Undergraduate Algebraic Geometry Miles Reid Math Inst., University of Warwick, 1st preprint edition, Oct 1985 2nd preprint edition, Jan 1988, LMS Student Texts 12, C.U.P., Cambridge 1988 2nd corrected printing, Oct 1989 first TeX edition, Jan 2007 October 20, 2013 4 Preface There are several good recent textbooks on algebraic geometry at the graduate level, but not (to my knowledge) any designed for an undergraduate course. Humble notes are from a course given in two successive years in the 3rd year of the Warwick under- graduate math course, and are intended as a self-contained introductory textbook. Contents 0 Woffle 11 0.1 What it’s about . 11 0.2 Specific calculations versus general theory . 12 0.3 Rings of functions and categories of geometry . 12 0.4 Geometry from polynomials . 13 0.5 “Purely algebraically defined” . 14 0.6 Plan of the book . 14 Course prerequisites . 15 Course relates to . 15 Exercises to Chapter 0 . 15 Books ............................................ 16 I Playing with plane curves 17 1 Plane conics 19 1.1 Example of a parametrised curve . 19 1.2 Similar example . 20 1.3 Conics in R2 ........................................ 21 1.4 Projective plane . 21 1.5 Equation of a conic . 23 ‘Line at infinity’ and asymptotic directions . 23 1.6 Classification of conics in P2 ................................ 24 1.7 Parametrisation of a conic . 25 1.8 Homogeneous form in 2 variables . 25 1.9 Easy cases of Bézout’s Theorem . 26 1.10 Corollary: unique conic through 5 general points of P2 . 27 1.11 Space of all conics . 28 1.12 Intersection of two conics . 29 1.13 Degenerate conics in a pencil . 30 1.14 Worked example . 30 Exercises to Chapter 1 . 32 5 6 CONTENTS 2 Cubics and the group law 35 2.1 Examples of parametrised cubics . 35 2.2 The curve (y2 = x(x − 1)(x − λ)) has no rational parametrisation . 36 2.3 Lemma . 37 2.4 Linear systems . 37 2.5 Lemma: divisibility by L or by Q ............................. 38 2.6 Proposition: cubics through 8 general points form a pencil . 39 2.7 Corollary: cubic through 8 points C1 \ C2 pass through the 9th . 40 2.8 Group law on a plane cubic . 40 2.9 Associativity “in general” . 42 2.10 Proof by continuity . 42 2.11 Pascal’s Theorem (the mystic hexagon) . 43 2.12 Inflexion, normal form . 44 2.13 Simplified group law . 45 Exercises to Chapter 2 . 46 2.14 Topology of a nonsingular cubic . 49 2.15 Discussion of genus . 51 2.16 Commercial break . 51 II The category of affine varieties 55 3 Affine varieties and the Nullstellensatz 57 3.1 Definition of Noetherian ring . 57 3.2 Proposition: Noetherian passes to quotients and rings of fractions . 58 3.3 Hilbert Basis Theorem . 58 3.4 The correspondence V ................................... 59 3.5 Definition: the Zariski topology . 59 3.6 The correspondence I ................................... 60 3.7 Irreducible algebraic set . 61 3.8 Preparation for the Nullstellensatz . 62 3.9 Definition: radical ideal . 62 3.11 Worked examples . 64 3.12 Finite algebras . 65 3.13 Noether normalisation . 66 3.14 Remarks . 68 3.15 Proof of (3.8) . 68 3.16 Separable addendum . 68 3.17 Reduction to a hypersurface . 69 Exercises to Chapter 3 . 70 4 Functions on varieties 73 4.1 Polynomial functions . 73 4.2 k[V ] and algebraic subsets of V .............................. 73 4.3 Polynomial maps . 74 4.4 Polynomial maps and k[V ] ................................. 75 CONTENTS 7 4.5 Corollary: f : V ! W is an isomorphism if and only if f ∗ is . 76 4.6 Affine variety . 77 4.7 Function field . 77 4.8 Criterion for dom f = V for f 2 k(V ) ........................... 78 4.9 Rational maps . 78 4.10 Composition of rational maps . 79 4.11 Theorem: dominant rational maps . 79 4.12 Morphisms from an open subset of an affine variety . 79 4.13 Standard open subsets . 80 4.14 Worked example . 81 Exercises to Chapter 4 . 82 III Applications 85 5 Projective and birational geometry 87 5.0 Why projective varieties? . 87 5.1 Graded rings and homogeneous ideals . 88 5.2 The homogeneous V -I correspondences . 89 5.3 Projective Nullstellensatz . 89 5.4 Rational functions on V .................................. 90 5.5 Affine covering of a projective variety . 91 5.6 Rational maps and morphisms . 92 5.7 Examples . 93 5.8 Birational maps . 94 5.9 Rational varieties . 95 5.10 Reduction to a hypersurface . 95 5.11 Products . 95 Exercises to Chapter 5 . 96 6 Tangent space and nonsingularity, dimension 101 6.1 Nonsingular points of a hypersurface . 101 6.2 Remarks . 102 6.3 Proposition: V nonsing is dense . 102 6.4 Tangent space . 103 6.5 Proposition: dim TP V is upper semicontinuous . 103 6.6 Corollary–Definition: dim TP V = dim V on a dense open set . 103 6.7 dim V = tr deg k(V ) – the hypersurface case . 104 6.8 Intrinsic nature of TP V ..................................104 6.9 Corollary: TP V only depends on P 2 V up to isomorphism . 105 6.10 Theorem: dim V = tr deg k(V ) ..............................106 6.11 Nonsingularity and projective varieties . 106 6.12 Worked example: blowup . 106 Exercises to Chapter 6 . ..