GEIRELLINGSRUD JOHNCHRISTIANOTTEM MAT4210 – ALGEBRAIC GEOMETRYI Contents 1 Algebraic sets and Nullstellensatz 7 Fields and the affine space 7 Closed algebraic sets 7 Hilbert’s Nullstellensatz 10 Hilbert’s Nullstellensatz—proofs 12 Examples 14 2 The Zariski topology 19 The Zariski topology 19 Irreducible topological spaces 21 Dimension 30 Polynomial maps between algebraic sets 34 Examples 36 3 Sheaves and varieties 45 Sheaves and ringed spaces 45 Functions on irreducible algebraic sets 49 The definition of a variety 53 The Hausdorff axiom 59 Products of varieties 61 An epilogue—about Zariski topologies. 68 2 4 Projective varieties 71 The projective spaces Pn 72 The Zariski topology and the projective Nullstellensatz 74 Regular functions on projective varieties 79 A simple criterion for being a morphism 84 5 Segre and Veronese varieties 91 Closed embeddings 91 Rational normal curves 94 The Segre embeddings 97 A Nullstellensatz for Pn ˆ Pm 101 The Veronese embeddings 107 Conics and the Veronese surface 112 Geometry of the spaces of polynomials 115 Appendix: Bihomogeneous polynomials 118 6 More on dimension 121 Some basic properties of polynomial maps 122 Noether’s Normalization Lemma 127 Krull’s Principal ideal theorem 132 System of parameters and fibres of morphisms 135 Applications to intersections 138 7 Non-singular varieties 143 Tangent spaces 143 The Zariski tangent space 145 7.2.1 Regular local rings 146 The Jacobian criterion in the projective case 150 7.3.1 The non-singular points are dense 151 Normal varieties 152 8 Birational maps and blowing up 155 Birational maps 158 Blowing up 161 3 9 Curves 169 Curves 169 Desingularizations of plane curves via blow-ups 176 Elliptic curves are not rational 177 10 Structure of Maps 179 Generic structure of morphisms 179 Morphisms from projective varieties 184 Generically finite maps 189 Curves over regular curves 196 11 Bézout’s theorem 199 Bézout’s Theorem 200 The local multiplicity 201 Hilbert functions and the degree of a variety 209 Proof of Bezout’s theorem 211 Applications of Bezout’s theorem 214 Plane cubics 229 Appendix: Depth, regular sequences and unmixedness 234 12 Varieties of lines 241 Grassmannians 241 Lines on quadric surfaces 244 27 lines on a cubic surface 244 13 Index 247 Introduction Algebraic geometry has many ramifications, but roughly speaking there are two main branches. One could be called the “geometric” branch where the geometry is the main objective. One studies geometric objects like curves, surfaces, threefolds and varieties of higher dimensions, defined by polynomials (or more generally algebraic functions). The aim is to understand their geometry. Frequently techniques from several other fields are used like from algebraic topology, differential geometry or analysis, and the theory is tightly connected with these other branches of mathematics. This makes it natural to work over the complex field C, even though other fields like function fields are also important. In fact, this interaction with other fields has been paramount in the development of algebraic geometry since the very beginning. The study of elliptic functions in the beginning of the 19th century, and subsequently of other algebraic functions, was the birth of modern algebraic geometry. The motivation and the origin was found in function theory, but the direction of research quickly took a geometric Figure 1: The affine Fer- route. Later Riemann surfaces and algebraic curves appeared together with their mat curve x3 ` y3 “ 1. function fields. The other branch one could call “arithmetic”. Superficially presented, one studies numbers by geometric methods. An ultra famous example is Fermat’s last theorem, now Andrew Wiles’ theorem, that the equation xn ` yn “ zn has no integral solutions except the trivial ones for n ¥ 3. The arithmetic branch also relies on techniques from other fields, like number theory, Galois theory and representation theory. One very commonly applied technique is reduction modulo a prime number p. Hence the importance of including fields of positive characteristic among the base fields. Of course another very natural base field Figure 2: The affine Fer- for many of these “arithmetic” studies is the field Q of algebraic numbers. mat curve x4 ` y4 “ 1. Algebraic geometry is to the common benefit a triple marriage of geometry, algebra and arithmetic. All of the spouses claim influence on the development of the field which makes the field quite abstract; but also an immensely beautiful part of mathematics. 6 mat4210 – algebraic geometry i A few words about categories See also Chapter 18 of CA. A category C is a set (or class) of objects Obj C and for each pair of objects A, B P Obj C a set of morphisms HomCpA, Bq. These have to satisfy the composition law: for f : B Ñ C and g : A Ñ B, we have a composition p f ˝ gq P HomCpA, Cq, and there are distinguished morphisms idA P HomCpA, Aq satisfying f ˝ pg ˝ hq “ p f ˝ gq ˝ h and g ˝ idA “ f and idB ˝g “ g, whenever these are defined. A covariant functor between two categories C and D is a map that takes covariant functor objects to objects and morphisms to morphisms. More precisely, there is a map kovariant funktor F : Obj C Ñ Obj D and for each pair A, B P Obj C a map F : HomCpA, Bq Ñ HomDpFpAq, FpBqq, such that Fp f ˝ gq “ Fp f q ˝ Fpgq and FpidAq “ idFpAq. A contravariant functor is defined similarly, but with F reversing the direction contravariant functor of the morphisms, i.e., there is a map kontravariant funktor F : HomCpA, Bq Ñ HomDpFpBq, FpAqq, satisfying Fp f ˝ gq “ Fpgq ˝ Fp f q and FpidAq “ idFpAq. Geir Ellingsrud/John Christian Ottem—versjon 1.29—23rd April 2021 at 4:52pm Chapter 1 Algebraic sets and Nullstellensatz Topics in Chapter 1: The correspondence between ideals and algebraic sets—weak and strong versions of Hilbert’s Nullstellensatz—the Rabinow- itsch trick—two proofs of the Nullstellensatz, one elementary, and another totally different—radical ideals—intuition, drawings and figures. 1.1 Fields and the affine space 1.1 We shall almost exclusively work over an algebraic closed field which we shall denote by k. In general we do not impose further constraints on k, except for a few results that require the characteristic to be zero. A specific field to have in mind would be the field of complex numbers C, but other important fields include the closures Q of the rational numbers Q and Fq, the algebraic closure of the finite field Fq with q elements. 1.2 The affine n-space An is, as a set, simply kn. The name change is here to affine n-space underline that there is more to An than just its set of elements – it is a topological det affine n-rommet space, and we have the notion of polynomial maps An Ñ k. Anyhow, in the beginning think about An as just kn. Often the ground field will be tacitly understood, but when we want to be precise about it, we shall write Anpkq. The ground field will always be algebraically closed unless the contrary is explicitly Figure 1.1: A one sheeted-hyperboloid. stated. 1.2 Closed algebraic sets The first objects we shall meet are the so-called closed algebraic sets, or simply Closed algebraic sets algebraic sets. They are the subsets of the affine space An defined by a set of lukkede algebraiske mengder polynomial equations. You have probably already seen examples of curves in the plane and maybe with some surfaces in 3-space – like conic sections, hyperboloids and paraboloids, for example. 1.3 Formally the definition of a closed algebraic set is as follows. If S is a subset a, a b, B c, C d, D e, E f F g G h H i I j J of the polynomial ring krx ,..., x s, one defines , , , , , 1 n k, K l, L m, m n, N o, O p, P q, Q r, R s, S t, T n ZpSq “ t x P A | f pxq “ 0 for all f P S u, u, U v, V w, W x, X y, Y z, Z Mathematicians are al- ways in shortage of sym- bols and use all kinds of alphabets. The ger- manic gothic letters are still in use in some con- text, like to denote ide- als in some text. 8 mat4210 – algebraic geometry i and the closed algebraic sets are the subsets of An obtained in this way. Note that any expression of the form bi fi, with bi polynomials and f1,..., fr P S, also vanishes at points of ZpSq. The ideal a generated by S therefore has the same ř zero set as S; that is, ZpSq “ Zpaq. We shall almost exclusively work with ideals and tacitly replace a set of polynomials by the ideal it generates. Hilbert’s basis theorem tells us that any ideal in krx1,..., xns is finitely gener- ated, so that a closed algebraic subset is described as the set of common zeros of finitely many polynomials. Examples 1.1 The polynomial ring krxs in one variable is a pid1, so if a is an ideal in krxs, 1 Recall that a ring is a then a “ p f pxqq. Because polynomials in one variable have only finitely many pid, or a principal ideal domain, if it is an inte- 1 1 zeros, the closed algebraic subsets of A are just the finite subsets of A . gral domain where ev- ery ideal is principal, i.e. 1.2 A more interesting example is the so-called Clebsch diagonal cubic; a surface generated by one ele- in A3pCq with equation ment.