MIT OpenCourseWare http://ocw.mit.edu 18.727 Topics in Algebraic Geometry: Algebraic Surfaces Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.� � ALGEBRAIC SURFACES, LECTURE 3 LECTURES: ABHINAV KUMAR 1. Birational maps continued Recall that the blowup of X at p is locally given by choosing x, y ∈ mp, letting U be a sufficiently small Zariski neighborhood of p (on which x and y are regular functions that vanish simultaneously only at the point p), and defining U˜ by xY − yX = 0 in U × P1 . If, for some q ∈ U, q = p, x(q) = 0, then Y = y X and x similarly if y ∈/ mq. So we obtain an isomorphism U˜ U at q, and U˜ π U fails→ →to be an isomorphism only at p, where π−1(p) = P1 is the exceptional divisor E. Note that the blowup X˜ does not depend on the choice of x, y. Proposition 1. If C is a curve passing through p ∈ X with multiplicity m, (1) π∗C = C˜+ mE Proof. Choose local coordinates x, y in a neighborhood of p s.t. y = 0 is not tangent to any branch of C at p. Then in Oˆ x,p we can expand the equation of C in a power series (2) f = fm(x, y) + fm+1(x, y) + · · · with fm(1, 0) = 0 and each fk a homogeneous polynomial of degree k. In a� ˜neighborhood of (p, = [1 : 0]) U U P1, we have local coordinates x y ∞ ∈ ⊂ ×and t = x and π∗f = f(x, tx) = xmfm(1, t) + xm+1fm+1(1, t) + ), giving the · · · desired formula. � Theorem 1. We have maps ˜ π∗ : Pic X → ˜ → Pic X, 1 � E givingPic X and Z ˜ = Pic X⊕Z. → rise to an isomorphism Pic X ∼If C, D ∈ Pic X, (π∗C) (π∗D) = C D,· ·while (π∗C) E = 0 and E E = −1. We further have that K ˜= π∗KX + E, so · · X 2K X˜ = (π∗KX )2 − 1. Proof. Note that Pic X ∼) ∼˜ Pic X˜= Pic (X�{p} = Pic (X�E) and we have Z → →Pic (X˜ � E) → 0. The first map is injective because E2 = −1, and π∗ splits the sequence to give the desired isomorphism. For the intersection formulae, move C, D so that they meet transversely and do not pass through p. Because π∗ is an isomorphism X˜ � E X � {p}, we get an equality of intersection numbers → 12 LECTURES: ABHINAV KUMAR as desired. Moreover, since C (possibly after moving) does not pass through p, (π∗C) E = 0. Next, taking a curve passing through p with multiplicity · 1, its strict transform meets E transversely at one point which corresponds to the tangent direction of p ∈ C, i.e. C˜ E = 1 and C˜= π∗C − E. Since · (π∗C) E) = 0, we get 1 = C˜ E = (π∗C − E) E = −E2 as desired. Finally, to · · · show the desired result about canonical divisors, we use the adjunction formula −2 = 2(0) − 2 = E(E + K ˜) = −1 + E K ˜= E KX˜= −1. By the previous X X· ⇒ · proposition, KX˜= π∗KX + nE = n = 1 (by taking intersection with E).⇒Note that we can see this latter fact more directly. Letting ω = dx ∧ dy be the top differential in local coordinates at p, then π∗ω = dx ∧ d(xt) = xdx ∧ dt = ⇒π∗KX + E = KX˜. � 1.1. Invariants of Blowing Up. iTheorem 2. πX = X and R πX = 0 for i > 0, so the two structure ∗O ˜O ∗O ˜sheaves have the same cohomology. Proof. π is an isomorphism away from E, π : X˜ � E → X � {pi}, so it is clear that OX → π∗O ˜ is an isomorphism except possibly at p, and R π∗OX˜ can only X be supported at p. By the theorem on formal functions, the completion at p of this sheaf is R�= lim Hi(E , ), where E is the closed subscheme iπ∗OX˜n OEn n defined on X by , ideal sheaf of E. We obtain an exact sequence 0 In/In+1 ˜ In I the ←− 2 In/In+1 → → OEn+1 → OEn → 0 with I/I= OE (1) = ⇒ = OE (n).∼ ∼Since E ∼= P1, we have Hi(E, OE(n)) = 0 for n, i > 0. Using the long exact sequence in cohomology, we find that Hi(En, OEn ) = 0 for all i > 0, n ≥ 1, so the above inverse limit vanishes. Riπ O ˜is concentrated at p and thus equals ∗ X its own completion, giving the desired vanishing of higher direct image sheaves. Also, π∗OX˜= π∗OX follows from the fact that X is normal and π is birational (trivial case of Zariski’s main theorem). The final statement follows from the spectral sequence associated to Hi and Riπ . �∗ This implies that the irregularity qX = h1(X, OX ) = q ˜and geometric genus X pg(X) = h2(X, OX ) = pg(X˜) are invariant under blowup. 2. Rational maps Let X, Y be varieties, X irreducible. Definition 1. A rational map X ��� Y is a morphism φ from an open subset U of X to Y . Note that if two morphisms U1, U2 → Y agree on some V ⊂ U ∩ U2, they agree on U1 ∩ U2, and thus each rational map has a unique maximal domain U. We say that φ is defined at x ∈ X if x ∈ U. Proposition 2. If X is nonsingular, Y projective, then X � U has codimension 2 or larger.3 ALGEBRAIC SURFACES, LECTURE 3 Proof. If φ is not defined on some irreducible curve C, then OC,X gives us a valuation vC : k(X) → Z. Let φ be given by (f0 : · · · : fn) with fi ∈ K(X) s.t. at least one fj has a pole along C. Take the fi s.t. v(fi) is the smallest, and divide by it. Then φ is defined on the generic point of C, a contradiction. � In particular, if X is a smooth surface and Y is projective, a rational map is defined on all but finitely many points F (those lying on the set of zeroes and poles of φ). If C is an irreducible curve on X, φ is defined on C � (C ∩F ), and we can set φ(C) = φ(C � (C ∩ F )) (and similarly φ(X) = φ(X � F )). Restriction gives us an isomorphism between Pic (X) and Pic (X � F ), so we can talk about the inverse image of a divisor D (or line bundle, or linear system) under φ. 3. Linear Systems For a divisor D, D is the set of effective divisors linearly equivalent to D, i.e. P(H0(X, …