Surveys on Recent Developments in Algebraic Geometry

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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. 𝐹-singularities versus Singularities of the MMP 5. Global Applications 6. Seshadri constants, 𝐹-pure centers and test ideals 7. Numerical Invariants 8. More on test ideals and 𝐹-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 \Mm𝑔𝑛 5. Construction of \ab𝑔 6. The Hodge classes and the Baily–Borel–Satake compactification 7. Kodaira dimension of \ab𝑔 and Mumford’s partial compactification 8. Toroidal compactifications of \ab𝑔 9. Torelli and Prym map 10. Curve models of abelian varieties of dimension 𝑔≤5 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 5. Meromorphic and higher order differentials References Cycles, derived categories, and rationality 1. Preliminaries on Chow groups 2. Preliminaries on semiorthogonal decompositions 3. Unramified cohomology and decomposition of the diagonal 4. Cubic threefolds and special cubic fourfolds 5. Rationality and 0-cycles 6. Categorical representability and rationality, the case of surfaces 7. 0-cycles on cubics 8. Categorical representability in higher dimension References Degenerations of Hodge structure 1. Introduction 2. Hodge structures and their generalizations 3. Nilpotent orbits 4. Classifications References Questions about Boij–Söderberg theory 1. Background on Boij–Söderberg Theory 2. Categorification 3. \BS theory and the tails of infinite resolutions 4. Exact sequences 5. Boij–Söderberg theory over a DVR 6. Non-commutative analogues 7. Connection with Stillman’s Conjecture 8. Extremal rays 9. More topics Acknowledgments References A primer for unstable motivic homotopy theory 1. Introduction 2. Classification of topological vector bundles 3. The construction of the Ź-homotopy category 4. Basic properties of Ź-algebraic topology 5. Classifying spaces in Ź-homotopy theory 6. Representing algebraic 𝐾-theory 7. Purity 8. Vista: classification of vector bundles 9. Further directions References Back Cover