INTRODUCTION TO ALGEBRAIC GEOMETRY Contents 1. Affine Geometry 2 1.1. Closed algebraic subsets of affine spaces 2 1.2. Regular functions 4 1.3. Regular maps 5 1.4. Irreducible subsets 8 1.5. Rational functions 10 1.6. Rational maps 11 1.7. Composition of rational maps 12 2. Projective Geometry 15 2.1. Closed subsets of projective space 15 2.2. Example of projective varieties 17 2.3. Regular functions and regular maps on quasiprojective algebraic sets 20 2.4. Rational functions and rational maps for quasiprojective varieties 24 2.5. Projective algebraic sets are universally closed 25 3. Finite maps 29 3.1. Local study of finite maps 31 4. Dimension 34 4.1. Dimension of intersection with a hypersurface 35 4.2. The dimension of the fibers of a regular map 36 4.3. Lines on surfaces 38 5. Nonsingular varieties 41 5.1. Tangent space 41 5.2. Singular points 47 5.3. Codimension one subvarieties 48 5.4. Nonsingular subvarieties of nonsingular varieties 49 6. Blow-ups 51 6.1. The blow-up of P2 at one point 51 7. Divisors and Class Group 53 8. B´ezout's Theorem 55 8.1. Arbitrary smooth projective surfaces 58 9. Appendix 61 9.1. Classical algebraic structures 61 9.2. Commutative algebra 62 9.3. Topology 72 9.4. Categories 73 References 75 1 2 INTRODUCTION TO ALGEBRAIC GEOMETRY 1. Affine Geometry 1.1. Closed algebraic subsets of affine spaces. Throughout this course, unless otherwise specified, we work over an algebraically closed field k = k¯. Let's see what space we will work in (for now): n Definition 1.1. The n-dimensional affine space over k is Ak . As a set this is just kn := k × ::: × k; | {z } n times but we will put more structure on it: a topology such that the only continuous functions n 1 n Ak ! Ak are polynomial. We also ignore the vector space structure on k . n A polynomial function on Ak is a polynomial P (X1;:::;Xn) with coefficients in k. The set of all such is the polynomial ring k[X1;:::;Xn]. We may also denote by P (X) when we don't want to write all indices X1;:::;Xn. Definition 1.2. An affine algebraic variety1 or closed subset of affine space is a n subset Y ⊂ Ak given by the vanishing of a family of polynomials Pi(X1;:::;Xn) and we denote Y = V ((Pi)i). Similarly we write V (T ) for the common vanishing locus of all polynomials in a set T ⊂ k[X]. We may allow non-algebraically closed fields k in this definition. Example 1.3. The following are examples of closed algebraic subsets: • A line in A2 is V (aX + bY + c). • In general, a d-dimensional linear subspace of An is given by the simultaneous van- ishing of d linear equations. • A curve in the affine plane is the set of zeros of one nonzero polynomial P (X; Y ). n • The union of closed subsets in Ak is also closed: If Y1 = V ((Pi)i), and Y2 = V ((Qj)j), then Y1 [ Y2 = V ((Pi · Qj)i;j). n • The intersection of close subsets in Ak is also closed: If Y1;Y2 are as above, then Y1 \ Y2 = V ((Pi)i; (Qj)j). n • If Y = V (f) in Ak is a hypersurface (given by the vanishing of just one polynomial), then n D(f) := Ak n Y n+1 is also an affine variety... but in Ak . In fact D(f) = V (Xn+1f − 1). We'll believe this more when we learn about morphisms and isomorphisms. • The closed subsets of A1 are: { ;. { Finite subsets of points. { A1. This is because a polynomial of degree n in one variable X has at most n zeros. Also, 1 if Y = fx1; : : : ; xng are n points in Ak, then P (X) = (X − x1) · ::: · (X − xn) is a polynomial with Y = V (P ). n Definition/Theorem 1.4. The Zariski topology on Ak is the topology whose closed sets n are all the closed algebraic subsets in Ak . 1Soon, affine algebraic variety will mean irreducible closed algebraic subset INTRODUCTION TO ALGEBRAIC GEOMETRY 3 Proof. What needs checking is that the union of two closed subsets is closed, the intersection of arbitrarily (indexed by any n family, not necessarily finite or countable) many closed subsets is closed, and that the empty set and Ak are both closed. All are clear. Question. How can we change the equations of an affine variety without changing the variety itself? Example 1.5. Let Y = V ((Pi)i). If we add to the given list of equations of Y one or P arbitrarily many equations of the form QjPj, where Qj are finitely many polynomials in k[X] and Pj are among fPigi2I , then we do not change Y . In particular, if we replace fPigi2I by the ideal (Pi)i that they generate inside k[X], we still get the same common vanishing locus Y . Related to this we make the following definition. n Definition/Theorem 1.6. If Y ⊂ Ak is a subset (usually an affine variety), the ideal of Y is the ideal I(Y ) ¢ k[X] containing all the polynomials P such that P vanishes on Y (i.e. Y ⊆ V (P )). Proof. All you need to check is that I(Y ) is indeed an ideal, and this is a consequence of the previous example. Directly from the definition we see Y ⊆ V (I(Y )). In fact equality holds, and several other strong results hold as well. Theorem 1.7. p (i) V (a) = V ( a) for any a ¢ k[X]. (ii) I(Y ) is a radical ideal, i.e. P r 2 I(Y ) ) P 2 I(Y ). (iii) If Y1 ⊂ Y2, then I(Y1) ⊃ I(Y2). (iv) If a ⊂ b, then V (a) ⊃ V (b). (v) More generally, I(Y1 [ Y2) = I(Y1) \ I(Y2) ⊇ I(Y1) · I(Y2). n (vi) If Y ⊂ Ak is an affine variety, then Y = V (I(Y )). n (vii) More generally, if Y ⊂ Ak is just a subset, then V (I(Y )) = Y is the closure of Y in the n Zariski topology on Ak . p (viii) Hilbert Nullstellensatz: If a ¢ k[X] is an ideal, then a = I(V (a)). Proof. Since (i) through (v) are easy and (vii) ! (vi), it remains to prove the Nullstellensatz and (vii). We prove the latter and leave (viii) for later. Clearly Y ⊂ V (I(Y )), hence Y ⊆ V (I(Y )). Conversely, let W be closed with W ⊇ Y . The definition of closed sets implies W = V (a) for some a ¢ k[X]. Then V (a) ⊇ Y implies a ⊆ I(V (a)) ⊆ I(Y ) and W = V (a) ⊃ V (I(Y )). In particular V (I(Y )) is the smallest closed subset containing Y , which is by definition Y . Corollary 1.8. There is a natural correspondence given by V (·) and I(·) between n fclosed subsets of Ak g fradical ideals of k[X]g: Corollary 1.9. If T and S are two sets of equations (polynomials in k[X]). Then they n p p describe the same affine variety Y ⊂ Ak (this means Y = V (T ) = V (S)) iff (T ) = (S), where (T ) and (S) are the ideals generated by T and S in k[X]. Question. OK, going the other way: If we have infinitely many equations for an affine variety Y , can we extract finitely many whose vanishing locus is still exactly Y ? More precisely: If T ⊂ k[X] is a an arbitrary subset of polynomials, does there exist a finite subset fP1;:::;Prg ⊂ T such that V (P1;:::;Pr) = V (T )? The answer is yes, and it comes from the: 4 INTRODUCTION TO ALGEBRAIC GEOMETRY Theorem 1.10 (Hilbert Basis Theorem). Any ideal I ¢ k[X] is generated by finitely many elements. Equivalently, k[X] is a Noetherian ring. Proof. See appendix x9.2.5. Assuming this theorem, then Y = V (T ) = V ((T )) = V (P1;:::;Pr), where P1;:::;Pr are a finite set of generators of (T ). n The theorem also shows that Ak is a Noetherian space (decreasing sequences of closed subsets are eventually constant). Indeed if Y1 ⊃ Y2 ⊃ ::: is a decreasing sequence of closed algebraic subsets, then I(Y1) ⊂ I(Y2) ⊂ ::: is an increasing sequence of ideals of k[X]. Since this is a Noetherian ring, this sequence is eventually constant. But Yi = V (I(Yi)) for all i and the conclusion follows. 1.2. Regular functions. n Definition 1.11. Let Y ⊂ Ak be a closed algebraic subset. A function f : Y ! k is called regular if there exists a polynomial F 2 k[X] such that F (y) = f(y) for all y 2 Y . In the above, if we know F then we know f, but if we know f, then usually there are 0 0 several options for F . More precisely, if F jY = F jY = f, then (F − F )jY = 0, which means that the polynomial F − F 0 vanishes on Y . By definition, this happens precisely when (F − F 0) 2 I(Y ). So two polynomials give the same regular function on Y when they are equl modulo I(Y ). n Definition/Theorem 1.12. The set of regular functions on the closed Y ⊂ Ak is k[X] k[Y ] := ; I(Y ) which is an algebra of finite type over k. n Example 1.13. • k[Ak ] = k[X] (0) = k[X] : • k[;] = k[X] (1) = 0: n • If Y = (1;:::; 1) is a point in Ak , then k[Y ] = k[X] I(Y ).