Math 6130 Notes. Fall 2002. 4. Projective Varieties ad their Sheaves of Regular Fuctios. These are the geometric objects associated to the graded domais: C[x 0,x 1,..., x ]/P (for homogeeous primes P ) defiig global complex algebraic geometry. Defiitio: (a) A subset V CP is algebraic if there is a homogeeous ideal I C[x 0,x 1,..., x ] for which V = V (I). (b) A homogeeous ideal I C[x 0,x 1,..., x ]isradical if I = I. Propositio 4.1: (a) Every algebraic set V CP is the zero locus: V = {F 1 (x 1,..., x )=F 2 (x 1,..., x )=... = F m (x 1,..., x )=0} of a fiite set of homogeeous polyomials F 1,..., F m C[x 0,x 1,..., x ]. (b) The maps V I(V ) ad I V (I) CP give a bijectio: {oempty alg sets V CP } {homog radical ideals I x 0,..., x } (c) A topology o CP, called the Zariski topology, results whe: U CP is ope Z := CP U is a algebraic set (or empty) Proof: Note the similarity with Propositio 3.1. The proof is the same, except that Exercise 2.5 should be used i place of Corollary 1.4. Remark: As i 3, the bijectio of (b) is iclusio reversig, i.e. V 1 V 2 I(V 1 ) I(V 2 ) ad irreducibility ad the features of the Zariski topology are the same. Example: A projective hypersurface is the zero locus: V (F )={(a 0 : a 1 :... : a ) CP F (a 0 : a 1 :... : a )=0} CP whose irreducible compoets, as i 3, are obtaied by factorig F, ad the basic ope sets of CP are, as i 3, the complemets of hypersurfaces. 1
Defiitio: A projective variety is a irreducible closed subset X CP. The Zariski topology o X is iduced from the Zariski topology o CP ad the hypersurfaces ad basic ope sets i X are the (proper) itersectios of X with hypersurfaces i CP ad their complemets i X, respectively. The homogeeous coordiate rig is the domai C[X] :=C[x 0,x 1,..., x ]/P (which is ow graded!) where P is the homogeeous prime ideal I(X). Observatio: Noempty closed subsets Z X correspod to homogeeous radical ideals I x 0,..., x C[X], ad irreducible closed subsets i X correspod to homogeeous prime ideals P x 0,..., x C[X]. But ow what should a regular fuctio o a projective variety X be? Costat fuctios we uderstad, but the o-costat elemets of C[X] are ot fuctios o X because the value: F (a 0 :... : a ) always depeds upo the choice of a represetative of the class (a 0 :... : a ) (but we do kow whether or ot F (a 0 :... : a )=0whe F is homogeeous). Defiitio: A ratioal fuctio o X is a elemet of degree zero i the field of fractios of C[X]. That is, φ is a ratioal fuctio if φ = F ad G F,G C[X] are homogeeous of the same degree. If G(a 0 :... : a ) 0 for some such ratio represetig φ the φ(a 0 :... : a ) C is well-defied, ad we say that φ is regular at (a 0 :... : a ) Observatio: The ratioal fuctios o X form a field, deoted C(X). Examples: (a) The field of ratioal fuctios o CP itself is: ( x1 C,..., x ) ( x0 =... = C,..., x ) ( x0 =... = C,..., x ) 1 x 0 x 0 x i x i x x C(x 0,..., x ) (b) Let F (x 0,..., x )=x 0 x 1 G(x 2,..., x ) be a quadratic homogeeous (irreducible) polyomial, ad let X = V (F ) CP. The: ( x2 C(X) =C,..., x ) ( x2 = C,..., x ) x 0 x 0 x 1 x 1 Clearly those fields are subfields of C(X), ad if A C(X), the we ca B remove all occurreces of x 1 (resp of x 0 ) by multiplyig top ad bottom by a sufficietly high power of x 0 (resp. x 1 ) ad usig the equatio. 2
This, of course, begs a iterestig questio: Questio: Are such projective hypersurfaces isomorphic to CP 1? We will aswer this i the affirmative i case = 2 ad i the egative i the other cases. We will see, however, i 7 that these quadric hypersurfaces always have a ope (dese) subset which is isomorphic to a ope (dese) subset of CP 1. We eed for ow to figure out what isomorphic meas. Defiitio: The sheaf O X of regular fuctios o a projective variety is: O X (U) ={φ C(X) φ is regular at all poits of U} C(X) This is the same defiitio as i 3, O X is agai a sheaf of commutative rigs with 1, ad if a ratioal fuctio is zero (as a fuctio) o a ope set, the agai it is zero i C(X) (same proof as Propositio 3.4). But recall that i the affie case there were lots of regular fuctios (ie. all of C[X]). I the projective case, by cotrast, there are very few: Propositio 4.2: The oly ratioal fuctios that are regular at all poits of a projective variety X are the costats (i.e. O X (X) =C C(X)). Proof: If the projective Nullstellesatz were the ordiary Nullstellesatz, this would be easy. Give φ C(X), the homogeeous deomiators i the expressios φ = F geerate a ideal I G φ, ad the ay (homogeeous) elemet of I φ appears as a deomiator, as i Propositio 3.3. By assumptio V (I φ )=, ad the ordiary Nullstellesatz would imply that 1 appears as a deomiator, ad the φ = a is a costat. But this requires more argumet, 1 sice applyig the projective Nullstellesatz (Corollary 2.1) oly gives us: φ = F 0 x N 0 = = F x N for some N ad homogeeous polyomials F i of degree N. We ca coclude from this that if d ( + 1)(N 1) + 1 ad if G C[X] d is arbitrary, the Gφ C[X] d (because each moomial i a represetative of G must have some x N i i it!) Now iterate this to coclude that Gφ m C[X] d for all m, ad thus: C[X] C[X]+φC[X] C[X]+φC[X]+φ 2 C[X] G 1 C[X] is a sequece of submodules of the fiitely geerated (i fact, pricipal!) C[X]-module G 1 C[X], which evetually stabilizes sice C[X] is Noetheria. 3
Thus for some m, wehave: φ m = f m 1 φ m 1 + f m 2 φ m 2 +... + f 0 for f 0,..., f m 1 C[X]. But this idetity must also hold whe we replace the f i by their degree 0 compoets, ad the φ satisfies a (moic) polyomial equatio with costat coefficiets. Sice C is algebraically closed, it follows that the polyomial factors, so that m i=1 (φ r i ) = 0 for some r i C. But if φ = F, the φ(a G 0 :... : a )=r i F (a 0 :... : a )=r i G(a 0 :... : a ), so the locus where φ r i is closed. Sice X is irreducible, it follows that oe of these loci must be all of X, ad the φ = r i C(X) for that r i C. Remark: Recall that the oly holomorphic fuctios o a compact complex maifold are the costats, sice o the oe had a cotiuous fuctio o a compact maifold must have a (global) maximum, ad o the other had, the absolute value of ay holomorphic fuctio is subharmoic meaig that it ca oly have a local maximum if it is locally costat! We will see i 6 that projective varieties are, i a very precise sese, aalogues i algebraic geometry of compact maifolds. Defiitio: A ope subset W X of a projective variety with sheaf: O W (U) :=O X (U) is a quasi-projective variety (which we will shorte to variety). A regular map is, as always, a cotiuous map that pulls back regular fuctios. Corollary 4.3: The oly regular maps Φ : X Y from a projective variety X to a affie variety Y are the costat maps. Proof: Suppose Y C has coordiate rig C[y 1,..., y ]/P. The the regular fuctios y 1,..., y O Y (Y ) must pull back to regular fuctios o X, which are costat by Propositio 4.2. If we let Φ (y i )=b i C, the Φ(a 0 :... : a )=(b 1,..., b ) (idepedet of (a 0 :... : a ) X) sice the ith coordiate is y i (Φ(a 0 :... : a )) = Φ (y i )((a 0 :... : a )) = b i. O the other had, there are plety of iterestig regular maps from oe projective variety to aother projective variety. The situatio is similar to but more subtle tha Propositio 3.5 for affie varieties. 4
Defiitio: If X CP m ad Y CP m are projective (or quasi-projective) varieties, the a mappig from a ope (dese) domai U X: Φ:X >Y (the broke arrow idicates that U may ot be all of X) isaratioal map if, for each (a 0 :... : a ) U, there is a degree d ad homogeeous polyomials F 0,..., F m C[x 0,x 1,..., x ] d such that ot every F i (a 0 :... : a ) = 0 ad: Φ(b 0 :... : b )=(F 0 (b 0 :... : b ):... : F m (b 0 :... : b )) for all (b 0 :... : b ) i the eighborhood X m i=0 V (F i) U of (a 0 :... : a ). Remark: If b =(b 0 :... : b ) CP, the (F 0 (b) :... : F m (b)) CP m is well-defied for b CP, provided that the F i are all homogeeous of the same degree ad ot all of the F i (b) are zero, sice i that case: (F 0 (λb) :... : F m (λb)) = (λ d F 0 (b) :... : λ d F m (b)) = (F 0 (b) :... : F m (b)) The subtlety here lies i the fact that the choice of F 0,..., F m may deped upo the poit (a 0 :... : a ), so that the domai U of Φ is usually larger tha X m i=0 (F i) for ay sigle set of F 0,..., F m. Example (The Projective Coic) The coic C := V (y 2 xz) CP 2 is isomorphic to CP 1, i the sese that there are ratioal maps: Φ:C > CP 1 ad Ψ : CP 1 >C that are defied everywhere ad are two-sided iverses of each other. The map Ψ is easy to costruct: Ψ(a : b) =(a 2 : ab : b 2 ) which is defied o all of CP 1 ad maps to C. The map Φ is i two parts: (a : b) oc (V (x) V (y)) Φ(a : b : c) = (b : c) oc (V (y) V (z)) ad oly take together do we see that C is the domai sice the poit (0:0:1) C missed by the first defiitio is covered by the secod, ad (a : b) =(b : c) wheever both defiitios apply, as a cosequece of y 2 = xz. Fially, oce easily checks that Φ ad Ψ are iverses of each other. 5
Here is the (quasi-)projective versio of Propositio 3.5: Propositio 4.4: If U U CP is a quasi-projective variety, the the regular maps Φ : U W W CP m to aother quasi-projective variety are the ratioal maps Φ : U > CP m with U i the domai ad Φ(U) W. Proof: Suppose Φ is a ratioal map, give locally by Φ = (F 0 :... : F m ). The Φ is cotiuous o ope sets i U m i=0 V (F i) sice Φ 1 (V (G)) = V (G(F 0,..., F m )), ad pulls back regular fuctios at b W to regular fuctios at each preimage of b, via ( ) G Φ = G(F 0,..., F m ) H H(F 0,..., F m ) (compare with Propositio 3.5). Thus Φ is a regular map, by defiitio! Coversely, give a regular map Φ : U W ad (a 0 :... : a ) U with Φ(a 0 :... : a )=(b 0 :... : b m ) W, the some b i 0 ad the ratioal fuctios y j C(W ) pull back to F j C(X) with each G y i G j (a 0 :... : a ) 0. j Φ ca the be writte: Φ=(F 0 G j :... : F i 1 G j : G j : F i+1 G j :... : F m G j ) j 0 j i 1 j j i+1 j m valid i a eighborhood of (a 0 :... : a ), fiishig the proof! Remark: The sheaf defiitio of a regular map is superior to the ratioal map defiitio because it is itrisic to the variety, ad does t deped upo the way U sits i CP (though it is easier to work with ratioal maps). It ca be tricky to fid the precise domai of a ratioal map i geeral, but whe X is projective space, we do have the followig: Propositio 4.5: If Φ : CP > CP m is a ratioal map ad F i Φ=(F 0 :... : F m ) o the ope set U = CP m i=0 V (F i) ad if the F i have o commo factor, the the domai of Φ is exactly U. Proof: Give aother expressio Φ = (G 0 :... : G m ), the some G i 0, ad F j because both are equal to Φ ( y j y i ). Thus G j F i = F j G i, ad the = G j G i sice the F i have o commo factor, every G j = HF j for some fixed H. But the i=0 V (G i)= i=0 V (F i) V (H) ad the Propositio is proved. 6
Corollary 4.6: If m<, the the oly regular maps: Φ:CP CP m are the costat maps. Proof: By Propositio 4.5, such a map would be give by: Φ=(F 0 :... : F m ) for oe choice of F 0,..., F m, but by Exercise 2.3, we kow that m i=0 V (F i) whe m<ad deg(f i ) > 0, so such a ratioal map caot be regular! We ve discussed maps from projective varieties to affie varieties ad to other projective varieties. This leaves us with maps from affie varieties to projective varieties, which are the most fudametal, as they allow us to regard a projective variety as a maifold which is locally affie: Propositio 4.7: A projective variety is always covered by affie varieties. Precisely, for a projective variety X CP, the particular basic ope sets: U i := X V (x i ) are either empty or isomorphic to affie varieties, ad X = i=0 U i. Coversely, if Y C is a affie variety, the the closure X = Y CP (addig the poits at ifiity) is a projective variety, for the iclusio: Y C CP ; (a 1,..., a ) (1 : a 1 :... : a ) ad the Y = U 0, so i particular every affie variety is quasi-projective. Proof: For the first part, by symmetry it suffices to assume i =0. If X = V (P ) CP for a homogeeous prime P C[x 0,x 1,..., x ], we defie Y := V (d 0 (P )) C for the dehomogeized ideal d 0 (P ) (see 2). The: U 0 = {(a 0 : a 1 :... : a ) X a 0 0} CP ad the bijectio: Φ : Y U 0 X; (a 1,..., a ) (1 : a 1 :... : a )isa isomorphism, with iverse (a 0 :... : a ) ( a 1 a 0,..., a a 0 ), as you may check. Ad sice i=0 U i = X i=0 V (x i) ad i=0 V (x i) = 0, we get the first part. For the secod part, we oly eed to kow that: Y C = Y But if P = I(Y ) C[ x 1 x 0,..., x x 0 ] is the ideal of Y, the the homogeized ideal h(p ) satisfies d 0 (h(p )) = P,soV(h(P)) CP is a closed set cotaiig Y, whose itersectio with C is Y, ad it follows that Y C = Y, as desired. 7
Example (Projectivizig a Affie Hypersurface): If we are give: V (f) C for f C[x 1,..., x ] of degree d the as i 2, we should reiterpret: f = f ( x1,..., x ) [ x1 C,..., x ] x 0 x 0 x 0 x 0 ad the homogeize to get: ( V (f) =V (F ) CP for F (x 0,..., x )=x d 0 f x1,..., x ) x 0 x 0 which is the set-theoretic uio: V (f) V (F (0,x 1,..., x )) C CP 1 = CP of the origial affie hypersurface ad a hypersurface at. We ca the cosider the cover of V (F ) by ope affie hypersurfaces: V (F )=V(f) V (f i ) for f i = F i=1 ( x0,..., x ) x i x i For example, cosider the affie elliptic curve (for ay λ 0, 1): E 0 := V (y 2 x(x 1)(x λ)) C 2 If we thik of x = x 1 x 0 ad y = x 2 x 0, we get the closure: E = V (x 2 2 x 0 x 1 (x 1 x 0 )(x 1 λx 0 )) = E 0 (0:1:0) CP 2 ad the E is covered by E 0 ad E 2 = E {(1:0 :0),(1:1:0),(1 : λ :0)} = ( x0 V x 1 ( x 1 x 0 )( x 1 λ x ) 0 ) x 2 x 2 x 2 x 2 x 2 x 2 8
Exercises 4. 1. (a) Describe the Zariski topology o CP 1. (b) Describe the Zariski topology o the elliptic curve E. (c) What are the irreducible closed sets for the Zariski topology o CP 2? 2. (a) Prove that the regular map from CP 1 to the ratioal ormal curve: (s : t) (s d : s d 1 t :... : t d 1 s : t d ) (see Exercise 2.2) is a isomorphism (geeralizig the coic example). (b) Prove that the regular map Φ : CP 1 CP 2 : (s : t) (s 3 : st 2 : t 3 ) is a homeomorphism to X = V (x 3 1 x 0x 2 ) but ot a isomorphism to X. 3. Cosider m + 1 liear forms y 0,..., y m C[x 0,x 1,..., x ] 1. (a) If m ad the y i are liearly idepedet, describe the domai of the ratioal map: Φ:CP > CP m ;Φ(a) =(y 0 (a) :... : y m (a)) ad for each b CP m, describe Φ 1 (b) (i the domai). (b) If m ad the forms spa C[x 0,x 1,..., x ] 1, describe the image of Φ ad prove that Φ is a isomorphism from CP to its image i CP m. (c) If Φ : CP CP ; Φ = (F 0 :... : F ) is ay regular map such that the F i have o commo factors, the show each pair F i,f j is relatively prime, ad eve more, that if G C[x 0,..., x ] e is ay polyomial that is ot divisible by x i, the gcd(f i,g(f 0,..., F )) = 1. Coclude that if d>1, the each: ( F1 C,..., F ) ( x1 C,..., x ) F i F i x i x i is a o-trivial field extesio, hece that the oly automorphisms of CP (i.e. isomorphisms from CP to itself) are give by + 1 liear forms. (d) Fid a isomorphism of groups: Aut(CP ) = PGL( +1, C) betwee the group Aut(CP ) of regular automorphisms of CP ad the group PGL( +1, C) of ivertible +1 + 1 matrices modulo scalars. 9
4. (a) If Φ : CP > CP m is oe of the ratioal maps of Exercise 3(a), prove that there is a automorphism α Aut(CP ) such that: (Φ α)(x 0 :... : x )=(x 0 :... : x m ) These maps are called the stadard projectios. (b) If Φ : CP CP m is oe of the regular maps from Exercise 3(b), prove that there are automorphisms α Aut(CP ) ad β Aut(CP m ) such that: (β Φ α) =(x 0 :... : x :0:... :0) (c) Prove that if y C[x 0,x 1,..., x ] 1 is ay ozero liear form ad X CP is a projective variety, the the basic ope subset U = X V (y) is either empty or else isomorphic to a ope dese affie variety. 5. (a) Prove the aalogue of Exercise 3.5 (b,c) for quasi-projective varieties. (b) Prove that CP is ot isomorphic to ay locally closed subset of CP. 6. Whe α GL(+1, C) acts o the space of liear forms C[x 0,x 1,..., x ] 1, the α also acts o each space C[x 0,x 1,..., x ] d by: α(x i 1 0...x i )=α(x 0) i 1...α(x ) i (a) Prove that for ay α ad F, the hypersurfaces V (F ) ad V (α(f )) are isomorphic. (We say that V (F )isprojectively equivalet to V (α(f )).) A quadratic homogeeous polyomial F C[x 0,x 1,..., x ] 2 ca be writte i a symmetric matrix form as: (a ij ) i a ii x 2 i + i<j a ij x i x j (b) Prove the actio of α traslates to cojugatio i this case: α T (a ij )α α( i a ii x 2 i + i<j a ij x i x j ) ad the prove that every quadric hypersurface V (F ) CP is projectively equivalet (hece isomorphic) to exactly oe of the followig: V (x 0 x 1 + x 2 2 +... + x2 i ) for i 2 (so C(X) = C(CP 1 ) for all quadric hypersurfaces). Defiitio: If i = above, the quadric V (F ) CP is o-degeerate. 10