نظرسنجی‌ها دربارهٔ تحولات اخیر در هندسه جبری: دوره آموزشی برای مؤسسه تحقیقاتی تابستانی ۲۰۱۵ در هندسه جبری، ۶-۱۰ ژوئیه ۲۰۱۵، دانشگاه یوتا، سالت‌لیک‌سیتی، یوتا

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. Cover 1 Title page 4 Contents 6 Preface 8 A snapshot of the Minimal Model Program 12 1. Introduction 12 2. Main idea of the MMP 13 3. Background 20 4. Main conjectures 21 5. Unconditional results 23 6. Conditional results 30 7. Additional results 31 References 34 Positive characteristic algebraic geometry 44 1. Introduction 44 2. Frobenius Splittings 45 3. Trace of Frobenius and Global Sections 50 4. F-singularities versus Singularities of the MMP 54 5. Global Applications 65 6. Seshadri constants, F-pure centers and test ideals 73 7. Numerical Invariants 79 8. More on test ideals and F-Singularities in Families 81 References 86 The geometry of the moduli space of curves and abelian varieties 92 1. The main players 92 2. Deligne–Mumford stable curves 94 3. Tautological classes on the moduli space of curves 95 4. Birational geometry of \Mmgn 96 5. Construction of \abg 99 6. The Hodge classes and the Baily–Borel–Satake compactification 101 7. Kodaira dimension of \abg and Mumford’s partial compactification 101 8. Toroidal compactifications of \abg 102 9. Torelli and Prym map 105 10. Curve models of abelian varieties of dimension g≤5 107 11. Curve models of abelian 6-folds 107 References 108 Birational geometry of moduli spaces of sheaves and Bridgeland stability 112 1. Introduction 112 2. Moduli spaces of sheaves 114 3. Properties of moduli spaces 121 4. Divisors and classical birational geometry 128 5. Bridgeland stability 139 6. Examples on \P2 147 7. The positivity lemma and nef cones 151 References 157 Gromov–Witten theory: From curve counts to string theory 160 1. Basic definitions 161 2. Computing Gromov–Witten invariants 166 3. A tour of applications and open questions 174 References 179 Teichmüller dynamics in the eyes of an algebraic geometer 182 1. Introduction 182 2. Preliminaries 183 3. Teichmüller curves 187 4. Affine invariant submanifolds 197 5. Meromorphic and higher order differentials 201 References 204 Cycles, derived categories, and rationality 210 1. Preliminaries on Chow groups 211 2. Preliminaries on semiorthogonal decompositions 215 3. Unramified cohomology and decomposition of the diagonal 222 4. Cubic threefolds and special cubic fourfolds 230 5. Rationality and 0-cycles 239 6. Categorical representability and rationality, the case of surfaces 244 7. 0-cycles on cubics 252 8. Categorical representability in higher dimension 258 References 269 Degenerations of Hodge structure 278 1. Introduction 278 2. Hodge structures and their generalizations 280 3. Nilpotent orbits 285 4. Classifications 287 References 293 Questions about Boij–Söderberg theory 296 1. Background on Boij–Söderberg Theory 296 2. Categorification 302 3. \BS theory and the tails of infinite resolutions 303 4. Exact sequences 304 5. Boij–Söderberg theory over a DVR 306 6. Non-commutative analogues 307 7. Connection with Stillman’s Conjecture 308 8. Extremal rays 310 9. More topics 311 Acknowledgments 312 References 312 A primer for unstable motivic homotopy theory 316 1. Introduction 316 2. Classification of topological vector bundles 319 3. The construction of the Å1-homotopy category 330 4. Basic properties of Å1-algebraic topology 342 5. Classifying spaces in Å1-homotopy theory 359 6. Representing algebraic K-theory 364 7. Purity 367 8. Vista: classification of vector bundles 371 9. Further directions 377 References 377 Back Cover 386 Cover -- Title page -- Contents -- Preface -- A snapshot of the Minimal Model Program -- 1. Introduction -- 2. Main idea of the MMP -- 3. Background -- 4. Main conjectures -- 5. Unconditional results -- 6. Conditional results -- 7. Additional results -- References -- Positive characteristic algebraic geometry -- 1. Introduction -- 2. Frobenius Splittings -- 3. Trace of Frobenius and Global Sections -- 4.-singularities versus Singularities of the MMP -- 5. Global Applications -- 6. Seshadri constants, -pure centers and test ideals -- 7. Numerical Invariants -- 8. More on test ideals and -Singularities in Families -- References -- The geometry of the moduli space of curves and abelian varieties -- 1. The main players -- 2. Deligne-Mumford stable curves -- 3. Tautological classes on the moduli space of curves -- 4. Birational geometry of \Mm -- 5. Construction of \ab -- 6. The Hodge classes and the Baily-Borel-Satake compactification -- 7. Kodaira dimension of \ab and Mumford's partial compactification -- 8. Toroidal compactifications of \ab -- 9. Torelli and Prym map -- 10. Curve models of abelian varieties of dimension d"--11. Curve models of abelian 6-folds -- References -- Birational geometry of moduli spaces of sheaves and Bridgeland stability -- 1. Introduction -- 2. Moduli spaces of sheaves -- 3. Properties of moduli spaces -- 4. Divisors and classical birational geometry -- 5. Bridgeland stability -- 6. Examples on \P2 -- 7. The positivity lemma and nef cones -- References -- Gromov-Witten theory: From curve counts to string theory -- 1. Basic definitions -- 2. Computing Gromov-Witten invariants -- 3. A tour of applications and open questions -- References -- Teichmüller dynamics in the eyes of an algebraic geometer -- 1. Introduction -- 2. Preliminaries -- 3. Teichmüller curves -- 4. Affine invariant submanifolds