**Proceedings of Symposia in Pure Mathematics**

Volume: 95;
2017;
370 pp;
Hardcover

MSC: Primary 14; 13; 19; 37;

Print ISBN: 978-1-4704-3557-8

Product Code: PSPUM/95

List Price: $126.00

Individual Member Price: $100.80

**Electronic ISBN: 978-1-4704-4121-0
Product Code: PSPUM/95.E**

List Price: $126.00

Individual Member Price: $100.80

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# Surveys on Recent Developments in Algebraic Geometry

Share this page *Edited by *
*Izzet Coskun; Tommaso de Fernex; Angela Gibney*

The algebraic geometry community has a tradition of running a
summer research institute every ten years. During these influential
meetings a large number of mathematicians from around the world
convene to overview the developments of the past decade and to outline
the most fundamental and far-reaching problems for the next. The
meeting is preceded by a Bootcamp aimed at graduate students and young
researchers. This volume collects ten surveys that grew out of the
Bootcamp, held July 6–10, 2015, at University of Utah, Salt Lake City,
Utah.

These papers give succinct and thorough introductions to some of
the most important and exciting developments in algebraic geometry in
the last decade. Included are descriptions of the striking advances in
the Minimal Model Program, moduli spaces, derived categories,
Bridgeland stability, motivic homotopy theory, methods in
characteristic \(p\) and Hodge theory. Surveys contain many examples,
exercises and open problems, which will make this volume an invaluable
and enduring resource for researchers looking for new directions.

#### Readership

Graduate students and researchers interested in new directions in algebraic geometry.

# Table of Contents

## Surveys on Recent Developments in Algebraic Geometry

- Cover Cover11
- Title page iii4
- Contents v6
- Preface vii8
- A snapshot of the Minimal Model Program 112
- Positive characteristic algebraic geometry 3344
- 1. Introduction 3344
- 2. Frobenius Splittings 3445
- 3. Trace of Frobenius and Global Sections 3950
- 4. 𝐹-singularities versus Singularities of the MMP 4354
- 5. Global Applications 5465
- 6. Seshadri constants, 𝐹-pure centers and test ideals 6273
- 7. Numerical Invariants 6879
- 8. More on test ideals and 𝐹-Singularities in Families 7081
- References 7586

- The geometry of the moduli space of curves and abelian varieties 8192
- 1. The main players 8192
- 2. Deligne–Mumford stable curves 8394
- 3. Tautological classes on the moduli space of curves 8495
- 4. Birational geometry of \Mm𝑔𝑛 8596
- 5. Construction of \ab𝑔 8899
- 6. The Hodge classes and the Baily–Borel–Satake compactification 90101
- 7. Kodaira dimension of \ab𝑔 and Mumford’s partial compactification 90101
- 8. Toroidal compactifications of \ab𝑔 91102
- 9. Torelli and Prym map 94105
- 10. Curve models of abelian varieties of dimension 𝑔≤5 96107
- 11. Curve models of abelian 6-folds 96107
- References 97108

- Birational geometry of moduli spaces of sheaves and Bridgeland stability 101112
- Gromov–Witten theory: From curve counts to string theory 149160
- Teichmüller dynamics in the eyes of an algebraic geometer 171182
- Cycles, derived categories, and rationality 199210
- 1. Preliminaries on Chow groups 200211
- 2. Preliminaries on semiorthogonal decompositions 204215
- 3. Unramified cohomology and decomposition of the diagonal 211222
- 4. Cubic threefolds and special cubic fourfolds 219230
- 5. Rationality and 0-cycles 228239
- 6. Categorical representability and rationality, the case of surfaces 233244
- 7. 0-cycles on cubics 241252
- 8. Categorical representability in higher dimension 247258
- References 258269

- Degenerations of Hodge structure 267278
- Questions about Boij–Söderberg theory 285296
- 1. Background on Boij–Söderberg Theory 285296
- 2. Categorification 291302
- 3. \BS theory and the tails of infinite resolutions 292303
- 4. Exact sequences 293304
- 5. Boij–Söderberg theory over a DVR 295306
- 6. Non-commutative analogues 296307
- 7. Connection with Stillman’s Conjecture 297308
- 8. Extremal rays 299310
- 9. More topics 300311
- Acknowledgments 301312
- References 301312

- A primer for unstable motivic homotopy theory 305316
- 1. Introduction 305316
- 2. Classification of topological vector bundles 308319
- 3. The construction of the Å¹-homotopy category 319330
- 4. Basic properties of Å¹-algebraic topology 331342
- 5. Classifying spaces in Å¹-homotopy theory 348359
- 6. Representing algebraic 𝐾-theory 353364
- 7. Purity 356367
- 8. Vista: classification of vector bundles 360371
- 9. Further directions 366377
- References 366377

- Back Cover Back Cover1386