ZAG Handbook of Algebraic Geometry

Ivan Cheltsov, Jesus Martinez-Garcia

• Publisher: Chapman and Hall/CRC
• Year: 2025
• eBook ISBN: 9781040378922
• Topic: Mathematics
• Subtopic: Algebra
• Index: Mathematics

The ZAG Handbook of Algebraic Geometry provides an extensive collection of extended summaries of all the research talks given at the worldwide ZAG (Zoom Algebraic Geometry) Seminar, as well as contributed short notes. Featuring contributions from all continents, from some of the most respected voices in the field, the book is aimed at researchers in algebraic geometry and related fields who want to gain a comprehensive understanding of state-of-the-art research in algebraic geometry. The book is broad and wide-ranging, offering material suitable to multiple levels of readership: from advanced undergraduate students in mathematics to active specialized researchers.

Features

· A showcase of the most important worldwide trends in algebraic geometry in recent years

· Over two hundred short summaries of state-of-the-art research in algebraic geometry accessible to other mathematicians

· A combination of contributions by renowned mathematicians, key researchers and new talent

· Links to recordings of hour-long presentations for each contribution

Ivan Cheltsov is a Professor at the University of Edinburgh, United Kingdom, and his research interests include birational geometry and K-stability of Fano varieties. He is the author of 3 books and 126 articles written together with 52 co-authors. Ivan has organized research meetings in Auckland, Beijing, Cambridge, Durham, Edinburgh, Fuerteventura, Istanbul, Levico Terme, London, Magadan, Melbourne, Moscow, New York, Oberwolfach, Pipa, Pohang, Easter Island, Shanghai, Sochi, Spitsbergen, Stony Brook, Sydney, Tianjin, Wakkanai, and Vladivostok. Since 1995, Ivan has been married to Elena Cheltsova: they have three children (Ivan, Fedor, Ekaterina) and a cat, Kuzya. Aside from Mathematics, Ivan enjoys traveling, reading, and cycling his Brompton bicycle.

Jesus Martinez-Garcia is a Senior Lecturer in Pure Mathematics at the University of Essex, United Kingdom. He obtained his PhD at the University of Edinburgh in 2013 and has held research positions at the University of Bath, United Kingdom, Johns Hopkins University, United States, and the Max Planck Institute for Mathematics, Germany. His research is in algebraic geometry with special emphasis on birational geometry, moduli spaces, K-stability, Fano varieties, and computational algebraic geometry. Martinez-Garcia has written one book, several research articles, and two software packages, and he has co-organized more than ten research meetings, including the annual international conference AGGITatE, as well as several research seminars.

Table of contents

• Cover Page
• Half-Title Page
• Title Page
• Copyright Page
• Contents
• Chapter 1 ▪ Algebraic Geometry in the Times of Covid
• Chapter 2 ▪ Models of Fano Threefolds
• Chapter 3 ▪ Weighted Ehrhart Polynomials and Series for the Standard Simplex
• Chapter 4 ▪ Advances in Moduli Theory
• Chapter 5 ▪ Normal Form of a Holomorphic Lagrangian Submanifold
• Chapter 6 ▪ Koszul Modules and Applications
• Chapter 7 ▪ Euler Characteristics of Aspherical Kähler Manifolds
• Chapter 8 ▪ Birational Geometry of Calabi-Yau Pairs and Cremona Transformations
• Chapter 9 ▪ Enumerating Punctured Log Maps via Wall-Crossing
• Chapter 10 ▪ Biregular and Birational Geometry of Rational Nodal Quartic Double Solids
• Chapter 11 ▪ Chern Character of Quantizable Sheaves
• Chapter 12 ▪ Vector Bundles on Fano Threefolds and K3 Surfaces
• Chapter 13 ▪ Cylinders in Del Pezzo Surfaces with Du Val Singularities
• Chapter 14 ▪ Approximation of Differentiable Submanifolds by Real Algebraic Subvarieties
• Chapter 15 ▪ Kähler-Einstein Metrics, Archimedean Zeta Functions and Phase Transitions
• Chapter 16 ▪ Geometry of Three-Dimensional Del Pezzo Fibrations in Positive Characteristic
• Chapter 17 ▪ Notes on Geometry of Polarised Varieties
• Chapter 18 ▪ On Properness of K-Moduli Spaces and Optimal Destabilizations
• Chapter 19 ▪ The Brasselet-Schürmann-Yokura conjecture on L-classes of singular varieties
• Chapter 20 ▪ A Surface with Ample Canonical Bundle and Kuranishi Space a Nonreduced Point
• Chapter 21 ▪ Projective Geometry Approach to the Jacobian Conjecture
• Chapter 22 ▪ Explicit Equations of Surfaces of General Type
• Chapter 23 ▪ General Type Results for Moduli of Hyperkähler Varieties
• Chapter 24 ▪ Geometric Structures on Spaces of Quadratic Differentials
• Chapter 25 ▪ Extending Tom and Jerry for Fano Threefolds
• Chapter 26 ▪ Supercycles and Stable Supermaps
• Chapter 27 ▪ Arcs and Singularities
• Chapter 28 ▪ Chern–Weil Theory and Hilbert–Samuel Theorem for Semipositive Singular Toroidal Metrics
• Chapter 29 ▪ Index 2 Fano Threefolds and Double Covers
• Chapter 30 ▪ Uniqueness of Enhancements of Triangulated Categories
• Chapter 31 ▪ On the Birational Geometry of Foliations
• Chapter 32 ▪ Effective Cones of Moduli Spaces of Stable Rational Curves and Blown Up Toric Surfaces
• Chapter 33 ▪ Nodal Surfaces, Coding Theory, and Cubic Discriminants
• Chapter 34 ▪ The Minimal Model Program for Arithmetic Surfaces Enriched by a Brauer Class
• Chapter 35 ▪ Low-Dimensional Components in the K-Moduli of Smoothable Fano Threefolds
• Chapter 36 ▪ Remarks on n-Folds of Type (1,n)
• Chapter 37 ▪ On the Distribution of Canonical Volumes and Moduli Spaces of Varieties of General Type
• Chapter 38 ▪ Singularities on Toric Fibrations
• Chapter 39 ▪ Subadditivity Theorem for Okounkov Bodies
• Chapter 40 ▪ Enumeration of Terracini Schemes
• Chapter 41 ▪ Numerical Characterization of Complex Torus Quotients
• Chapter 42 ▪ Slope Inequalities and Ample Cone of KSB Moduli Spaces
• Chapter 43 ▪ The Cohomology of the Moduli Space of Sheaves on Surfaces
• Chapter 44 ▪ Orbifold Quot Schemes Via the Le Bruyn–Procesi Theorem
• Chapter 45 ▪Deformations of Exterior Differential Ideals and Applications
• Chapter 46 ▪ Log Minimal Model Program for Kähler Threefolds
• Chapter 47 ▪ cscK Metrics on Rank One Spherical Fano Fourfolds
• Chapter 48 ▪ K-stability of P3 Blown Up Along the Disjoint Union of a Twisted Cubic Curve and a Line
• Chapter 49 ▪ Stability of Fibrations
• Chapter 50 ▪ Wall-Crossing for K-Moduli Spaces of Plane Curves
• Chapter 51 ▪ Dynamical Filtrations
• Chapter 52 ▪ The Bottleneck Degree of Algebraic Varieties
• Chapter 53 ▪ Roth's Theorem for Adelic Curves
• Chapter 54 ▪ Enumerative Geometry, Fredholm Analysis and Moduli Spaces of Surfaces of General Type
• Chapter 55 ▪ Bounding the Complexity of Two-Loop Feynman Integrals
• Chapter 56 ▪ K-Stability of P3 Blown Up Along Smooth Curves of Genus 4 and Degree 6
• Chapter 57 ▪ Birational Geometry and Cylindricity of Severi-Brauer Varieties
• Chapter 58 ▪ Rigidity of Affine Brieskorn-Pham Threefolds
• Chapter 59 ▪ Okawa's Theorem and Wide Subcategories of Coherent Sheaves on Curves
• Chapter 60 ▪ Old Mixed Hodge Structure Decomposition of Alexander Modules into Generalized Eigenspaces
• Chapter 61 ▪ Local Stability Threshold of del Pezzo Surfaces of Degree 2
• Chapter 62 ▪Rational Simple Connectedness and the Quintic del Pezzo Threefold
• Chapter 63 ▪ K3 Structures from Singular Fano Varieties
• Chapter 64 ▪ An Overview of Homotopy Path Algebras
• Chapter 65 ▪ Quasi-invariants and Free Arrangements
• Chapter 66 ▪ Notes on Motivic Integration on Berkovich Spaces
• Chapter 67 ▪ Counting Sheaves by Counting Curves
• Chapter 68 ▪ On the boundedness of elliptically fibred Calabi–Yau threefolds
• Chapter 69 ▪ Connected Algebraic Groups Acting on Fano Fibrations Over P1
• Chapter 70 ▪ Maximal Automorphism Groups of Surfaces
• Chapter 71 ▪ A Canonical Hodge Theoretic Projective Structure on Compact Riemann Surfaces
• Chapter 72 ▪ On Extremal Contractions of Log Canonical Pairs
• Chapter 73 ▪ On the Singular Loci of Higher Secant Varieties of Veronese Embeddings
• Chapter 74 ▪ The Bourbaki Degree of a Plane Curve
• Chapter 75 ▪ Compactifications of the Moduli Space of Marked Cubic Surfaces
• Chapter 76 ▪ Intrinsic Mirrors for Minimal Adjoint Orbits (ICM G&T)
• Chapter 77 ▪ On a Generalized Batyrev's Cone Conjecture
• Chapter 78 ▪ Complex Curves in Hypercomplex Nilmanifolds
• Chapter 79 ▪ Projective Flatness Over Klt Spaces and Characterisation of Finite Quotients of Projective Spaces
• Chapter 80 ▪ Distinguishing Triangle-Free Level-One Phylogenetic Networks using Computational Algebraic Geometry
• Chapter 81 ▪ Open FJRW Theory and Mirror Symmetry: A ZAG Lecture
• Chapter 82 ▪ Recent Progress in the MMP for Threefolds and Fourfolds in Char p>0
• Chapter 83 ▪ Rationality Questions on Seshadri Constants
• Chapter 84 ▪ Log Symplectic Pairs and Mixed Hodge Structures
• Chapter 85 ▪ Gaps Between lc Pairs and Klt Pairs in a Viewpoint of the Minimal Model Theory
• Chapter 86 ▪ Constructing New Q-Fano Threefolds Using Laurent Inversion
• Chapter 87 ▪ Fano Manifolds Such that the Tangent Bundle is (not) Big
• Chapter 88 ▪ Geometric Extension
• Chapter 89 ▪ An Algebro-geometric Higher Szemeredi Lemma
• Chapter 90 ▪ A Dynamical Approach to Generalized Weil's Riemann Hypothesis and Semisimplicity
• Chapter 91 ▪ Torelli Problem on Logarithmic Sheaves
• Chapter 92 ▪ Minimal Rational Curves and 1-Flat Cone Structures
• Chapter 93 ▪ A Bound of the Number of Weighted Blow-ups to Compute the Minimal Log Discrepancy for Smooth Threefolds
• Chapter 94 ▪ The McKay Correspondence
• Chapter 95 ▪ Shafarevich's Conjecture for Canonically Polarized Varieties Revisited
• Chapter 96 ▪ Symmetries of Fano Varieties
• Chapter 97 ▪ Minimal Log Discrepancies in Dimension Three
• Chapter 98 ▪ Equilateral Convex Triangulations of RP2 with Three Conical Points of All Possible Defects
• Chapter 99 ▪ One-Dimensional K-Moduli Spaces of Fano Threefolds
• Chapter 100 ▪ A Brief Survey on Stability of Tangent Bundles on Fano Manifolds
• Chapter 101 ▪ Are Klt Fano Varieties of Small Volume Stably Rational?
• Chapter 102 ▪ New Examples of Surgery Invariant Counts in Real Algebraic Geometry
• Chapter 103 ▪ Hodge Sheaves for Singular Families
• Chapter 104 ▪ Secant Varieties of Real Curves
• Chapter 105 ▪ Sextic Double Solids
• Chapter 106 ▪ On Effective Cones of Rational Surfaces
• Chapter 107 ▪ The Rational Chow Rings of M7, M8 and M9
• Chapter 108 ▪ A Néron–Ogg–Shafarevich Criterion for K3 Surfaces
• Chapter 109 ▪ Derived Categories and Motives of Moduli Spaces of Vector Bundles on Curves
• Chapter 110 ▪ Dominant Rational Maps from a Very General Hypersurface
• Chapter 111 ▪ Valuative Stability of Polarised Varieties ZAG Online Seminar, June 15, 2021
• Chapter 112 ▪ Rational Curves on Del Pezzo Surfaces in Positive Characteristic
• Chapter 113 ▪ Codimension Two Cycles of Classifying Spaces of Low-Dimensional Algebraic Tori
• Chapter 114 ▪ Locally Free Twisted Sheaves of Infinite Rank
• Chapter 115 ▪ Rational Curves on K3 Surfaces
• Chapter 116 ▪ Report on the Talk: Automorphisms of Projective Hypersurfaces
• Chapter 117 ▪ Motivic Invariants of Birational Maps Zoom Algebraic Geometry Talk on May 26, 2022
• Chapter 118 ▪ A Motivic Version of Topological Mirror Symmetry
• Chapter 119 ▪ Poisson and Symplectic Geometry of the Moduli Spaces of Higgs Bundles
• Chapter 120 ▪ On Non-Rationality of Degenerations of Del Pezzo Surfaces
• Chapter 121 ▪ A Conjecture on D Algebras
• Chapter 122 ▪ Fano Threefolds of Picard Rank 2 and Volume 28
• Chapter 123 ▪ Birational Involutions of the Real Projective Plane Fixing an Irrational Curve
• Chapter 124 ▪ Unirationality of Instanton Moduli Space for Small Charges
• Chapter 125 ▪ The Defect of a Cubic Threefold
• Chapter 126 ▪ The Determinant of Cohomology and Moduli of λ-Connections
• Chapter 127 ▪ Two Remarks on Asymptotically Log Fano Pairs
• Chapter 128 ▪ On K-stability of Calabi-Yau Fibrations
• Chapter 129 ▪ On the (Uni)rationality Problem for Hypersurfaces
• Chapter 130 ▪ Transformations of Transfinite Diameter
• Chapter 131 ▪ Perverse-Hodge Octahedron
• Chapter 132 ▪ A Special Rational Surface
• Chapter 133 ▪ On the Top-Weight Rational Cohomology of Ag
• Chapter 134 ▪ Jordan Property for Automorphism Groups
• Chapter 135 ▪ Geometry and Arithmetic of Equivariant Compactifications of the Affine Space
• Chapter 136 ▪ Special Variational Hodge Conjecture for Toric Varieties
• Chapter 137 ▪ Tropical Geometry and Phylogenetic Diversity
• Chapter 138 ▪ ZAG Report: Fundamental Groups of Singularities of the MMP
• Chapter 139 ▪ A Moduli Space in the Differential Geometry World
• Chapter 140 ▪ Interview with David Mumford
• Chapter 141 ▪ The Kawamata–Viehweg Vanishing Theorem for Schemes
• Chapter 142 ▪ Minimal Exponent and Hodge Filtrations
• Chapter 143 ▪ Inversion of Adjunction for Quotient Singularities
• Chapter 144 ▪ On Compatifying Moduli and Degenerations of K-Trivial Varieties
• Chapter 145 ▪ K-Stability and Birational Superrigidity for Fano Threefold Weighted Hypersurfaces
• Chapter 146 ▪ Moduli Space of Semiorthogonal Decompositions
• Chapter 147 ▪ Stable Irrationality of the Very General Quartic Fivefold
• Chapter 148 ▪ Contact in Algebraic and Tropical Geometry
• Chapter 149 ▪ Extended Abstract of Counting Divisorial Contractions with Centre a cAn-Singularity
• Chapter 150 ▪ Fine Compactified Jacobians of Nodal Curves
• Chapter 151 ▪ On Polynomial Automorphisms Commuting with a Simple Derivation
• Chapter 152 ▪ K-Moduli for log Fano Complete Intersections
• Chapter 153 ▪ Cayley Octads, Plane Quartic Curves, del Pezzo Surfaces of Degree 2, and Double Veronese Cones
• Chapter 154 ▪ A Castelnuovo–Mumford Regularity Bound for Threefolds with Rational Singularities
• Chapter 155 ▪ Moderately Discontinuous Algebraic Topology and Singularities
• Chapter 156 ▪ Generic Flexibility of Cubic Cones
• Chapter 157 ▪ Serre Functors of Semiorthogonal Components
• Chapter 158 ▪ Deformations of Toric Singularities and Applications to K-Moduli of Fano Varieties
• Chapter 159 ▪ Campana Points on Fano Varieties
• Chapter 160 ▪ On Canonical Threefolds Near the Noether Line
• Chapter 161 ▪ K-Stability and Space Sextic Curves of Genus Three
• Chapter 162 ▪ Alena Pirutka, Courant Institute, New York University
• Chapter 163 ▪ Conic-Line Arrangements in the Complex Projective Plane
• Chapter 164 ▪ Hyperelliptic Limits of Quadrics Through Canonical Curves and the Super-Schottky Locus
• Chapter 165 ▪ General Elephants for Threefold Extremal Contractions
• Chapter 166 ▪ Rationally Connected Rational Double Covers of Primitive Fano Varieties
• Chapter 167 ▪ Singularities and Divisors in the Moduli Space of Surfaces
• Chapter 168 ▪ Sheaf Counting of Hilbert Schemes of Points on C4
• Chapter 169 ▪ Wall-Crossing Pathologies in Three Dimensions
• Chapter 170 ▪ K-Polystability of Smooth Fano SL2-Threefolds
• Chapter 171 ▪ Hodge-Riemann Classes and Schur Polynomials
• Chapter 172 ▪ The Splendour of Asymptotically log del Pezzos
• Chapter 173 ▪ Smoothing Toroidal Crossing Fano Threefolds
• Chapter 174 ▪ An Overview on Mordell–Weil Rank Jumps for Fibres of Elliptic Surfaces
• Chapter 175 ▪ Blow-ups with Log Canonical Singularities
• Chapter 176 ▪ On Birational Boundedness of Some Calabi—Yau Hypersurfaces
• Chapter 177 ▪ Equivariant Solidity of The Three-dimensional Projective Space
• Chapter 178 ▪ Near-rationality Properties of Norm Varieties
• Chapter 179 ▪ Generalized Hyperpolygons and Applications
• Chapter 180 ▪ Equality in the Bogomolov–Miyaoka–Yau Inequality in the Non-general Type Case
• Chapter 181 ▪ Rational Curves on Enriques Surfaces, But Only Few
• Chapter 182 ▪ Cohomology of the Moduli of Higgs Bundles and the Hausel–Thaddeus Conjecture
• Chapter 183 ▪ Survey of the Article on Atiyah Class and Sheaf Counting on Local Calabi Yau Fourfolds
• Chapter 184 ▪ Automorphism Groups of Complex Elliptic Surfaces
• Chapter 185 ▪ LeBrun-Salamon Conjecture from the Effective Nonvanishing for Fano Manifolds
• Chapter 186 ▪ Partial Order on Involutive Permutations and Double Schubert Cells
• Chapter 187 ▪ Geometric Aspects of Kähler-Einstein Metrics on Klt Pairs: A Remark on Optimal Bogomolov-Miyaoka-Yau Inequality
• Chapter 188 ▪ Weak del Pezzo Surfaces with Global Vector Fields
• Chapter 189 ▪ On the Normalised Volume of Toric Singularities
• Chapter 190 ▪ An O-Acyclic Variety of Even Index
• Chapter 191 ▪ Higher-Order Minimal Families of Rational Curves on Fano Manifolds
• Chapter 192 ▪ The Mukai Model of M―7
• Chapter 193 ▪ Local Effectivity in Projective Spaces
• Chapter 194 ▪ Fermat-Type Arrangements and the Containment Problem
• Chapter 195 ▪ Birational Properties of Base Spaces of Smooth Projective Families of Good Minimal Models
• Chapter 196 ▪ Key Varieties for Prime Q-Fano Threefolds of Codimension Four
• Chapter 197 ▪ Deformations of F-Pure and F-Regular Singularities
• Chapter 198 ▪ On Mori Fibre Spaces in Positive Characteristic
• Chapter 199 ▪ Rational Curves on Fano Threefolds
• Chapter 200 ▪ ZAG Report: Delta Invariants of Quasi-Smooth Hypersurfaces
• Chapter 201 ▪ Real Structures on Almost Homogeneous Varieties
• Chapter 202 ▪ Mirror Symmetry for Fibrations and Degenerations
• Chapter 203 ▪ On a Logarithmic version of a Theorem of Enriques
• Chapter 204 ▪ Equivariant Birational Types
• Chapter 205 ▪ Vector Fields on Canonically Polarized Surfaces
• Chapter 206 ▪ Noncommutative del Pezzo Surfaces
• Chapter 207 ▪ Exceptional Collections on Σ2
• Chapter 208 ▪ Counts of Secant Planes to Varieties, Degenerations, and Universal Polynomials
• Chapter 209 ▪ Actions of Cremona Groups on CAT(0) Cube Complexes
• Chapter 210 ▪ Artin-Mumford Counterexample, with Generalizations, via Enriques Surfaces
• Chapter 211 ▪ On Quadratic Points on Intersections of Two Quadrics
• Chapter 212 ▪ Higher Fano Manifolds
• Chapter 213 ▪ Global Brill–Noether Theory over the Hurwitz Space
• Chapter 214 ▪ Positivity of the Second Exterior Power of the Tangent Bundles
• Chapter 215 ▪ Pinched Handle Decompositions of Finite Simplicial Complexes
• Chapter 216 ▪ Concurrent Exceptional Curves on del Pezzo Surfaces of Degree One and Torsion Points on Elliptic Fibrations
• Chapter 217 ▪ Rational Maps via C * Actions
• Chapter 218 ▪ On Weighted del Pezzo Hypersurfaces
• Chapter 219 ▪ Equivariant Birational Geometry of Linear Actions
• Chapter 220 ▪ Equivariant Birational Rigidity: Around One Question of Cheltsov and Kollár
• Chapter 221 ▪ Stringy Motives and Local Fundamental Groups of KLT Surface Singularities in Arbitrary Characteristic
• Chapter 222 ▪ Degenerations, Fibrations and Mirror Symmetry
• Chapter 223 ▪ Stability of Pencils of Plane Curves
• Chapter 224 ▪ Jordan Properties of Automorphism Groups of Algebraic Varieties and Complex Manifolds
• Chapter 225 ▪ Basis Divisors and Balanced Metrics
• Chapter 226 ▪ The Moduli Space of Cubic Surface Pairs via the Intermediate Jacobians of Eckardt Cubic Threefolds
• Chapter 227 ▪ Equivariant Geometry of Singular Cubic Threefolds
• Chapter 228 ▪ Topological SYZ Fibrations with Discriminant in Codimension 2
• Chapter 229 ▪ Compact Kähler Threefolds with the Action of an Abelian Group of Maximal Dynamical Rank
• Chapter 230 ▪ Equivariant K-stability Under Finite Group Action
• Chapter 231 ▪ K-Stability of Fano Varieties via Admissible Flags
• Chapter 232 ▪ Unbounded Connected Algebraic Subgroups of Birational Transformations
• Chapter 233 ▪ Finite Quotients of Cremona Groups