 2
1 Introduction
2 Preliminaries: Geometric Actions on CAT(0) Spaces
3 Finiteness Theorem
4 Two Examples
5 Appendix
References
Anticanonical Volumes of Fano 4-Folds
1 Introduction
2 Proof of Results
References
Constant Scalar Curvature Sasaki Metrics and Projective Bundles
1 The Projective Bundles
2 The Orbifold Boothby-Wang Construction
3 The Topology of the Orbifolds
3.1 The Orbifold Cohomology Groups
3.2 The Cohomology of the Branched Covers
3.3 The Regular Case

3.4 The General Case
Branched Covers
3.5 When M7 is an S3w-Join
4 Admissible Projective Bundles
4.1 Orbifolds
4.2 Connection with S3w-Joins
4.3 Connection with Yamazaki's Fiber Joins
4.4 Ricci Solitons and Kähler-Einstein
4.5 Extremal and CSC Kähler Metrics
4.6 Weighted Extremal Metrics
References
Toric Sarkisov Links of Toric Fano Varieties
1 Introduction
2 The Context
2.1 Overview of the Sarkisov Program
2.2 Hard Fano Varieties from Easy Ones
2.3 The Midpoints of Sarkisov Links
3 Operations on a Simplicial Fan
4 Extending Prokhorov's Web

4.1 Links from a Smooth Point
4.2 Links from the Eighth Toric Mori-Fano 3-Fold
4.3 Links from 1-Dimensional Centres
4.4 Bounding Links from a Smooth Centre
4.5 Higher Rank Fano 3-Folds, Links and Relations
5 The Web of 4-Dimensional Toric Sarkisov Links
5.1 Blowups of mathbbP(1,2,3,4,5)
5.2 Links Between Fake Weighted Projective Spaces
5.3 Blowups of mathbbP4
6 Further Related Problems
6.1 High-Discrepancy Blowups
6.2 Non-terminal Singularities
6.3 Running the Sarkisov Program on Toric n-Folds
6.4 Higher Picard Rank

6.5 Fano 3-Folds and 4-Folds More Generally
References
Du Val Singularities
1 Introduction
2 Finite Subgroups of SL2(mathbbC)
3 Description of Du Val Singularities
4 A1-Singularity
5 E6-Singularity
6 Two-Dimensional McKay Correspondence
References
K-Polystability of Two Smooth Fano Threefolds
1 Introduction
2 Smooth Divisor in mathbbP1timesmathbbP1timesmathbbP2 of Degree (1,1,1)
3 Blow up of a Complete Intersection of Two Quadrics in a Conic
References
A Note on Families of K-Semistable Log-Fano Pairs
1 Introduction