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 su ciently small Zariski neighborhood of p on which x and y are regular functions that vanish simultaneously only at the point p and de ning U by p x q 0 then Y xy X and xY yX 0 in U P1 If for some q U q similarly if y mq So we obtain an isomorphism U U at q and U U fails to be an isomorphism only at p where 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 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 O 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 and t xy and f f x tx xm fm 1 t xm 1 fm 1 1 t giving the desired formula and Z Pic X 1 Theorem 1 We have maps Pic X Pic X E giving rise to an isomorphism Pic X Pic X Z If C D Pic X C D C D while C E 0 and E E 1 We further have that KX KX E so 2 KX KX 2 1 E and we have Z Pic X Proof Note that Pic X Pic X p Pic X Pic X E 0 The rst map is injective because E 2 1 and 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 is an isomorphism X E X p we get an equality of intersection numbers 1 2 LECTURES ABHINAV KUMAR as desired Moreover since C possibly after moving does not pass through p 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 C E Since C E 0 we get 1 C E C E E E 2 as desired Finally to show the desired result about canonical divisors we use the adjunction formula 2 2 0 2 E E KX 1 E KX E KX 1 By the previous proposition KX KX nE n 1 by taking intersection with E Note that we can see this latter fact more directly Letting dx dy be the top di erential in local coordinates at p then dx d xt xdx dt KX E KX 1 1 Invariants of Blowing Up Theorem 2 OX OX and Ri OX 0 for i 0 so the two structure sheaves have the same cohomology Proof is an isomorphism away from E X E X p so it is clear that OX OX is an isomorphism except possibly at p and Ri OX can only be supported at p By the theorem on formal functions the completion at p i i O of this sheaf is R X lim H En OEn where En is the closed subscheme by I n I the ideal sheaf of E We obtain an exact sequence 0 de ned on X n n 1 I I OEn 1 OEn 0 with I I 2 OE 1 I n I n 1 OE n 1 i Since E P we have H E OE n 0 for n i 0 Using the long exact sequence in cohomology we nd that H i En OEn 0 for all i 0 n 1 so the above inverse limit vanishes Ri OX is concentrated at p and thus equals its own completion giving the desired vanishing of higher direct image sheaves Also OX OX follows from the fact that X is normal and is birational trivial case of Zariski s main theorem The nal statement follows from the spectral sequence associated to H i and Ri This implies that the irregularity qX h1 X OX qX and geometric genus pg X h2 X OX pg X are invariant under blowup 2 Rational maps Let X Y be varieties X irreducible De nition 1 A rational map X Y is a morphism 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 is de ned at x X if x U Proposition 2 If X is nonsingular Y projective then X U has codimension 2 or larger ALGEBRAIC SURFACES LECTURE 3 3 Proof If is not de ned on some irreducible curve C then OC X gives us a valuation vC k X Z Let 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 is de ned on the generic point of C a contradiction In particular if X is a smooth surface and Y is projective a rational map is de ned on all but nitely many points F those lying on the set of zeroes and poles of If C is an irreducible curve on X is de ned on C C F and we can set C C C F and similarly X 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 3 Linear Systems For a divisor D D is the set of e ective divisors linearly equivalent to D i e P H 0 X OX D A hyperplane in this projective space pulls back to give a divisor equivalent to D For f H 0 OX D let D be the divisor of zeroes of f A complete linear system is such a space P H 0 X OX D while a linear system P is simply a linear subspace of such a system One dimensional linear systems are called pencils A component C of P is called xed if every divisor of P contains C i e all the elements of the corresponding subspace of H 0 X OX D vanish along C The xed part of P is the biggest divisor F which is contained …