**Proceedings of Symposia in Pure Mathematics**

Volume: 97;
2018;
655 pp;
Hardcover

MSC: Primary 14; 53;

**Print ISBN: 978-1-4704-3577-6
Product Code: PSPUM/97.1**

List Price: $133.00

AMS Member Price: $106.40

MAA Member Price: $119.70

**Electronic ISBN: 978-1-4704-4678-9
Product Code: PSPUM/97.1.E**

List Price: $133.00

AMS Member Price: $106.40

MAA Member Price: $119.70

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# Algebraic Geometry: Salt Lake City 2015

Share this page *Edited by *
*Tommaso de Fernex; Brendan Hassett; Mircea Mustaţă; Martin Olsson; Mihnea Popa; Richard Thomas*

A co-publication of the AMS and Clay Mathematics Institute

This is Part 1 of a two-volume set.

Since Oscar Zariski organized a meeting in 1954, there has
been a major algebraic geometry meeting every decade: Woods Hole
(1964), Arcata (1974), Bowdoin (1985), Santa Cruz (1995), and Seattle
(2005). The American Mathematical Society has supported these summer
institutes for over 50 years. Their proceedings volumes have been
extremely influential, summarizing the state of algebraic geometry at
the time and pointing to future developments.

The most recent Summer Institute in Algebraic Geometry was held
July 2015 at the University of Utah in Salt Lake City, sponsored by
the AMS with the collaboration of the Clay Mathematics Institute. This
volume includes surveys growing out of plenary lectures and seminar
talks during the meeting. Some present a broad overview of their
topics, while others develop a distinctive perspective on an emerging
topic.

Topics span both complex algebraic geometry and arithmetic
questions, specifically, analytic techniques, enumerative geometry,
moduli theory, derived categories, birational geometry, tropical
geometry, Diophantine questions, geometric representation theory,
characteristic \(p\) and \(p\)-adic tools, etc. The
resulting articles will be important references in these areas for
years to come.

#### Readership

Graduate students and researchers working in algebraic geometry and its applications.

# Table of Contents

## Algebraic Geometry: Salt Lake City 2015

Table of Contents pages: 1 2

- Cover Cover11
- Title page iii4
- Contents v6
- Preface vii8
- Scientific program ix10
- Part 1 118
- Wall-crossing implies Brill-Noether: Applications of stability conditions on surfaces 320
- 1. Introduction 320
- 2. The heart of the matter 522
- 3. Geometric stability 825
- 4. Moduli spaces of stable objects 1027
- 5. Brill-Noether and the moduli space of torsion sheaves 1128
- 6. Hitting the wall 1229
- 7. Conclusion 1633
- 8. Geometry of the Brill-Noether locus and birational geometry of the moduli space 1835
- 9. Birational geometry of moduli spaces of sheaves: a quick survey 2037
- References 2441

- Kähler–Einstein metrics, canonical random point processes and birational geometry 2946
- 1. Introduction 2946
- 2. Outline of the proof of Theorem 1.1 and relations to statistical mechanics 3552
- 3. Analytic setup 4057
- 4. The LDP for temperature deformed determinantal point processes 4764
- 5. Canonical random point processes on varieties of positive Kodaira dimension and log pairs 5370
- 6. Fano manifolds and Gibbs stability 6279
- 7. Appendix: Probabilistic background 6683
- Acknowledgments 7087
- References 7188

- Hall algebras and Donaldson-Thomas invariants 7592
- The Cremona group 101118
- 1. An introduction based on examples 101118
- 2. Algebraic subgroups of \Cr_{𝑛}(\bfk) 105122
- 3. Generating sets and relations 111128
- 4. An infinite dimensional hyperbolic space 114131
- 5. The Cremona group is thin 121138
- 6. Finitely generated subgroups 126143
- 7. Small cancellation and normal subgroups 128145
- 8. Zimmer program 132149
- 9. Growths 135152
- References 137154

- Mori dream spaces and blow-ups 143160
- 1. Introduction 143160
- 2. Mori dream spaces 145162
- 3. Examples 148165
- 4. Structure theory 150167
- 5. Blow-ups of surfaces of Picard number one 152169
- 6. Blow-ups of weighted projective planes 153170
- 7. Blow-ups of higher dimensional toric varieties 155172
- 8. Blow-ups of toric surfaces 158175
- References 164181

- The space of arcs of an algebraic variety 169186
- Stability of algebraic varieties and Kähler geometry 199216
- Syzygies of projective varieties of large degree: Recent progress and open problems 223240
- Stable gauged maps 243260
- Uniformisation of higher-dimensional minimal varieties 277294
- Part OT1OT1cmrcmrmmnncmrcmrI. Techniques 286303
- Part OT1OT1cmrcmrmmnncmrcmrII. Proof of the main results 297314
- Boundedness of varieties of log general type 309326
- Θ-stratifications, Θ-reductive stacks, and applications 349366
- Bimeromorphic geometry of Kähler threefolds 381398
- Moduli of stable log-varieties—an update 403420
- Enumerative geometry and geometric representation theory 419436
- A calculus for the moduli space of curves 459476
- Frobenius techniques in birational geometry 489506
- Singular Hermitian metrics and positivity of direct images of pluricanonical bundles 519536
- Positivity for Hodge modules and geometric applications 555572
- Notes on homological projective duality 585602

Table of Contents pages: 1 2