Lecture 9: Chow’s Lemma, Blowups

Lecture 9 18.725 Algebraic Geometry I Lecture 9: Chow’s Lemma, Blowups Last time we showed that projective varieties are complete. The following result from Wei-Liang Chow gives a partial converse. Recall that a birational morphism between two varieties is an isomorphism on some pair of open subsets. Lemma 1 (Chow’s Lemma). If X is a complete, irreducible variety, then there exists a projective variety X̃ that is birational to X. Proof. This proof is a standard one. Here we follow the proof presented by [SH77]. Choose an affine covering X = U1 ∪ . . . ∪ Y Un , and let Yi ⊇ Ui be\ projective Y varieties containing Ui as open subsets. Now consider n ∆:U →U → Ui → Y where U = Ui , Y = Yi , and φ : U → X × Y be induced by the standard i i i ˜ be the closure of φ(U ), and π1 gives a map f : X ˜ → X. This map is inclusion U → X and ∆. Let X birational because f −1 (U ) = φ(U ), and on U the map π1 ◦ φ is just identity. (To see the first claim, note ˜ = φ(U ), i.e. φ(U ) is closed in U × Y , which is true because φ(U ) in U × Y is just that it means (U × Y ) ∩ X the graph of ∆, which is closed as Y is separated.) ˜ is projective. We show this by showing that the restriction of π2 : X×Y → Y So it remains to check that X ˜ which we write as g : X ˜ → Y , is a closed embedding. Let Vi = p−1 (Ui ), where pi is the projection to X, i ˜ which easily follow from the statement that map from Y to Yi . First we claim that π2−1 (Vi ) cover X, π2−1 (Vi ) = f −1 (Ui ), since Ui cover X. Consider W = f −1 (U ) = φ(U ) as an open subset in f −1 (Ui ): on W we have f = pi g, so the same holds on f −1 (Ui ) and the covering property follows. ˜ ∩ Vi → Ui are closed embeddings. Noting that Vi = Y1 × . . . × Yi−1 × Ui × It remains to show that X pi Yi+1 × . . . × Yn , we write Zi to denote the graph of Vi −→ Ui ,→ X, and note that it is closed and isomorphic ˜ ∩ Vi → Ui to Vi via projection. Noting that φ(U ) ⊆ Zi and that Zi is closed, taking closure we see that X is closed in Zi . cn = Bl0 (An ) ⊆ Blowing up of a point in An The blow-up of the affine n-space at the origin is defined as A n n−1 n n−1 A ×P = {(x, L) : x ∈ A , L ∈ P , x ∈ L}. It is a variety defined by equations xi tj = xj ti . We have a n n c projection π : A → A . Atop 0 there is an entire Pn−1 , and on the remaining open set the projection is an isomorphism. Now consider X an closed subset of An , such that {0} is not a component. The proper transform of ˜ is the closure of the preimage of X \ 0 under π. Suppose X X (a.k.a. the blowup of X at 0), denoted X, −1 n−1 ˜ ˜ because dim(Pn−1 ) ≥ dim(X). ˜ If X is contains 0, then π (X) = X ∪ P . If X ( An , then Pn−1 6⊆ X −1 n−1 ˜ ˜ irreducible, then X is the irreducible component of π (X) other than P . The preimage of 0 within X is called the exceptional locus. cn is covered by n affine charts. More explicitly, A cn i ⊆ An−1 × An has coNext, observe that A i i i i i ordinates (t1 , . . . , ti−1 , ti+1 , . . . , tn ). On there, the defining equation becomes xj = tij xi for j 6= i, so cn i ∼ A = An with coordinates (ti1 , . . . , tii−1 , xi , tii+1 , . . . , tin ). In other words, if P (x1 , . . . , xn ) ⊆ IX , then i P (t1i xi , . . . , ti−1 xi , xi , . . .) ⊆ IX∩A n . ˜ c i Example 1. Let X = (y 2 = x3 + x2 ) ⊆ An . Suppose y = tx, then t2 x2 = x3 + x2 =⇒ t2 = x + 1, so the ˜ → X is not 1-to-1, though preimage of (0, 0) is {(t = ±1, x = 0)}. Thus X is not normal because the map X ˜ → X) = 1 (recall that a finite birational morphism to a normal variety is isomorphism). deg(X ˜ x to denote X ˜ for an embedding Definition 1. Let X an affine variety, x ∈ X, we write Blx (X) = X X ⊆ An where x 7→ 0. Remark 1. Blx (X) contains X \ x as an open set, so this generalizes to any variety X. Proposition 1. Suppose X embeds via two embeddings i1 , i2 to An and Am respectively, such that there exists some x such that i1 (x) = i2 (x) = 0, then X˜1 = X˜2 for two blowups at x. In particular, this tells us that blowup is an intrinsic operation that does not depend on the embedding. 1 Lecture 9 18.725 Algebraic Geometry I Proof. First consider the special case X = An , i1 = id, and i2 given by (x1 , . . . , xn ) 7→ (x1 , . . . , xn , f ) for n+1 n [ [ n+1 = n+1 \ {(0 : 0 : . . . : 0 : 1) ∈ Pn }. [ [ some polynomial f . Write A Ain+1 , and observe that An+1 =A i i=1 i=1 ∼ An ⊆ A cn (Locally write Call that point ∞, then one can check that ∞ ∈ / A˜n . Now note that A˜n ∩ An+1 = i i it as tn+1 xi = f (t1 xi , . . . , xi , . . . , tn xi ), and observe we have a xi on both sides so the closure would be of shape tn+1 = f 0 (t1 , . . . , xi , . . . , tn ), which gives an entire An ), so together we see that the blowup is nothing cn . Second, consider X = An , i1 = id, i2 : An ,→ An+m being a graph of a morphism An → Am . but A This can be reduced to the first case by induction on m (or really, just the exactly same argument applied several times). Now consider the general case of arbitrary i1 , i2 . First extend the embedding i2 : X → Am to a map An → Am by lifting each generator (one can switch to the algebraic side, suppose X = Spec A, then we get two surjective maps ψ1 : k[x1 , . . . , xm ] → A and ψ2 : k[y2 , . . . , yn ] → A, lift ψ1 to ψ2 ◦ φ for φ : k[x1 , . . . , xm ] → k[y1 , . . . , yn ] where we map each xi into A then lift), then one can use part 2. (x 7→ i1 (x) 7→ i1 (x) has the same blowup as x 7→ i1 (x) 7→ (i1 (x), i2 (x)), which has the same blowup as x 7→ i2 (x) 7→ i2 (x) by the same argument applied on the other direction.) As an application, consider an example of a complete non-projective surface: start with P1 × P1 , blow it up at (0, 0), consider the projection to the second factor. For any x 6= 0, the preimage of x is a projective line; for x = 0, the preimage is the union of two projective lines (one can see this by passing to affine chart then consider closure). Consider two copies of this blow up, call them X, Y , and call the two exceptional lines L1 , L2 for both of them, Now consider the disjoint union of X and Y where we identify L1 of X with the fiber of ∞ of Y , and vise versa. References [SH77] Igor Rostislavovich Shafarevich and Kurt Augustus Hirsch. Basic algebraic geometry. Vol. 1. Springer, 1977. 2