COMPLEX ALGEBRAIC SURFACES CLASS 7 RAVI VAKIL CONTENTS 1. How basic aspects of surfaces change under blow-up 1 2. Rational maps of surfaces, linear systems, and elimination of indeterminacy 3 3. The universal property of blowing up 4 3.1. Applications of the universal property of blowing up 4 Recap of last time. Last time we began discussing blow-ups: ¼ ¼ Ô ¾ Ë Ë = BÐ Ë : Ë ! Ë Given , there is a surface Ô and a morphism , unique up to ½ ´Ë fÔgµ isomorphism, such that (i) the restriction of to is an isomorphism onto ½ ½ ½ fÔg ´Ôµ È ´Ôµ Ô Ë , and (ii) is isomorphic to . is called the exceptional divisor , and is called the exceptional divisor. A key example, and indeed the analytic-, formal-, or etale-local situation, was given by ¾ = A blowing up Ë at the origin, which I’ll describe again soon when it comes up in a proof. For the definition, complex analytically, you can take the same construction. Then you need to think a little bit about uniqueness. There is a more intrinsic definition that works ¼ d Á Ë = ÈÖÓ j ¨ Á ¼ algebraically, let be the ideal sheaf of the point. Then d . 1. HOW BASIC ASPECTS OF SURFACES CHANGE UNDER BLOW-UP ×ØÖicØ Ë C C Definition. If C is a curve on , define the strict transform of to the the closure of ÔÖÓÔ eÖ £ ´C \ Ë Ôµ Ë Ô j C the pullback on , i.e. ¼ . The proper transform is given by the Ë E £ ¼ ¼ Ç ´C µ=Ç ´C µ Ë pullback of the defining equation, so for example Ë . ×ØÖicØ ÔÖÓÔ eÖ ×ØÖicØ £ C · ÑE C Ñ Ô C = C · ÑE C Lemma. If has multiplicity at , then , i.e. = . Ñ Proof. The multiplicity of C being means that in local coordinates, the defining equation Ñ Ñ has terms of degree Ñ, but not lower. (Better: the defining equation lies in but not Ñ·½ Ü Ý Ñ .) Analytically, this means that the leading term in and has degree m. Date: Wednesday, October 23. 1 Í = We do this by local calculation which will be useful in general. (Draw picture.) ¼ f´´Ü ;Ý µ; [½; Ú ]µ : Ý = Ü Ú g = ËÔ ec k [Ü ;Ý ;Ú]=Ý = Ü Ú = ËÔ ec k [Ü ;Ú] ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ . The exceptional E Ý =¼ divisor is given by ¼ (after morphism). Í = f´´Ü ;Ý µ; [Ù; ½]µ : Ü = Ý Ùg = ËÔ ec k [Ü ;Ý ;Ù]=Ü = Ý Ù = ËÔ ec k [Ý ;Ù] ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ . The E Ý =¼ exceptional divisor is given by ½ . ´Ü; Ý µ = ËÔ ec k [Ü; Ý ] ´Ü ;Ý ;Úµ 7! ´Ü ;Ý µ ´Ü ;Ý ;Úµ 7! ´Ü ;Ý µ ¼ ¼ ¼ ½ ½ ½ ½ Map down to . ¼ , . f ´Ü; Ý µ=¼ Í f ´Ü; Ý µ= f ´Ü; Ý µ · higheÖ = f ´Ü ; Ü Úµ · Ñ Ñ ¼ ¼ Given a function . Pull it back to ¼ : ¡¡¡ higheÖ · . Exercise. To see if you understood that, do the same calculation on patch 2. ¼ ¼ : Ë ! Ë Ë Ô E Ë Theorem. Suppose is a blow up of at , with exceptional curve . Let ¼ £ £ ¼ ¼ £ ¾ D Ë D ¡ D = D ¡ D E ¡ D =¼ E = ½ D and be divisors on . Then , , . ½ ½ Remark. A curve on a smooth surface that is isomorphic to È and has self-intersection ½µ is called a ´ -curve. Proof. The first we did yesterday. The second: by Serre’s moving lemma, we can move Ô C Ô D away from , then pull back. For the third: choose a curve passing through with ×ØÖicØ C ¡ E = ½ multiplicity 1. (How to do this: hyperplane section of Ë .) Then . Also ÔÖÓÔ eÖ ×ØÖicØ ÔÖÓÔ eÖ ¡ E =¼ C · E = C C .as , we’re done. ¼ £ Ë ¨ Z ! Èic Ë ´D; Òµ 7! D · ÒE Theorem. (a) There is an isomorphism Èic defined by . ÆË (b) The same with Èic replaced by . Proof. The arguments are the same for both parts, so I’ll do (a). It is surjective: the divisors E upstairs are either E or strict transforms (which are proper transforms plus ’s). It is £ D · ÒE = ¼ E Ò = ¼ injective: if , then intersect with to see that ; then apply £ to see =¼ that D . £ ¼ = à · E Ã Ë Theorem. Ë . £ ¼ Ã Ñ E à = = à · ÑE Ë E Proof. Clearly Ë for some . By the adjunction formula for , ¼ ´E µj à E Ë . Taking degrees: £ ¾= ´ à · ÑE · E µ ¡ E = Ñ ½: Ë Exercise/Remark. If you want practice with the canonical bundle in local coordinates, Ã Ô take a meromorphic section of Ë that has neither zero nor pole at (possible by Serre’s f ´Ü; Ý µdÜ ^ dÝ Í moving lemma), write it as , and pull it back to the open set ½ to see that f ´Ü ;Ü Ú µdÜ ^ d´Ü Ú µ= f ´Ü ;Ü Ú µÜ dÜ ^ dÚ ¼ ¼ ¼ ¼ ¼ ¼ ¼ you get ¼ . 2 2. RATIONAL MAPS OF SURFACES, LINEAR SYSTEMS, AND ELIMINATION OF INDETERMINACY 99 ÃX X A rational map Ë , where is a variety, means a morphism from an dense open C set of Ë . Recall that a rational map from a curve to a projective variety can always be extended to a morphism. Similarly, a rational map from a surface Ë to a projective variety can be extended over most points; the set of indeterminacy is a finite set of points. More Ò : Ë 99 ÃÈ Ò ·½ precisely, given a map . This is given by sections of some line bundle. It makes sense except where the sections are all zero. This will be in codimension 2. ´Ë F µ Ë ´Ë µ Let F be this finite set. We’ll denote the image of , and denote it . (I’m Ë ´C F µ not sure we need to take the closure.) If C is a curve on , then we’ll denote the ´C µ image of C , and denote it . Here we definitely need to take the closure. Ë Î Ò Now suppose you have a divisor D on . Given a subspace of dimension of ¼ £ ´Ë; Ç ´D µµ ÈÎ À , we might hope to get a map to projective space . (This is called a linear system of dimension Ò; I should have introduced this notation earlier.) If it is base point free, we do. If it has base points, the locus could have components of dimension 1. Such a compo- Î nent is called a fixed component of the linear system Î . The fixed part of is the biggest F D F divisor contained in every element of Î . So if this fixed part is , then has no fixed components. (I’m not happy with how I explained the previous paragraphs in class. I hope this is clearer.) Lemma. If the linear systems has no fixed part, then it has only a finite number of fixed points. Proof. Take two general sections, and look at their two zero-sets. Where do they intersect? ¾ At a bunch of points. Hence we get at most D ? We’ve basically shown that there is a bijection between: Ò : Ë 99 ÃÈ ´Ë µ g (i) f rational maps such that is contained in no hyperplane Ë Ò g (ii) f linear systems on without fixed part and of dimension (Explain the correspondence.) : Ë 99 à X Theorem (Elimination of indeterminacy). Let be a rational map from a ¼ ¼ : Ë ! Ë surface to a projective variety. Then there exists a surface Ë , a morphism ¼ : Ë ! X which is the composite of a finite number of blow-ups, and a morphism f such 3 that the diagram ¼ Ë f º ² Ë 99 à X is commutative. ¾ Proof. Idea: blow up fixed points, show that D decreases. Ñ È ´Ë µ We immediately reduce to the case where X is , and isn’t contained in any Ñ Î jD j Ò Ë hyperplane of È . Then corresponds to a linear system of dimension on , with no fixed component. If Î has no base point, then we’re done. Ü Ë ! Ë Ü Otherwise, we blow up a base point , and consider ½ at (and hence a rational Ë 99 ÃË map ½ ). The exceptional curve is now in the fixed part of the linear system, with ½ kE some multiplicity k . So we subtract to get rid of the fixed part, i.e. get a new linear £ Î j D kE j : Ë 99 ÃË D = D kE ½ ½ ½ system ½ , to get the same rational map , given by . If this is a morphism, we win, otherwise we keep going. At some point, this process must stop (and hence we win in the long run). We prove ¾ = i i i = ¼ this is the case when D , by induction on . Base case, : the number of fixed ¾ = ¼ i > ¼ points is bounded by D , so there aren’t any. Inductive step: Now . Then ¾ D = we blow-up once, and we get a new surface with divisor class. On this surface, ½ ¾ ¾ ¾ D kE µ´D kE µ = D k < D ´ . So by the inductive hypothesis, the process will terminate on this new surface, completing the induction. 3. THE UNIVERSAL PROPERTY OF BLOWING UP : X ! Ë Theorem (Universal property of blowing up). Let f be a birational morphism ½ Ô Ë f of surfaces, and suppose that the rational map f is undefined at a point of . Then factorizes as g ~ f : X ! Ë := BÐ Ë ! Ë Ô Ô where g is a birational morphism and is the blow-up at . Proof: next day. 3.1. Applications of the universal property of blowing up. Two theorems. f : Ë ! Ë Theorem (all birational morphisms factor into blow-ups). Let ¼ be a birational : Ë ! Ë k ½;:::;Ò k k ½ morphism of surfaces. Then there is a sequence of blow-ups k ( ) Ù : Ë ! Ë f = Æ ¡¡¡ Æ Æ Ù ½ Ò and an isomorphism Ò such that .