Massachusetts Institute of Technology Notes for 18.721 INTRODUCTION TO ALGEBRAIC GEOMETRY (This is a preliminary draft. Please don’t reproduce.) 1 . 2 TABLE OF CONTENTS Chapter 1: PLANE CURVES 1.1 The Affine Plane 1.2 The Projective Plane 1.3 Plane Projective Curves 1.4 Tangent Lines 1.5 Nodes and Cusps 1.6 Transcendence Degree 1.7 The Dual Curve 1.8 Resultants 1.9 Hensel’s Lemma 1.10 Coverings of the Projective Line 1.11 Genus 1.12 Bézout’s Theorem 1.13 The Plücker Formulas Chapter 2: AFFINE ALGEBRAIC GEOMETRY 2.1 Rings and Modules 2.2 The Zariski Topology 2.3 Some Affine Varieties 2.4 The Nullstellensatz 2.5 The Spectrum of a Finite-type Domain 2.6 Morphisms of Affine Varieties 2.7 Finite Group Actions Chapter 3: PROJECTIVE ALGEBRAIC GEOMETRY 3.1 Projective Varieties 3.2 Homogeneous Ideals 3.3 Product Varieties 3.4 Morphisms and Isomorphisms 3.5 Affine Varieties 3.6 Lines in Projective Three-Space Chapter 4: STRUCTURE OF VARIETIES I: DIMENSION 4.1 Dimension 4.2 Proof of Krull’s Theorem 4.3 The Nakayama Lemma 4.4 Integral Extensions 4.5 Normalization 4.6 Geometry of Integral Morphisms 4.7 Chevalley’s Finiteness Theorem 4.8 Double Planes 3 Chapter 5: STRUCTURE OF VARIETIES II: CONSTRUCTIBLE SETS 5.1 Localization, again 5.2 Valuations 5.3 Smooth Curves 5.4 Constructible sets 5.5 Closed Sets 5.6 Fibred Products 5.7 Projective Varieties are Proper 5.8 Fibre Dimension Chapter 6: MODULES 6.1 The Structure Sheaf 6.2 O-Modules 6.3 The Sheaf Property 6.4 Some O-Modules 6.5 Direct Image 6.6 Twisting 6.7 Proof of Theorem 6.3.2 Chapter 7: COHOMOLOGY 7.1 Cohomology of O-Modules 7.2 Complexes 7.3 Characteristic Properties 7.4 Construction of Cohomology 7.5 Cohomology of the Twisting Modules 7.6 Cohomology of Hypersurfaces 7.7 Finiteness of Cohomology 7.8 Bézout’s Theorem Chapter 8: THE RIEMANN-ROCH THEOREM FOR CURVES 8.1 Branched Coverings 8.2 Modules 8.3 Divisors 8.4 The Riemann-Roch Theorem I 8.5 The Birkhoff-Grothendieck Theorem 8.6 Differentials 8.7 Trace, and proof of Riemann-Roch 8.8 The Riemann-Roch Theorem II 8.9 Using Riemann-Roch 4 Chapter 1 PLANE CURVES 1.1 The Affine Plane 1.2 The Projective Plane 1.3 Plane Projective Curves 1.4 Tangent Lines 1.5 Nodes and Cusps 1.6 Transcendence Degree 1.7 The Dual Curve 1.8 Resultants 1.10 Coverings of the Projective Line 1.11 Genus 1.12 Bézout’s Theorem 1.13 The Plücker Formulas Plane curves were the first algebraic varieties to be studied, so we begin with them. They provide helpful examples, and we will see in Chapter 5 how they control varieties of arbitrary dimension. Chapters 2 - 7 are about varieties of arbitrary dimension. We come back to curves in Chapter 8. 1.1 The Affine Plane The n-dimensional affine space An is the space of n-tuples of complex numbers. The affine plane A2 is the two-dimensional affine space. Let f(x1; x2) be an irreducible polynomial in two variables with complex coefficients. The set of points of the affine plane at which f vanishes, the locus of zeros of f, is called a plane affine curve. Let’s denote this locus by X. Using vector notation x = (x1; x2), (1.1.1) X = fx j f(x) = 0g The degree of the curve X is the degree of its irreducible defining polynomial f. 1.1.2. 3 2 1 0 -1 -2 -3 -2 -1 0 1 2 The Cubic Curve y2 = x3 − x (real locus) 5 1.1.3. Note. In contrast with polynomials in one variable, most complex polynomials in two or more variables are irreducible – they cannot be factored. This can be shown by a method called “counting constants”. For instance, quadratic polynomials in x1; x2 depend on the six coefficients of the monomials of degree at most two. Linear polynomials ax1 +bx2 +c depend on three coefficients, but the product of two linear polynomials depends on only five parameters, because a scalar factor can be moved from one of the linear polynomials to the other. So the quadratic polynomials cannot all be written as products of linear polynomials. This reasoning is fairly convincing. It can be justified formally in terms of dimension, which will be discussed in Chapter 4. We will get an understanding of the geometry of a plane curve as we go along, and we mention just one important point here. A plane curve is called a curve because it is defined by one equation in two variables. Its algebraic dimension is one. But because our scalars are complex numbers, it will be a surface, geometrically. This is analogous to the fact that the affine line A1 is the plane of complex numbers. 1 One can see that a plane curve X is a surface by inspecting its projection to the affine x1-line A . One writes the defining polynomial as a polynomial in x2, whose coefficients ci = ci(x1) are polynomials in x1: d d−1 f(x1; x2) = c0x2 + c1x2 + ··· + cd Let’s suppose that d is positive, i.e., that f isn’t a polynomial in x1 alone (in which case, since it is irreducible, it would be linear). The fibre of a map X ! Z over a point p of Z is the inverse image of p, the set of points of X that map to 1 p. The fibre of the projection X ! A over the point x1 = a is the set of points (a; b) such that b is a root of the one-variable polynomial d d−1 f(a; x2) = c0x2 + c1x2 + ··· + cd with ci = ci(a). There will be finitely many points in this fibre, and the fibre won’t be empty unless f(a; x2) is a constant. So the curve X covers most of the x1-line, a complex plane, finitely often. (1.1.4) changing coordinates We allow linear changes of variable and translations in the affine plane A2. When a point x is written as t 0 0 0 t the column vector (x1; x2) , the coordinates x = (x1; x2) after such a change of variable will be related to x by the formula (1.1.5) x = Qx0 + a t where Q is an invertible 2×2 matrix with complex coefficients and a = (a1; a2) is a complex translation vector. This changes a polynomial equation f(x) = 0, to f(Qx0 + a) = 0. One may also multiply a polynomial f by a nonzero complex scalar without changing the locus ff = 0g. Using these operations, all lines, plane curves of degree 1, become equivalent. An affine conic is a plane affine curve of degree two. Every affine conic is equivalent to one of the loci 2 2 2 (1.1.6) x1 − x2 = 1 or x2 = x1 The proof of this is similar to the one used to classify real conics. The two loci might be called a complex 2 2 ’hyperbola’ and ’parabola’, respectively. The complex ’ellipse’ x1 + x2 = 1 becomes the ’hyperbola’ when one multiplies x2 by i. On the other hand, there are infinitely many inequivalent cubic curves. Cubic polynomials in two variables 2 2 3 2 2 3 depend on the coefficients of the ten monomials 1; x1; x2; x1; x1x2; x2; x1; x1x2; x1x2; x2 of degree at most 3 in x. Linear changes of variable, translations, and scalar multiplication give us only seven scalars to work with, leaving three essential parameters. 1.2 The Projective Plane n The n-dimensional projective space P is the set of equivalence classes of nonzero vectors x = (x0; x1; :::; xn), the equivalence relation being 6 0 0 0 0 (1.2.1) (x0; :::; xn) ∼ (x0; :::; xn) if (x0; :::; xn) = (λx0; :::; λxn) for some nonzero complex number λ. The equivalence classes are the points of Pn, and one often refers to a point by a particular vector in its class. Points of Pn correspond bijectively to one-dimensional subspaces of Cn+1. When x is a nonzero vector, the vectors λx, together with the zero vector, form the one-dimensional subspace of the complex vector space Cn+1 spanned by x. The projective plane P2 is the two-dimensional projective space. Its points are equivalence classes of nonzero vectors (x0; x1; x2). (1.2.2) the projective line 1 Points of the projective line P are equivalence classes of nonzero vectors (x0; x1). If x0 isn’t zero, we −1 may multiply by λ = x0 to normalize the first entry of (x0; x1) to 1, and write the point it represents in a unique way as (1; u), with u = x1=x0. There is one remaining point, the point represented by the vector (0; 1). The projective line P1 can be obtained by adding this point, called the point at infinity, to the affine u-line, which is a complex plane. Topologically, P1 is a two-dimensional sphere. (1.2.3) lines in projective space A line in projective space Pn is determined by a pair of distinct points p and q. When p and q are represented by specific vectors, the set of points frp + sqg, with r; s in C not both zero is a line L. Points of L correspond bijectively to points of the projective line P1, by (1.2.4) rp + sq ! (r; s) A line in the projective plane P2 can also be described as the locus of solutions of a homogeneous linear equation (1.2.5) s0x0 + s1x1 + s2x2 = 0 1.2.6.