An Invitation to Modern Enumerative Geometry

Description

This book is based on a series of lectures given by the author at SISSA, Trieste, within the PhD coursesTechniques in enumerative geometry (2019)and Localisation in enumerative geometry (2021). The goal of this book is to provide a gentle introduction, aimed mainly at graduate students, to the fast-growing subject of enumerative geometry and, more specifically, counting invariants in algebraic geometry. In addition to the more advanced techniques explained and applied in full detail to concrete calculations, the book contains the proofs of several background results, important for the foundations of the theory. In this respect, this text is conceived for PhD students or research “beginners” in the field of enumerative geometry or related areas. This book can be read as an introduction to Hilbert schemes and Quot schemes on 3-folds but also as an introduction to localisation formulae in enumerative geometry. It is meant to be accessible without a strong background in algebraic geometry; however, three appendices (one on deformation theory, one on intersection theory, one on virtual fundamental classes) are meant to help the reader dive deeper into the main material of the book and to make the text itself as self-contained as possible. Preface Acknowledgements Contents 1 Introduction 2 Counting in Algebraic Geometry 2.1 Asking the Right Question 2.2 Counting the Points on a Moduli Space 2.3 Transversality, and Counting Lines Through Two Points Two More Words on Excess Intersection 2.4 Before and After the Virtual Class 2.5 Warming Up: Counting Weierstrass Points 3 Background Material 3.1 Varieties, Schemes, Morphisms 3.1.1 Schemes and Their Basic Properties 3.1.2 Varieties, Fat Points and More Morphisms 3.1.3 Schemes with Embedded Points 3.2 Sheaves and Supports 3.2.1 Coherent Sheaves, Projective Morphisms 3.2.2 Properties of Sheaves: Torsion Free, Pure, Reflexive, Flat 3.2.3 Supports 3.2.3.1 Another Notion of Support: Fitting Ideals 3.2.4 Derived Category Notation 3.2.5 Dualising Complexes, Cohen–Macaulay and Gorenstein Schemes 3.3 Degeneracy Loci and Chern Classes 3.3.1 The Thom–Porteous Formula 3.4 Critical loci 3.5 Representable Functors 3.6 More Notions of Representability, and GIT Quotients 3.6.1 Fine Moduli Spaces and Automorphisms 3.6.2 More Notions of Moduli Spaces 3.6.3 Quotients in Algebraic Geometry in a Nutshell 4 Informal Introduction to Grassmannians 4.1 The Grassmannian as a Projective Variety 4.2 Schubert Cycles 4.3 The Chow Ring of G(1,3) 4.3.1 Codimension 4.3.2 Codimension 4.3.3 Codimension 4.3.4 A Famous Intersection Number on G(1,3) 4.4 The Leibniz Rule and the Degree of G(1,n+1) 5 Relative Grassmannians, Quot, Hilb 5.1 Relative Grassmannians 5.1.1 The Grassmann Functor and Its Representability 5.2 Quot and Hilbert Schemes 5.2.1 The Quot Functor and Grothendieck's Theorem 5.2.2 The Hilbert Scheme of a Quasiprojective Family 5.2.3 Hilbert Polynomials, Universal Families of Hilbert Schemes 5.3 Tangent Space to Hilb and Quot 5.4 Examples of Hilbert Schemes 5.4.1 Plane Conics 5.4.2 Curves in 3-Space 5.4.2.1 Twisted Cubics 5.4.2.2 A Line and a Point 5.4.2.3 A Plane Conic (and No Point) 5.4.2.4 A More General Example 5.4.3 The Hilbert Scheme of a Jacobian 5.5 Lines on Hypersurfaces: The Fano scheme 6 The Hilbert Scheme of Points 6.1 Subschemes and 0-Cycles 6.1.1 The Hilbert–Chow Morphism 6.1.2 The Punctual Hilbert Scheme 6.2 The Hilbert Scheme of Points on Affine Space 6.2.1 Equations for the Hilbert Scheme of Points 6.2.2 Equations for the Quot Scheme of Points 6.2.3 Quot-to-Chow Revisited 6.2.4 Varieties of Commuting Matrices: What's Known 6.3 The Special Case of Hilbn A2 6.3.1 The Motive of the Hilbert Scheme 6.3.2 Nakajima Quiver Varieties 6.3.3 The Hilbert Scheme as a Nakajima Quiver Variety 6.4 The Special Case of Hilbn A3 6.4.1 Critical Locus Description 6.4.2 A Quiver Description 7 Equivariant Cohomology 7.1 Universal Principal Bundles and Classifying Spaces 7.1.1 Classifying Spaces in Topology 7.1.2 First Examples of Classifying Spaces 7.2 Definition of Equivariant Cohomology 7.2.1 Preview: How to Calculate via Equivariant Cohomology 7.3 Approximation Spaces 7.4 Equivariant Vector Bundles 7.5 Two Computations on Pn-1 7.5.1 Equivariant Cohomology of Pn-1 7.5.2 The Tangent Representation 8 The Atiyah–Bott Localisation Formula 8.1 A Glimpse of the Self-Intersection Formula 8.2 Equivariant Pushforward 8.3 Trivial Torus Actions 8.4 Torus Fixed Loci 8.5 The Localisation Formula 9 Applications of the Localisation Formula 9.1 How Not to Compute the Simplest Intersection Number 9.2 The Lines on a Smooth Cubic Surface 9.3 The 2875 lines on the Quintic 3-Fold 9.4 The Degree of G(1,n+1) and Its Enumerative Meaning 9.5 The Euler Characteristic of the Hilbert Scheme of Points 9.5.1 The Torus Action 9.5.2 Euler Characteristic of Hilbert Schemes 10 The Toy Model for the Virtual Class and Its Localisation 10.1 Obstruction Theories on Vanishing Loci of Sections 10.2 Basic Theory of Equivariant Sheaves 10.2.1 The Category of Quasicoherent Equivariant Sheaves 10.2.2 Forgetful Functor 10.3 Virtual Localisation Formula for the Toy Model 10.3.1 Equivariant Obstruction Theories 10.3.2 Statement of the Virtual Localisation Formula 10.3.3 Proof of the Virtual Localisation Formula in the Case of Vanishing Loci 11 Degree 0 DT Invariants of a Local Calabi–Yau 3-Fold 11.1 Preliminary Tools 11.1.1 The Trace Map of a Perfect Complex 11.1.2 The Huybrechts–Thomas Atiyah Class of a Perfect Complex 11.1.3 Calabi–Yau 3-Folds 11.2 The Perfect Obstruction Theory on 11.2.1 Useful Vanishings 11.2.2 Dimension and Point-Wise Symmetry 11.2.3 Construction of the Obstruction Theory 11.2.4 Symmetry of the Obstruction Theory 11.2.5 Equivariance of the Obstruction Theory 11.3 Virtual Localisation for 11.3.1 K-Theory Notation 11.3.2 Projective Toric 3-Folds 11.3.3 Donaldson–Thomas Invariants via Virtual Localisation 11.4 Evaluating the Virtual Localisation Formula 11.4.1 General Formula for the Degree 0 DT Vertex 11.4.2 Specialisation to Local Calabi–Yau Geometry 11.5 Two Words on Some Refinements 12 DT/PT Correspondence and a Glimpse of Gromov–Witten Theory 12.1 DT/PT for a Projective Calabi–Yau 3-Fold 12.2 DT/PT on the Resolved Conifold 12.2.1 Point Contribution 12.2.2 PT Side 12.2.3 DT Side 12.3 Relation with Multiple Covers in Gromov–Witten Theory 12.3.1 Moduli of Stable Maps 12.3.2 The Problem of Multiple Covers 12.3.3 Relation with Gopakumar–Vafa Invariants 12.3.4 Gromov–Witten/Pairs Correspondence 12.4 An Overview of Gromov–Witten Theory of a Point 12.4.1 Witten's Conjecture 12.4.2 A Descendent Integral on A Deformation Theory A.1 The General Problem A.2 Liftings A.3 Tangent-Obstruction Theories A.3.1 Definitions and Main Examples A.3.2 Applications to Moduli Problems B Intersection Theory B.1 Chow Groups: Pushforward, Pullack, Degree B.2 Operations on Bundles: Formularium B.3 Refined Gysin Homomorphisms B.3.1 An Example: Localised Top Chern Class B.3.2 More Properties of f! and Relation with Bivariant Classes B.3.3 Compatibilities of Refined Gysin Homomorphisms B.3.3.1 Bivariant Classes C Perfect Obstruction Theories and Virtual Classes C.1 Cones C.1.1 Definition of Cones C.1.2 A Short Digression on Gradings and A1-Actions C.1.3 Abelian Cones and Hulls C.2 The Truncated Cotangent Complex C.2.1 The Cotangent Complex for Algebraic Stacks C.3 The Idea Behind Obstruction Theories C.4 The Intrinsic Normal Cone C.4.1 The Absolute Case C.4.2 The Relative Case C.5 Virtual Fundamental Classes C.5.1 The Absolute Case C.5.2 The Relative Case C.5.2.1 Pullback of Obstruction Theories C.5.2.2 Compatibility, Take I C.5.2.3 Compatibility, Take II C.6 The Example of Stable Maps C.6.1 Virtual Dimension C.6.2 The Scheme of Morphisms C.6.3 Obstruction Theory on Moduli of Stable Maps References