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The Geometry of Schemes

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The Geometry of Schemes David Eisenbud Joe Harris Springer . the end of all our exploring Will be to arrive where we started And know the place for the first time. – T. S. Eliot, “Little Gidding” (Four Quartets) Contents Introduction 1 I Basic Definitions 7 I.1 Affine Schemes . 7 I.1.1 Schemes as Sets . 9 I.1.2 Schemes as Topological Spaces . 10 I.1.3 An Interlude on Sheaf Theory . 11 References for the Theory of Sheaves . 18 I.1.4 Schemes as Schemes (Structure Sheaves) . 18 I.2 Schemes in General . 21 I.2.1 Subschemes . 23 I.2.2 The Local Ring at a Point . 26 I.2.3 Morphisms . 28 I.2.4 The Gluing Construction . 33 Projective Space . 34 I.3 Relative Schemes . 35 I.3.1 Fibered Products . 35 I.3.2 The Category of S-Schemes . 39 I.3.3 Global Spec . 40 I.4 The Functor of Points . 42 II Examples 47 II.1 Reduced Schemes over Algebraically Closed Fields . 47 II.1.1 Affine Spaces . 47 II.1.2 Local Schemes . 50 II.2 Reduced Schemes over Non-Algebraically Closed Fields . 53 viii Contents II.3 Nonreduced Schemes . 57 II.3.1 Double Points . 58 II.3.2 Multiple Points . 62 Degree and Multiplicity . 65 II.3.3 Embedded Points . 66 Primary Decomposition . 67 II.3.4 Flat Families of Schemes . 70 Limits . 71 Examples . 72 Flatness . 75 II.3.5 Multiple Lines . 80 II.4 Arithmetic Schemes . 81 II.4.1 Spec Z ......................... 82 II.4.2 Spec of the Ring of Integers in a Number Field . 82 II.4.3 Affine Spaces over Spec Z .............. 84 II.4.4 A Conic over Spec Z .................. 86 1 II.4.5 Double Points in AZ .................. 88 III Projective Schemes 91 III.1 Attributes of Morphisms . 92 III.1.1 Finiteness Conditions . 92 III.1.2 Properness and Separation . 93 III.2 Proj of a Graded Ring . 95 III.2.1 The Construction of Proj S .............. 95 III.2.2 Closed Subschemes of Proj R ............. 100 III.2.3 Global Proj . 101 Proj of a Sheaf of Graded OX -Algebras . 101 The Projectivization P(E ) of a Coherent Sheaf E . 103 III.2.4 Tangent Spaces and Tangent Cones . 104 Affine and Projective Tangent Spaces . 104 Tangent Cones . 106 III.2.5 Morphisms to Projective Space . 110 III.2.6 Graded Modules and Sheaves . 118 III.2.7 Grassmannians . 119 III.2.8 Universal Hypersurfaces . 122 III.3 Invariants of Projective Schemes . 124 III.3.1 Hilbert Functions and Hilbert Polynomials . 125 III.3.2 Flatness II: Families of Projective Schemes . 125 III.3.3 Free Resolutions . 127 III.3.4 Examples . 130 Points in the Plane . 130 P3 Examples: Double Lines in General and in K . 136 III.3.5 B´ezout’s Theorem . 140 Multiplicity of Intersections . 146 III.3.6 Hilbert Series . 149 Contents ix IV Classical Constructions 151 IV.1 Flexes of Plane Curves . 151 IV.1.1 Definitions . 151 IV.1.2 Flexes on Singular Curves . 155 IV.1.3 Curves with Multiple Components . 156 IV.2 Blow-ups . 162 IV.2.1 Definitions and Constructions . 162 An Example: Blowing up the Plane . 163 Definition of Blow-ups in General . 164 TheBlowupasProj.................. 170 Blow-ups along Regular Subschemes . 171 IV.2.2 Some Classic Blow-Ups . 173 IV.2.3 Blow-ups along Nonreduced Schemes . 179 Blowing Up a Double Point . 179 Blowing Up Multiple Points . 181 The j-Function . 183 IV.2.4 Blow-ups of Arithmetic Schemes . 184 IV.2.5 Project: Quadric and Cubic Surfaces as Blow-ups . 190 IV.3 Fano schemes . 192 IV.3.1 Definitions . 192 IV.3.2 Lines on Quadrics . 194 Lines on a Smooth Quadric over an Algebraically Closed Field . 194 Lines on a Quadric Cone . 196 A Quadric Degenerating to Two Planes . 198 More Examples . 201 IV.3.3 Lines on Cubic Surfaces . 201 IV.4 Forms . 204 V Local Constructions 209 V.1 Images . 209 V.1.1 The Image of a Morphism of Schemes . 209 V.1.2 Universal Formulas . 213 V.1.3 Fitting Ideals and Fitting Images . 219 Fitting Ideals . 219 Fitting Images . 221 V.2 Resultants . 222 V.2.1 Definition of the Resultant . 222 V.2.2 Sylvester’s Determinant . 224 V.3 Singular Schemes and Discriminants . 230 V.3.1 Definitions . 230 V.3.2 Discriminants . 232 V.3.3 Examples . 234 x Contents V.4 Dual Curves . 240 V.4.1 Definitions . 240 V.4.2 Duals of Singular Curves . 242 V.4.3 Curves with Multiple Components . 242 V.5 Double Point Loci . 246 VI Schemes and Functors 251 VI.1 The Functor of Points . 252 VI.1.1 Open and Closed Subfunctors . 254 VI.1.2 K-Rational Points . 256 VI.1.3 Tangent Spaces to a Functor . 256 VI.1.4 Group Schemes . 258 VI.2 Characterization of a Space by its Functor of Points . 259 VI.2.1 Characterization of Schemes among Functors . 259 VI.2.2 Parameter Spaces . 262 The Hilbert Scheme . 262 Examples of Hilbert Schemes . 264 Variations on the Hilbert Scheme Construction . 265 VI.2.3 Tangent Spaces to Schemes in Terms of Their Func- tors of Points . 267 Tangent Spaces to Hilbert Schemes . 267 Tangent Spaces to Fano Schemes . 271 VI.2.4 Moduli Spaces . 274 References 279 Index 285 Introduction What schemes are The theory of schemes is the foundation for algebraic geometry formu- lated by Alexandre Grothendieck and his many coworkers. It is the basis for a grand unification of number theory and algebraic geometry, dreamt of by number theorists and geometers for over a century. It has strength- ened classical algebraic geometry by allowing flexible geometric arguments about infinitesimals and limits in a way that the classic theory could not handle. In both these ways it has made possible astonishing solutions of many concrete problems. On the number-theoretic side one may cite the proof of the Weil conjectures, Grothendieck’s original goal (Deligne [1974]) and the proof of the Mordell Conjecture (Faltings [1984]). In classical alge- braic geometry one has the development of the theory of moduli of curves, including the resolution of the Brill–Noether–Petri problems, by Deligne, Mumford, Griffiths, and their coworkers (see Harris and Morrison [1998] for an account), leading to new insights even in such basic areas as the the- ory of plane curves; the firm footing given to the classification of algebraic surfaces in all characteristics (see Bombieri and Mumford [1976]); and the development of higher-dimensional classification theory by Mori and his coworkers (see Koll´ar [1987]). No one can doubt the success and potency of the scheme-theoretic meth- ods. Unfortunately, the average mathematician, and indeed many a be- ginner in algebraic geometry, would consider our title, “The Geometry of Schemes”, an oxymoron akin to “civil war”. The theory of schemes is widely 2 Introduction regarded as a horribly abstract algebraic tool that hides the appeal of ge- ometry to promote an overwhelming and often unnecessary generality. By contrast, experts know that schemes make things simpler. The ideas behind the theory — often not told to the beginner — are directly related to those from the other great geometric theories, such as differential ge- ometry, algebraic topology, and complex analysis. Understood from this perspective, the basic definitions of scheme theory appear as natural and necessary ways of dealing with a range of ordinary geometric phenomena, and the constructions in the theory take on an intuitive geometric content which makes them much easier to learn and work with. It is the goal of this book to share this “secret” geometry of schemes. Chapters I and II, with the beginning of Chapter III, form a rapid intro- duction to basic definitions, with plenty of concrete instances worked out to give readers experience and confidence with important families of ex- amples. The reader who goes further in our book will be rewarded with a variety of specific topics that show some of the power of the scheme- theoretic approach in a geometric setting, such as blow-ups, flexes of plane curves, dual curves, resultants, discriminants, universal hypersurfaces and the Hilbert scheme. What’s in this book? Here is a more detailed look at the contents: Chapter I lays out the basic definitions of schemes, sheaves, and mor- phisms of schemes, explaining in each case why the definitions are made the way they are. The chapter culminates with an explanation of fibered products, a fundamental technical tool, and of the language of the “functor of points” associated with a scheme, which in many cases enables one to characterize a scheme by its geometric properties. Chapter II explains, by example, what various kinds of schemes look like. We focus on affine schemes because virtually all of the differences between the theory of schemes and the theory of abstract varieties are encountered in the affine case — the general theory is really just the direct product of the theory of abstract varietiesa ` la Serre and the theory of affine schemes. We begin with the schemes that come from varieties over an algebraically closed field (II.1). Then we drop various hypotheses in turn and look successively at cases where the ground field is not algebraically closed (II.2), the scheme is not reduced (II.3), and where the scheme is “arithmetic” — not defined over a field at all (II.4). In Chapter II we also introduce the notion of families of schemes. Families of varieties, parametrized by other varieties, are central and characteristic aspects of algebraic geometry.
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