MATH 631 NOTES ALGEBRAIC GEOMETRY KAREN SMITH Contents 1. Algebraic sets, affine varieties, and the Zariski topology 4 1.1. Algebraic sets 4 1.2. Hilbert basis theorem 4 1.3. Zariski topology 5 2. Ideals, Nullstellensatz, and the coordinate ring 5 2.1. Ideal of an affine algebraic set 5 2.2. Hilbert’s Nullstellensatz 6 2.3. Irreducible spaces 6 2.4. Sept. 10 warmup 7 2.5. Some commutative algebra 7 2.6. Review of Hilbert’s Nullstellensatz 7 2.7. Irreducible algebraic sets 7 2.8. Aside on non-radical ideals 8 3. Regular functions, regular maps, and categories 8 3.1. Regular functions 8 3.2. Properties of the coordinate ring 8 3.3. Regular mappings 8 3.4. Category of affine algebraic sets 9 3.5. Sep. 14 quiz question 11 3.6. Convention on algebraic sets 11 3.7. Hilbert’s Nullstellensatz and the Zariski topology 11 4. Rational functions 12 4.1. Function fields and rational functions 12 4.2. Regular points 12 4.3. Sheaf of regular functions on V 12 5. Projective space, the Grassmannian, and projective varieties 13 5.1. Projective space 13 5.2. Homogeneous coordinates 13 5.3. More about projective space 14 5.4. Projective algebraic sets 14 5.5. Projective algebraic sets, continued 14 5.6. The projective Nullstellensatz 15 5.7. Projective closure 15 6. Mappings of projective space 16 6.1. Example: Second Veronese embedding 16 6.2. Geometric definition 17 Date: Fall 2012. Notes taken by Daniel Hast. 1 2 KARENSMITH 6.3. Example: The twisted cubic 17 6.4. Example: A conic in P2 17 6.5. Projection from a point in Pn onto a hyperplane 17 6.6. Homogenization of affine algebraic sets 18 7. Abstract and quasi-projective varieties 18 7.1. Basic definition and examples 18 7.2. Quasi-projective varieties are locally affine 18 7.3. The sheaf of regular functions 19 7.4. Main example of regular functions in projective space 19 7.5. Morphisms of quasi-projective varieties 20 8. Classical constructions 20 8.1. Twisted cubic and generalization 20 8.2. Hypersurfaces 20 8.3. Segre embedding 21 8.4. Products of quasi-projective varieties 22 8.5. Conics 23 8.6. Conics through a point 23 9. Parameter spaces 24 9.1. Example: Hypersurfaces of fixed degree 24 9.2. Philosophy of parameter spaces 24 9.3. Conics that factor 24 10. Regular maps of projective varieties 25 10.1. Big theorem on closed maps 25 10.2. Preliminary: Graphs 25 10.3. Proof of Theorem 10.1 26 11. Function fields, dimension, and finite extensions 27 11.1. Commutative algebra: transcendence degree and Krull dimension 27 11.2. Function field 27 11.3. Dimension of a variety 28 11.4. Noether normalization 28 11.5. Dimension example 29 11.6. Facts about dimension 29 11.7. Dimension of hyperplane sections 30 11.8. Equivalent formulations of dimension 31 12. Families of varieties 31 12.1. Family of varieties (schemes) 31 12.2. Incidence correspondences 31 12.3. Lines contained in a hypersurface 32 12.4. Dimension of fibers 32 12.5. Cubic surfaces 33 13. Tangent spaces 33 13.1. Big picture 33 13.2. Intersection multiplicity 34 13.3. Tangent lines and the tangent space 34 13.4. Smooth points 35 13.5. Differentials, derivations, and the tangent space 35 13.6. The Zariski tangent space 37 13.7. Tangent spaces of local rings 38 14. Regular parameters 39 14.1. Local parameters at a point 39 MATH 631 NOTES ALGEBRAIC GEOMETRY 3 14.2. Nakayama’s lemma 40 14.3. Embedding dimension 40 14.4. Transversal intersection at arbitrary points 40 14.5. Philosophy of power series rings 41 14.6. Divisors and ideal sheaves 41 15. Rational maps 43 15.1. Provisional definition 43 15.2. Definition of rational map 43 15.3. Examples of rational maps 44 15.4. Rational maps, composition, and categories 44 15.5. Types of equivalence 45 15.6. Dimension of indeterminacy 45 15.7. Dimension of indeterminacy, continued 46 15.8. Images and graphs of rational maps 47 16. Blowing up 47 16.1. Blowing up a point in An 47 16.2. Resolution of singularities 48 16.3. More about blowups 48 16.4. Blowing up in general 49 16.5. Hironaka’s theorem 50 17. Divisors 50 17.1. Main definitions 50 17.2. The divisor of zeros and poles 51 17.3. Order of vanishing 51 17.4. Alternate definitions of order of vanishing 52 17.5. Divisors of zeros and poles, continued 53 17.6. Divisor class group, continued 53 17.7. Divisors and regularity 54 17.8. Commutative algebra digression 54 17.9. Divisors and regularity, continued 55 18. Locally principal divisors 56 18.1. Locally principal divisors 56 18.2. The Picard group 57 18.3. Summary of locally principal divisors 57 18.4. Pulling back locally principal divisors 57 18.5. The Picard group functor 59 18.6. Moving lemma 59 19. Riemann–Roch spaces and linear systems 60 19.1. Riemann–Roch spaces 60 19.2. Riemann–Roch spaces, continued 61 19.3. Complete linear systems 63 19.4. Some examples 63 19.5. Linear systems 64 19.6. Linear systems and rational maps 65 20. Differential forms 67 20.1. Sections 67 20.2. Differential forms 67 20.3. Regular differential forms 67 20.4. Rational differential forms and canonical divisors 69 20.5. Canonical divisors, continued 69 4 KARENSMITH 20.6. The canonical bundle on X 70 Index 72 1. Algebraic sets, affine varieties, and the Zariski topology List of topics: (1) Algebraic sets (2) Hilbert basis theorem (3) Zariski topology 1.1. Algebraic sets. Fix a field k. Consider kN, the set of N-tuples in k. Definition 1.1. An affine algebraic subset of kN is the common zero locus of a collection of polynomials in k[x ,...,x ]. 1 N That is: Fix S ⊆ k[x ,...,x ] any subset. Then 1 N V(S) = (cid:8)p = (λ1,...,λN) ∈ kN (cid:12)(cid:12) f(p) = 0 ∀f ∈ S(cid:9). Example 1.2. (1) Lines in R2: V(y−mx−b) ⊆ R2. (2) Rational points on a cone (arithmetic geometry): V(x2+y2−z2) ⊆ Q3 (3) All linear subspaces of kN are affine algebraic sets. (4) V(det(x )−1) = SL (C) = {n×n matrices /C of det1} ⊆ Cn2 ij n (cid:26)(cid:18) (cid:19) (cid:12) (cid:27) (5) sl2(R) = xz wy (cid:12)(cid:12)(cid:12) trace = 0 ⊆ R2×2 (6) Point in kN: {(a ,...,a )} = V(x −a ,...,x −a ). 1 N 1 1 N N (cid:16) (cid:17) (7) V(x,y) = (0,0) = V (cid:8)xn+y,yn+17(cid:9) ⊆ R2 n∈N ≥30 Remark 1.3. S ⊆ T ⊆ k[x ,...,x ] =⇒ V(S) ⊇ V(T). 1 N 1.2. Hilbert basis theorem. Theorem 1.4 (Hilbert basis theorem). Every affine algebraic set in kN can be defined by finitely many polynomials. Proof requires a lemma: Lemma 1.5. Let {f } ⊆ k[x ,...,x ] and let I ⊆ k[x ,...,x ] be the ideal generated by the λ λ∈Λ 1 N 1 N {f } . Then V(S) = V(I). λ λ∈Λ Proof. We know V(S) ⊇ V(I). Take p ∈ V(S). We want to show that given any g ∈ I, we have g(p) = 0. Take g ∈ I, so g = r f +···+r f , where f ∈ S and r ∈ k[x ,...,x ]. So 1 1 t t i i 1 N g(p) = r (p)f (p)+···+r (p)f (p) = 0 1 1 t t since f (p) = 0 for i = 1,...,t. Hence p ∈ V(I). (cid:3) i Proof of Theorem 1.4. Take any S ⊆ k[x ,...,x ], I = (cid:104)S(cid:105) ideal generated by S. We have V(S) = 1 N V(I) by Lemma 1.5. But every ideal in a polynomial ring in finitely many variables is finitely generated. Hence V(S) = V(I) = V(g ,...,g ), 1 t where g ,...,g generate I. (cid:3) 1 t Remark 1.6 (Algebra black box). • R is Noetherian if every ideal is f.g. • Thm: R Noetherian =⇒ R[x] Noetherian. ∼ • k[x ,...,x ][x ] = k[x ,...,x ], use induction. 1 N−1 N 1 N MATH 631 NOTES ALGEBRAIC GEOMETRY 5 1.3. Zariski topology. Definition 1.7 (topology). A topology on a set X is a collection of distinguished subsets, called closed sets, satisfying: (1) ∅ and X are closed. (2) An arbitrary intersection of closed sets is closed. (3) A finite union of closed sets is closed. Example 1.8. (1) On R, the Euclidean topology. (2) On R, cofinite: closed sets are finite sets, and R,∅. Definition 1.9 (Zariski topology). The Zariski topology on kN is defined as the topology whose closed sets are affine algebraic sets. 1.3.1. Proof that affine algebraic sets form closed sets on a topology on kN. (1) ∅ = V(1), kN = V(0). (2) WTS: {V } closed sets =⇒ (cid:84) V closed. Write V = V(I ). Then λ λ∈Λ λ λ λ (cid:92) (cid:92) (cid:16)(cid:91) (cid:17) (cid:16)(cid:88) (cid:17) V = V(I ) = V I = V I . λ λ λ λ λ∈Λ λ∈Λ λ∈Λ λ∈Λ (3) WTS: Finite union of closed sets are closed. By induction, suffices to show V(f ,...,f )∪ 1 t V(g ,...,g ) is an algebraic set. 1 s Note: V(f ,...,f )∪V(g ,...,g ) = V(cid:0){f g } (cid:1). 1 t 1 s i j i∈{1,...,t} j∈{1,...,s} Proof on quiz. Example 1.10. Zariski topology on k1 is the cofinite topology. Since k[x] is a PID, V = V((cid:104)f ,...,f (cid:105)) = V(f) = {roots of f}, 1 t which is finite if f (cid:54)= 0. 2. Ideals, Nullstellensatz, and the coordinate ring Today: (1) ideal of V (2) Hilbert’s Nullstellensatz (3) Regular functions (4) coordinate ring 2.1. Ideal of an affine algebraic set. Affine algebraic subset of kN: V = V((f ,...,f )) ⊆ kN. 1 t Consider the map {ideals in k[x ,...,x ]} −→ (cid:8)(affine) algebraic subsets of kN(cid:9) 1 N I (cid:55)−→ V(I). Note 2.1. • This map is order reversing: I ⊆ J =⇒ V(J) ⊆ V(I). • Surjective. • Not injective: e.g., (x,y),(cid:0)x2,y2(cid:1). Remark 2.2 (algebra). R commutative ring, I ⊆ R any ideal. Definition 2.3. The radical of I is the ideal RadI = (cid:8)f ∈ R (cid:12)(cid:12) fN ∈ I for some N(cid:9). 6 KARENSMITH • Sanity check: show this is an ideal. • I is radical if RadI = I. Lemma 2.4. Let I ⊆ k[x ,...,x ]. Then 1 N V(I) = V(RadI). Proof. I ⊆ RadI =⇒ V(RadI) ⊆ V(I). So take p ∈ V(I) ⊆ kN. Need to show ∀f ∈ RadI that f(p) = 0. We have f ∈ RadI =⇒ fN ∈ RadI, so (cid:0)f(p)(cid:1)N = fN(p) = 0 =⇒ f(p) = 0. (cid:3) Now is the map I (cid:55)−→ V(I) injective? Example 2.5. (cid:0)x2+y2(cid:1) ∈ R[x,y]. V(x,y) = (0,0) = V(x2+y2) ⊆ R2. We have 2 radical ideals defining the same algebraic set. Definition 2.6. Let V ⊆ kN be an affine algebraic set. The ideal of V is I(V) = (cid:8)f ∈ k[x1,...,xN] (cid:12)(cid:12) f(p) = 0 ∀p ∈ V(cid:9). Note 2.7. I(V) is a radical ideal, and is the largest ideal defining V. Proposition 2.8. V = V(I(V)). Proof. Say V = V(I). Since I ⊆ I(V), we have V(I(V)) ⊆ V(I) = V. Take p ∈ V. Need to show ∀g ∈ I(V) that g(p) = 0, which is true by definition of I(V). (cid:3) This shows that I is a right inverse of V. Example 2.9. Going back to our previous example, we should really view V(cid:0)x2+y2(cid:1) in C2 rather than R2: V(cid:0)x2+y2(cid:1) = V(cid:0)(x+iy)(x−iy)(cid:1) = V(x+iy)∪V(x−iy). 2.2. Hilbert’s Nullstellensatz. Theorem 2.10 (Hilbert’sNullstellensatz). Let k = k (i.e., assume k is algebraically closed). There is an order-reversing bijection {radical ideals in k[x ,...,x ]} ←→ (cid:8)affine algebraic subsets of kN(cid:9) 1 N I (cid:55)−→ V(I) I(V) →−(cid:55) V. Remark 2.11. Points in affine space kN correspond to maximal ideals in the polynomial ring k[x ,...,x ]. 1 N 2.3. Irreducible spaces. Definition 2.12. AtopologicalspaceX isirreducible ifX isnottheunionoftwononemptyproper closed sets. Example 2.13. The cofinite topology on R is irreducible. MATH 631 NOTES ALGEBRAIC GEOMETRY 7 2.4. Sept. 10 warmup. • Draw V(xy,xz) ⊆ R3. • Prove Lemma: For I,J ⊆ k[x ,...,x ], 1 N V(I ∩J) = V(I)∪V(J). Proof 1. I ∩J ⊆ I,J =⇒ V(I)∪V(J) ⊆ V(I ∩J). Take p ∈ V(I ∩J). Need p ∈ V(I) or V(J). If p ∈/ V(I), then ∃f ∈ I such that f(p) (cid:54)= 0. Now: ∀g ∈ J, look at fg ∈ IJ. Because p ∈ V(I ∩J), f(p)g(p) = (fg)(p) = 0, hence g(p) = 0 and p ∈ V(J). (cid:3) (cid:16)√ (cid:17) (cid:16)√ (cid:17) Proof 2. V(I ∩J) = V I ∩J = V IJ = V(IJ) = V(I)∪V(J). (cid:3) 2.5. Some commutative algebra. R commutative ring. • I,J radical =⇒ I ∩J radical. • p ⊆ R is prime ⇐⇒ R/p is a domain ⇐⇒ if fg ∈ p, then f ∈ p or g ∈ p. • If R is Noetherian, I radical, then I = p ∩···∩p 1 t uniquely, where the p are prime (irredundant). i 2.6. Review of Hilbert’s Nullstellensatz. The mappings I and V are mutually inverse, giving us an order-reversing bijection I (cid:47)(cid:47) (cid:8)affine algebraic subsets of kN(cid:9) (cid:111)(cid:111) {radical ideals of k[x ,...,x ]}. 1 N V kN ←→ 0 ∅ ←→ (1) = k[x ,...,x ] 1 N {points} ←→ {maximal ideals} (a ,...,a ) ←→ (x −a ,...,x −a ) 1 N 1 1 N N {irreducible algebraic sets} ←→ Speck[x ,...,x ] = {prime ideals} 1 N 2.7. Irreducible algebraic sets. Definition 2.14. An algebraic set V ⊆ kN is irreducible if it cannot be written as the union of two proper algebraic sets contained in V. [If V = V ∪V , then V = V or V = V .] 1 2 1 2 Exercise 2.15. V(I) is irreducible ⇐⇒ I is prime, where I is radical. Observation 2.16. I ⊆ k[x ,...,x ] radical (k not necessarily algebraically closed), write I = 1 N p ∩···∩p , where p are prime (unique!). 1 t i V(I) = V(p )∪···∪V(p ) 1 t are the (unique) irreducible components of V(I). The point is: Proposition 2.17. Every algebraic set in kN is a union of its irreducible components. 8 KARENSMITH 2.8. Aside on non-radical ideals. We also have V(I)∩V(J) = V(I ∪J). However, I ∪J is not usually an ideal, and I +J is not necessarily radical. Non-radical ideals lead into scheme theory: V(y−x2)∩V(y) = V(y−x2,y) = V(y,x2). We should somehow keep track of the multiplicity. 3. Regular functions, regular maps, and categories 3.1. Regular functions. Fix V ⊆ kN algebraic set, k = k. Definition 3.1. A function V −→ k is regular if it agrees with the restriction to V of some polynomial function on the ambient kN. Proposition–Definition 3.2. The set of all regular functions on V has a natural ring structure (where addition and multiplication are the functional notions). This is the coordinate ring of V, denoted k[V]. Example 3.3. On kN, k[kN] = k[x ,...,x ]. 1 N Remark 3.4. (1) k = k =⇒ k is infinite. (2) If k is infinite, then there is no ambiguity in the word “polynomial”. Example 3.5. Consider V(y − x2) ⊆ R2. This is the set of all points (t,t2). The function “y” outputs the y-coordinate (projection to y-axis), and “x2” is the same function in V. Example 3.6. Consider V(xy−1) ⊆ Q2. Is 1 regular? y Yes: 1 = x on V(xy−1). y Observation 3.7. The restriction map gives a natural ring surjection k[x ,...,x ] −→ k[V] 1 N (cid:12) ϕ (cid:55)−→ ϕ(cid:12) V whose kernel is I(V). In particular, k[x ,...,x ] ∼ 1 N k[V] = . I(V) 3.2. Properties of the coordinate ring. The coordinate ring k[V] has the following properties: (1) k[V] is a f.g. k-algebra generated by the images of x ,...,x . 1 N (2) reduced (the only nilpotent element is 0) (3) domain ⇐⇒ V is irreducible. (4) The maximal ideals of k[V] correspond to points of V (need k = k). Note 3.8 (commutative algebra). Maximal ideals in k[V] ∼= k[x ,...,x ]/I(V) correspond to max- 1 N imal ideals in k[x ,...,x ] containing I(V). By the Nullstellensatz, these correspond to points on 1 N V. 3.3. Regular mappings. Definition 3.9. Let V ⊆ kn and W ⊆ km be affine algebraic sets. A regular mapping of affine algebraic sets ϕ : V −→ W is any mapping ϕ which agrees with a polynomial map Ψ on the ambient kn −→ km: Ψ (cid:0) (cid:1) x = (x ,...,x ) (cid:55)−→ Ψ (x),...,Ψ (x) , 1 n 1 m where Ψ are polynomials. i MATH 631 NOTES ALGEBRAIC GEOMETRY 9 Note 3.10. If W = k, then a regular map is a regular function. Note 3.11. We can describe a regular map V −ϕ→ W ⊆ km by giving regular functions ϕ ,...,ϕ ∈ 1 m k[V]: p (cid:55)−→ (cid:0)ϕ (p),...,ϕ (p)(cid:1) ∈ W ⊆ km. 1 m Example 3.12. k −→ V(y−x2) ⊆ k2 t (cid:55)−→ (t,t2) is a regular map from k to V(y−x2). The projection V(y−x2) ⊆ k2 −→ k (x,y) (cid:55)−→ x is the inverse to the map t (cid:55)−→ (t,t2). ϕ Definition 3.13. An isomorphism of affine algebraic sets is a regular map V −→ W which has a ψ regular map W −−→ V inverse: ψ◦ϕ = id and ϕ◦ψ = id . V W Example 3.14. Let V ,V ⊆ kn be linear subspaces (defined by some collection of linear polynomi- 1 2 ∼ als). Then V = V as algebraic sets ⇐⇒ dimV = dimV . 1 2 1 2 Example 3.15 (diagonal map). Give kn×kn coordinates x ,...,x ,y ,...,y . 1 n 1 n kn −−∆→ kn×kn p (cid:55)−→ (p,p) Image is the “diagonal” D = V(x −y ,...,x −y ) ⊆ kn×kn. 1 1 n n The map kn −−∆→ D ⊆ kn×kn is an isomorphism of affine algebraic sets. Example 3.16. X,Y ⊆ kn algebraic sets. View X ⊆ kn with coordinates x ,...,x and Y ⊆ kn 1 n with coordinates y ,...,y . 1 n kn ∆ (cid:47)(cid:47) kn×kn ⊆ ⊆ ∼= (cid:47)(cid:47) X ∩Y (X ×Y)∩D p(cid:55)−→(p,p) 3.4. Category of affine algebraic sets. Key idea: The category of affine algebraic sets over k = k is “the same” (anti-equivalence, duality) as the category of f.g. reduced k-algebras. ϕ Point: Given a regular map V −→ W of affine algebraic sets, there is a naturally induced k- ϕ∗ g algebraic homomorphism k[W] −−→ k[V] given for g ∈ k[W], W −→ k by ϕ (cid:47)(cid:47) g (cid:47)(cid:47) V W (cid:55)(cid:55) k g◦ϕ (cid:0) (cid:1) (cid:0) (cid:1) x = (x ,...,x ) (cid:55)−→ ϕ (x),...,ϕ (x) (cid:55)−→ g ϕ (x),...,ϕ (x) ∈ k[V], 1 n 1 m 1 m where ϕ ,...,ϕ are polynomials in x ,...,x . 1 m 1 n 10 KARENSMITH Theorem 3.17. For k = k, there is an anti-equivalence1 of categories (cid:110)affine algebraic sets over k(cid:111) (cid:110)f.g. reduced k-algebras with(cid:111) ←→ with regular maps k-algebra homomorphisms V (cid:55)−→ k[V] (cid:32) ϕ∗ (cid:33) ϕ k[W] −−→ k[V] (V −→ W) (cid:55)−→ g (cid:55)−→ g◦ϕ k[x ,...,x ] kn ⊇ V(I) →−(cid:55) R ∼= 1 n . I Proof. Note 3.18. The assignment V (cid:55)−→ k[V] is functorial: Given f (cid:47)(cid:47) g (cid:47)(cid:47) V W (cid:54)(cid:54) X, h there is f∗,g∗,h∗ and a commutative diagram (cid:111)(cid:111) f∗ (cid:111)(cid:111) g∗ k[V] (cid:106)(cid:106) k[W] k[X], h∗ i.e., (g◦f)∗ = f∗◦g∗. (Make sure this is obvious to you.) Problem: Given a reduced, f.g. k-algebra R, how to cook up V? Fix a k-algebra presentation for R: k[x ,...,x ] 1 n R = . I Because R is reduced, I is radical. Let V = V(I) ⊆ kn. By the Nullstellensatz, I(V(I)) = I, so k[x ,...,x ] k[x ,...,x ] ∼ 1 n 1 n k[V] = = = R. I(V) I What about homomorphisms of k-algebras? ϕ (cid:47)(cid:47) R S ϕ (cid:47)(cid:47) k[y ,...,y ]/I k[x ,...,x ]/J 1 m 1 n Let ϕ = ϕ(y ) ∈ k[V] for i = 1,...,m. This uniquely defines ϕ. i i Need to construct kn ⊇ V(J) −−Ψ→ V(I) ⊆ km (cid:0) (cid:1) x = (x ,...,x ) (cid:55)−→ ϕ (x),...,ϕ (x) . 1 n 1 m We have that Ψ is a map V −→ km. Need to check that (1) the image is in W, (2) Ψ∗ = ϕ. 1An anti-equivalence of categories C,D is an equivalence of C and the opposite category Dop.