The structure and slope of bielliptic fibrations
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1 THE STRUCTURE AND SLOPE OF BIELLIPTIC FIBRATIONS Miguel A. BARJA 1 Departament de Matematica Aplicada I ETSEIB. Universitat Politecnica de Catalunya Diagonal 647, 08028 Barcelona, Spain e-mail: b[email protected] c.es 1991 Mathematics Sub ject Classication: Primary 14H10, Secundary 14J29 A la memoria de Fernando 0 Intro duction Let : S ;! B be a bration , i.e. a surjective morphism with connected bres, from a smo oth surface S onto a smo oth curve B . A bration is said to b e relatively minimal when it has no vertical ( ; 1)-curve. Let g denote the genus of a general bre and b the genus of B . Let ! S=B = ! S ( ! ; 1 B ) b e the relative canonical bundle and let ( ):= deg ( ! S=B ). It is known that ( ) 0 and that ( ) = 0 if and only if is lo cally trivial. Assume is not lo cally trivial. Then we dene the slop e of as ( ):= ! 2 S=B . ( ) (see 19]). There are several results on the lower slop e of relatively minimal brations of genus g 2. First of all wehave 4 ; 4 g (see 8], 12], 13], 18] for the hyp erelliptic case and 19] for the general case) and equality holds only in the hyp erelliptic case (9]). There are improvements in the non-hyp erelliptic case for g 5 (see 4], 7], 9], 11], 14]) but the presently known techniques seem to have some limitations to extend these results to higher genus. Recently Konno is trying to nd go o d b ounds dep ending on some extra numerical invariants of the general bre, such as the Cliord index. In 10], Konno nds b etter b ounds for trigonal and plane quintic brations (so Cliord index 1), although they do not seem to b e sharp. Also in 11] he gets general b ounds dep ending on the Cliord index in some cases. 1 Partially supp orted by CICYT PS93-0790 and HCM pro ject n.ERBCHRXCT-940557
2 MIGUEL A. BARJA In this pap er we deal with the case of bielliptic brations (i.e., when the general bre has a 2-to-1 map onto an elliptic curve). In the study of general brations it turns out to b e relevant the prop erties of the canonical emb eding of the general bre with resp ect to quadrics. A rst example is the trigonal case where Konno uses the fact that sucha curve lies in a rational normal scroll which is the intersection of the quadrics containing it. In our case the role of the quadrics through the canonical bielliptic curve can not b e used directly. A dierent approach using relative Brill-No ether lo ci shows Theorem 2.4 Let : S ;! B be a biel liptic bration of genus g 6 . Then S is, birational ly, a double cover of an el liptic smooth surface V over B . In chapter 3 we prove that the conclusions of the ab ove theorem hold for genus 5 bielliptic brations after a base change and wegive an example where a nontrivial base change is needed. As a by-pro duct we obtain that every smo oth bielliptic bration of genus g 5is isotrivial (see prop osition 2.6 and corollary 3.3). Finally, using theorem 2.4 and canonical resolution of singularities for double covers we get the following sharp b ound for the slop e of bielliptic brations Theorem 4.1 Let : S ;! B bearelatively minimal biel liptic bration of genus g 6 . Let V be the relative minimal model of the el liptic bration obtainedin theorem 2.4. Then (a) ( ) 4+ 2( g ; 5) XO V ( ) 4 . (b) ( )= 4 if and only if S is the minimal desingularization of a double cover S 0 ;! V ofasmooth el liptic surface such that All the bres of the el liptic bration : V ;! B aresmooth and isomorphic. The branch divisor of the double cover has only negligeable singularities. In particular, the bound is sharp. The author want to thank, among others, professor Juan Carlos Naranjo for his encouragementand interesting comments chapter 3 grew from fruitful conversations with him. During the nal revision of this pap er the advisor of the author, professor Fernando Serrano, passed away.Iwould like to thank him heartfully for his supp ort and continuous help, not only during the preparation of this work but also during the last years when I enjoyed his teachings and friendship. Notations and conventions All throughout this pap er wework over the eld of complex numb ers C . If X is a scheme and G is a sheaf of graded algebras and E is a lo cally free coherent sheaf on X we dene Pro j G and P ( E ) as in 5],I I.7.
THE STRUCTURE AND SLOPE OF BIELLIPTIC FIBRATIONS 3 1 Some generalities on bred surfaces We recall some basic facts ab out bred surfaces (see 9]). The following are some easy but useful results. Lemma 1.1 Let X be a smooth variety and ' : X ;! B a morphism onto a curve. Let X t be the breof ' over t 2 B . For any coherent sheaf F on X and for any a 2 Pic ( B ) let F ( a )= F ' ( a ) . Suppose that a is ample enough and that F satises the fol lowing technical condition: for general t 2 B the sequence 0 ;!F ( ; t ) ;!F ;!F j X t ;! 0 is exact. Then, the natural morphism H 0 ( X F ( a )) ;! H 0 ( X t F j X t ) is surjective for general t 2 B . Pro of. Consider the exact sequence 0 ;!F ( a ; t ) ;!F ( a ) ;!F j X t ;! 0 for general t 2 B .Taking cohomology we get 0 ;! H 0 ( X F ( a ; t )) ;! H 0 ( X F ( a )) m t ;! H 0 ( X t F j X t ) If a is ample enough then h 1 ( B ( ' F ) ( a ; t )) = 0 and then h 0 ( X F ( a ; t )) = h 0 ( B ( ' F ) ( a ; t )) do es not dep end on t by the Hirzebruch-Riemann-Ro ch theorem for coherent sheaves on B . Furthermore dim Im ( m t ) = h 0 ( X F ( a )) ; h 0 ( X F ( a ; t )) = = h 0 ( B ( ' F ) a ) ; h 0 ( B ( ' F ) ( a ; t )) = = d + r ( a +1 ; b ) ; ( d + r ( a ; 1+1 ; b )) = r where d =deg ' F a =deg a r = rank ( ' F )= h 0 ( X t F j X t )for t 2 B general 2 Lemma 1.2 Let : S ;! B beabration. Let L2 Pic S . If a 2 Pic B is ample enough, then the natural map h : L ( a ) ;!L ( a ) is an epimorphism just except at the base points of the linear system jL ( a ) j . Moreover, if for a general bre F of , the linear system jL j F j is base-point free, then such base points areconcentrated on a nite number of bres.
4 MIGUEL A. BARJA Pro of. From the sequence of maps S ;! B ;! Sp ec C we can consider the following natural commutative diagram e L ( a ) k =( ) ( ) L ( a )= H 0 ( S L ( a )) C O S / / L ( a ) s s h g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g L ( a ) Since k is surjectivefor a ample enough, it follows that surjectivityof h is equivalent to surjectivityof e . This fails to b e an epimorphism precisely at the base p oints of jL ( a ) j . Finally, using lemma 1.1 one has that jL ( a ) j has no base p oints on a general bre F of . 2 Then we can construct the relative canonical morphism of S . Consider L = ! S=B = ! S ! ; 1 B then L j F = ! F which has no base p oints for smo oth F if g 2. L ( a )has only base p oints on singular bres of ,for a ample enough by lemma 1.2 and the natural map h : L ( a ) ;!L ( a )(1) is an epimorphism away from such a base p oints. In fact wehave e h :( L ( a )) L ( ; a ) ;!O S (2) with Im e h = I Z = O S ( ; R ) I ; b eing the ideal sheaf of such base p oints, endowed with some scheme strcture, where O S ( ; R ) is the ideal sheaf of its divisorial part and I ; is the ideal sheaf of its discrete part (see 14]). Consider a sequence of blow-ups : e S ;! S such that the base lo cus of j L ( a ) j is of pure co dimension 1. Then, in particular wehave an epimorphism I ; ;! ; 1 I ; O e S = O e S ( ; E ) where E is a divisor on e S exceptional with resp ect to . Thus from L ( a ) ;!L ( a ) O S ( ; R ) I ; (epimorphism) weget ( L ( a )) ;! L ( a ) O S ( ; R ) I ; ;! L ( a ) O S ( ; R ) O e S ( ; E )(3) which is also an epimorphism. We shall call O e S ( M ( a )) = L ( a ) O S ( ; R ) O e S ( ; E )the moving part of L ( a ) O e S ( Z ( a )) = O S ( R ) O e S ( E )the xed part of L ( a ) (as one can easily see ( ) ( ) O e S ( M ( a )) ;!O e S ( M ( a )) is an epimorphism so, in view of lemma 1.2, j M ( a ) j hasnobasepoints).
THE STRUCTURE AND SLOPE OF BIELLIPTIC FIBRATIONS 5 Then (3) leads to e S = = = = = = = = S / / ___ P / / = P w w ' p p p p p p p p p p p p p p B where P = P B ( L ( a )) P = P B ( L ) is the natural isomorphism such that O P (1) = O e S ( M ( a )). If wecall = weget ( O P (1) ' ( a )) = O e S ( M ( a )) : We shall call the image of e S by . In fact is the closure in P of the image of the birational map induced on S by h in (1), so do esn't dep end on a .Thenwecan change a 2 Pic B if needed. We shall call the relative canonical image of S and the relative canonical map . We remark that for F S a smo oth bre of , F = ( F ) = ( ( F )), so we denote it always by F if no confusion arises. For t 2 B wecall P t = ' ; 1 ( t ) = P g ; 1 . We shall call O P ( T ) the tautological line bundle on P . Remark 1.3 Consider the sheaf F = I P O P (2 T ) ' (2 a ), where I P is the ideal sheaf of in P .For t 2 B wehave F j P t = I F t P t (2). We claim that F veries the hyp othesis of lemma 1.1 and so that for a ample enough wehave an epimorphism H 0 ( P I P O P (2 T ) ' (2 a )) ;! H 0 ( P g ; 1 I F t P t (2)) : Indeed, just consider 0 0 0 ? ? ? ? ? y ? ? ? ? ? y ? ? ? ? ? y 0 ;;;! I P ' ( ; t ) ;;;! O P ( ' ( ; t )) ;;;! O ( ' ( ; t )) ;;;! 0 ? ? ? ? ? y ? ? ? ? ? y ? ? ? ? ? y 0 ;;;! I P ;;;! O P ;;;! O ;;;! 0 ? ? ? ? ? y ? ? ? ? ? y ? ? ? ? ? y 0 ;;;! I F t P t ;;;! O P t ;;;! O F t ;;;! 0 ? ? ? ? ? y ? ? ? ? ? y ? ? ? ? ? y 0 0 0
6 MIGUEL A. BARJA with the three rows and the tworight columns trivially exact. Then the snake lemma makes the left hand side column exact. Now just tensor with O P (2 T ), which is lo cally free. 2 Bielliptic brations of genus g 6 From nowon we consider bielliptic brations only, i.e. brations such that the general bre admits a 2-to-1 morphism onto an elliptic curve. It seems reasonable to think that all those maps can b e glued together to yield a global 2-to-1 map from S onto an elliptic surface over B . A similar case, very much studied (see 8], 12], 13], 18]) is that of hyp erelliptic brations, i.e. when bres are hyp erelliptic. Here the hyp erelliptic involutions glue together to yield a birational double cover of a ruled surface in an easy way: just consider the relative canonical morphism of : S ;! B . The case where the general bres are trigonal also globalizes to a 3-to-1 map from S onto a ruled surface over B (this fact is implicit in 10]). Weproveinthis chapter that such a double cover actually exists for any bielliptic bration of genus at least 6. To see this we need, rst of all, a canonical way of constructing the double cover of an elliptic curvefrom F . Recall (1] Ch.IV) that for any smo oth curve F wehavea variety W r d ( F ) inside Pic d ( F ) parametrizing the complete linear series on F of degree d and dimension at least r . Given a bielliptic involution : F ;! E there is a natural wayof embedding E in W 1 4 ( F ). This map is given by the comp osition of with the hyp erelliptic involutions coming from E . On the other hand W 0 4 ( F ), which has dimension four, is singular precisely along W 1 4 ( F ) if g 5 (see 1] p.160). Take L 2 W 1 4 ( F ) we denote T L = P T L W 0 4 ( F ) the pro jectivized tangentconeto W 0 4 ( F )at L ,which ts canonically in P ( H 0 ( F ! F ) ) = P g ; 1 .If g 5 wehaveby the Kempf 's Singularity Theorem (1] p.241) that T L is a threefold of degree ( g ; 3) in P g ; 1 containing F . For any eective divisor D on F wedenoteby <D> the linear subspace of P ( H 0 ( F ! F ) ) spanned by D , in the sense of 1] p.12. Lemma 2.1 Let : F ;! E be a biel liptic cover. Then (a) For g 4 the union of the lines < ( p ) > ,with p 2 E , is an el liptic normal cone R of degree ( g ; 1) ,containing F . (b) For g 6 the biel liptic involution : F ;! E is unique, W 1 4 ( F ) = E and if L 1 L 2 2 W 1 4 ( F ) are two distinct points, we have R = T L 1 \ T L 2 . Pro of. If q 1 :::q i 2 E are distinct general p oints, Riemann-Ro chon E and F shows that dim < ( q 1 )+ ::: + ( q i ) > = i for i +1 g: (4) Then, for i = 3 and g 4, it follows that all the lines < ( p ) > ,for p 2 E , meet at the same p oint q 2 P g ; 1 . So pro jection from q gives the elliptic curve E as an elliptic normal curveof P g ; 2 , whichproves (a).
THE STRUCTURE AND SLOPE OF BIELLIPTIC FIBRATIONS 7 Let 1 : F ;! E b e a bielliptic involution. Assume there exists a 4-to-1 map 2 : F ;! P 1 distinct from those pro duced by 1 . Then the map : F ;! E P 1 given by ( p )=( 1 ( p ) 2 ( p )) is birational. If we denote a = E f p 2 g , b = f p 1 g P 1 for p 1 2 E , p 2 2 P 1 wehave ( F ) 2 a +4 b and hence, byadjuntion in E P 1 ,we get g ( F ) p a ( ( F )) = 5. So if g 6 the bielliptic involution is unique and W 1 4 ( F ) = E . For g 6wehave from (4) that if L j = j ( q j 1 + q j 2 ) j q i k 2 E j =1 2 are two dierent p oints in W 1 4 ( F ) then dim < ( q 1 1 + q 1 2 ) > \ < ( q 2 1 + q 2 2 ) > =0 or 1(5) dep ending on whether f q 1 1 q 1 2 g\f q 2 1 q 2 2 g is empty or has one p oint, resp ectively. Then, again by Kempf 's Singularity Theorem, wehave the set-theoretic equality T L i = D 2 L i <D > and thus T L 1 \ T L 2 = D 2 L 1 D 0 2 L 2 ( <D > \ <D 0 > )= p 2 E < ( p ) > = R using (5), if L 1 6 = L 2 . 2 Theorem 2.2 Let : S ;! B be a biel liptic bration of genus g 6 . Let P = P ( ! S=B ) be the relative canonical image of S as in section 1. Then there exists a threefold W such that (a) W P . (b) For t 2 B such that the bre F t is smooth we have W \ P t = R t (the el liptic normal cone containing F t ). Pro of. Weprove the theorem in several steps. Step 1. We can assume that the bration has enough sections . Indeed, consider a very ample line bundle L on S and let B b e a global smo oth section of L . Consider the diagramm S / / > > > > > > > > b S b / / S B / / B where = j B , b is given by at base change and : S ;! b S is a minimal desingularization of b S . In this situation we remark that b , and hence , has a section. Indeed, this section is given by f ( b b ) 2 B B B S B B = b S g which is a comp onentof B B B .We can rep eat this construction in order to get as many sections as we need.
8 MIGUEL A. BARJA Then wehave b ( ( ! S )) = ( ( ! S )) by at base change 0 ;! ( ( ! S )) ;! ! S by ramication formula ( ( ( ! S ))) = b ( ( ! S )) = b ( ( ! S ) O S ) by pro jection formula 0 ;!O b S ;! O S dominant and then 0 ;! ( ! S ) ;! ( ! S ) : Being b oth lo cally free sheaves of the same rank we get a birational map given bya sequence of elementary transformation on certain bres P B ( ! S ) ;;; ' P B ( ! S ) which is an isomorphism for a general bre. So nally we get a generically nite rational map : P B ( ! S= B ) ;;; ! P B ( ! S=B ) given as the comp osite P B ( ! S= B ) = P B ( ! S ) ;;; ! ' P B ( ! S ) ;! P B ( ! S ) = P B ( ! S=B ) which is linear on bres and restricts to the natural map from the relative canonical image of : S ;! B onto the relative canonical image of : S ;! B . Then supp ose there exists W as in the theorem for the bielliptic bration : S ;! B . We had that for smo oth F t , W \ P t = R t .Now just consider W = ( W )whichveries the desired conditions. Indeed, for t 2 B suchthat F t is smo oth wehave W \ P t = ( t 0 )= t ( W \ P t 0 )= ( t 0 )= t ( R t 0 )= R t b ecause ( R t 0 ) is an elliptic normal cone containing F t = ( F t 0 ), whichisuniqueby lemma 2.1. Step 2. We can assume that is smo oth and has enough disjoint sections . Once wehave enough sections we can restrict the bration to the nonempty Zariski op en set where is smo oth and the sections do not meet (weonly haveto avoid a nite numb er of bres). If there exists W U P U ( i ( ! S=B )), where i : U ;! B is the natural inclusion, verifying the theorem, then as W it is enough to take the closure of W U inside P = P B ( ! S=B ). Step 3. Existence of W . From the previous steps wehave : S ;! B a smo oth bration over a, p ossibly non complete, curve with enough disjoint sections. Under these assumptions (see 16) there exist schemes W r d ( ), Pic d ( )over B suchthat (i) For every t 2 B ,( W r d ( )) t = W r d ( F t ), (Pic d ( )) t =Pic d ( F t ).
THE STRUCTURE AND SLOPE OF BIELLIPTIC FIBRATIONS 9 (ii) (Base change prop erty) If : B ;! B is a base change and : S ;! B satises the same go o d prop erties as , then W r d ( )= W r d ( ) and Pic d ( )=Pic d ( ) (where, as always, X denotes the base change of X ;! B ). Then we can consider W 1 4 ( ) W 0 4 ( ) Pic 4 ( ). Moreover we remark that W 1 4 ( ) ;! B is an elliptic bration which, up to base change, we can assume has at least two sections (by the base change prop ertythiswould corresp ond to a base change for that, as proven at step 1, can always b e done). We also remark that, if we set J =Pic 4 ( ) f ;! B , then the sheaf of relative dierentials of f is just 1 J=B = f ( ! S=B ) (see 17] p.2). Then we can pro ceed as follows. Consider W 1 4 ( ) W 0 4 ( ) i Pic 4 ( )= J f B f f s M M M M M M M M M M M M M M M M M M M M M M M M M M M M where s is a section of f j W 1 4 ( ) (we are assuming that suchan s exists). Let e B b e the image of s .Ifwe call I 1 , I 2 the ideal sheaves of e B in W 0 4 ( )and J resp ectively, the natural epimorphism I 2 ;! i I 1 induces the epimorphisms S I 2 . I 2 2 ;!S i I 1 . ( i I 1 ) 2 = S I 1 . I 2 1 ;! i I i 1 . I i +1 1 (where S denotes the symmetric algebra) and then also the inclusions Z 1 = Pro j i I i 1 . I i +1 1 , ! Z 2 = P e B I 1 . I 2 1 , ! Z 3 = P e B I 2 . I 2 2 : Since W 0 4 ( ) is singular along W 1 4 ( ), hence along e B ,wehave that Z 1 is the relative pro jectivized tangent cone of W 0 4 ( )along e B which ts canonically into Z 2 and Z 3 , the relative pro jectivized Zariski tangent spaces to W 0 4 ( )and J , resp ectively, along e B . On the other hand, since e B J is smo oth, wehave an exact sequence 0 ;! I 2 . I 2 2 ;! 1 J=B O e B ;! 1 e B=B ;! 0 k 0 that leads to I 2 . I 2 2 = 1 J=B O e B .Then Z 3 = P e B I 2 . I 2 2 = P e B 1 J=B O e B = P B s 1 J=B = = P B s f ! S=B = P B ! S=B
16 MIGUEL A. BARJA Although this example shows that theorem 2.4 is not true for genus 5 bielliptic brations wehave Prop osition 3.2 Let : S ;! B be a biel liptic bration of genus 5. Then, there exists abase change e : e S ;! e B of such that e S is, birational ly, a double cover of an el liptic bration e : e V ;! e B . If the general breof has only one biel liptic involution, then we can take e = . Pro of. As in step 3 of theorem 2.2 wehave after a base change W 1 4 ( b ) ;! b B (at least over a non-empty op en subset of b B ). Let b L b e an irreducible comp onentof W 1 4 ( b ) such that for general t 2 b B b L t has only elliptic comp onents. Let L b e a desingularization of L . After a base change we can get e L / / L e B / / b B such that e L ;! e B has irreducible general bres and at least two sections. If e : e S ;! e B is the induced bration wehavethat e L is an irreducible comp onentof W 1 4 ( e ) ;! e B . Then step 3 in theorem 2.2 works and weget e S ;; > e f W P e B ( e ! e S= e B ). The existence of f W is all what we need to use the same pro of of theorem 2.4. If the general bre of has only one bielliptic involution (hence there exists only one elliptic cone containing F P 4 ) then from the existence of e f W we can deduce the existence of W in the same way as step 1 in theorem 2.2 (uniqueness of elliptic cones containing the general bre is all what we need). 2 This result is enough to extend prop osition 2.6 to the genus 5 case. Corollary 3.3 Let : S ;! B be a genus 5 bration such that al l bres are smooth and biel liptic. Then is isotrivial. Pro of. We can check isotriviallity after a base change. The result follows then from prop osition 3.2 and the pro of of prop osition 2.6. 2 4 Double covers and the slop e of bielliptic brations We recall some basic facts ab out double covers (see 6], 3]). By a double cover we mean a nite, degree twomapbetween surfaces, f 0 : S 0 ;! V 0 . This map is determined by a divisor Z 0 on V 0 (the branch divisor) and a line bundle L 0 such that L 2 0 = O V 0 ( Z 0 ). If V 0 is smo oth, S 0 is normal (resp ectively smo oth) if and only if Z 0 is reduced (resp ectively smo oth). Consider a double cover as ab ovewith S 0 normal and V 0 smo oth. Then there exists a canonical resolution of singularities for S 0 which consists on a nite sequence of maps
THE STRUCTURE AND SLOPE OF BIELLIPTIC FIBRATIONS 17 S k k ;;;;;;;! S k ; 1 ;;;;;;;! ::: ;;;;;;;! S 1 1 ;;;;;;;! S 0 f k ? ? ? ? ? ? y f k ; 1 ? ? ? ? ? ? y ::: ? ? ? ? ? ? y f 1 ? ? ? ? ? ? y f 0 V k ;;;;;;;! k V k ; 1 ;;;;;;;! ::: ;;;;;;;! V 1 ;;;;;;;! 1 V 0 satisfying: (i) j is the blow-up of V j ; 1 at a singular p oint p j ; 1 of Z j ; 1 (the branching divisor of f j ; 1 ). (ii) f j is the double cover of V j dened by L 2 j = O ( Z j ), with Z j = j ( Z j ; 1 ) ; 2 m j ; 1 E j , L j = j ( L j ; 1 ) O V j ( ; m j ; 1 E j ), where E j is the exceptional divisor of j and p j ; 1 is a singular p ointof Z j ; 1 of multiplicity2 m j ; 1 or 2 m j ; 1 +1. (iii) j is a birational morphism induced by the cartesian diagram of j and f j ; 1 . (iv) Z k is smo oth and, hence, S k is a smo oth surface. Nowwe can use this as follows. Recall from section 2 that wehave obtained f : e S ;! V a generically 2-to-1 morphism (we can supp ose that f is everywhere dened up to blowups) from a blow-up of S onto an elliptic bration V over B whichwe can supp ose relatively minimal after some blow-downs. Supp ose that is relatively minimal. Now consider e S + + u W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W S = S k { { x x x x x x x x x x x f k / / ::: / / S 0 f 0 S ::: V = V k / / ::: / / V 0 = V s s g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g B where: f = f 0 u is the Stein factorization of f ,with u birational, f 0 nite (so it is a double cover) and S 0 normal. f k : S k ;! V k is the canonical resolution of singularities of f 0 : S 0 ;! V 0 . : S k ;! S is the birational morphism dened by the relativeminimalityof . Theorem 4.1 Let : S ;! B bearelatively minimal biel liptic bration of genus g 6 . Let V be the relative minimal model of the el liptic bration obtainedin theorem 2.4. Then
18 MIGUEL A. BARJA (a) ! 2 S=B ; 4( ) 2( g ; 5) XO V .Inparticular, if is not local ly trivial ( ) 4+ 2( g ; 5) XO V ( ) 4 (b) ( )= 4 if and only if S is the minimal desingularization of a double cover S 0 ;! V ofasmooth el liptic surface such that All the bres of the el liptic bration : V ;! B aresmooth and isomorphic. The branch divisor of the double cover has only negligeable singularities (i.e., al l the multiplicities m j in the above process are 2 or 3 (see 13], 17])). In particular, the bound is sharp. Pro of. (a) First of all wehave ! 2 S=B ; 4( )= ( K 2 S ; 4 XO S ) ; 4( b ; 1)( g ; 1) ( K 2 S ; 4 XO S ) ; 4( b ; 1)( g ; 1) : (6) For smo oth double covers f k : S ;! V wehave (see 3] p.183): XO S = 2 XO V + 1 2 L k K V + 1 2 L k L k K 2 S = 2 K 2 V +4 L k K V +2 L k L k so wehave K 2 S ; 4 XO S =2 K 2 V k ; 4 XO V k ]+ 2 L k K V k : (7) Moreover, in each blow-up j : V j ;! V j ; 1 we get XO V j = XO V j ; 1 K V j = j K V j ; 1 + E j L j = j L j ; 1 ; m j ; 1 E j : Then 2 K 2 V j ; 4 XO V j ]+ 2 L j K V j =2 K 2 V j ; 1 ; 4 XO V j ; 1 ]+ 2 L j ; 1 K V j ; 1 +2( m j ; 1 ; 1) 2 K 2 V j ; 1 ; 4 XO V j ; 1 ]+ 2 L j ; 1 K V j ; 1 : (8) Finally as : V ;! B is an elliptic minimal bration, numerically wehave K V h 2( b ; 1) + XO V + P i ( n i ; 1) n i i E (3] p.162) where E denotes a smo oth bre of and f n i g are the multiplicities of singular bres of . In particular K 2 V 0. As L 2 0 = O V 0 ( Z 0 )and Z 0 is the branch divisor of f 0 weget L 0 E =( g ; 1) by Hurwitz formula. So 2 K 2 V 0 ; 4 XO V 0 ]+ 2 L 0 K V 0 = ; 8 XO V 0 +(9) +2 L 0 E " 2( b ; 1) + XO V 0 + X i ( n i ; 1) n i # 4( b ; 1)( g ; 1) + 2( g ; 5) XO V :
THE STRUCTURE AND SLOPE OF BIELLIPTIC FIBRATIONS 19 Then (a) follows from (6), (7), (8) and (9) and from the fact that XO V 0 for elliptic brations. (b) Lo oking at the pro of of (a) we see that =4 i XO V = 0 and equality holds in (6), (7), (8) and (9). So wehave =4 i S is the minimal desingularization of a double cover of an elliptic, relatively minimal, bration : V ;! B suchthat: has no multiple bres ( 8 i n i = 1). XO V =0. The branch divisor Z 0 of the double cover has only negligeable singularities (see 13], 17]), i.e. all the multiplicites of the singularities of the branch divisors in the pro cess of canonical resolution are 2 or 3. But the rst two conditions are equivalent to the fact that is smo oth and isotrivial (see 15 thms. 6,7 Ch.IV). This allows us to construct examples with ( )= 4 which are esentially the same as in 19] example 4.3. So the b ound is sharp. 2 Remark 4.2 Although we cannot use double covers for the case of bielliptic brations of genus 5 we already knowthat 4 also holds for such brations (see 9] thm.5.1, 11]). References. 1. E. Arbarello, M. Cornalba., P.A. Gri"ths, J. Harris. Geometry of algebraic curves , vol I. Grund. Math. Wiss. 267 . Springer-Verlag 1985. 2. M. Beltrametti, P.Francia. Threefolds with negative Kodaira dimension and positive irregularity .Nagoya Math. J. vol 91 (1983), 163{172. 3. W. Barth, C. Peters, A. Van de Ven. Compact complex surfaces . Ergebnisse der Mathematik und ihrer Grenzgebiete 4 . Springer-Verlag (1984). 4. Z. Chen. On the bound of the slope of a non-hyperel liptic bration of genus 4 .Intern. J. Math., vol 4 , No.3 (1993), 367{378. 5. R. Hartshorne. Algebraic Geometry . Graduate Texts in Mathematics 52 . SpringerVerlag (1977). 6. E. Horikawa. On deformation of Quintic Surfaces .Inventiones math. 31 (1975), 43{85. 7. E. Horikawa. Notes on canonical surfaces .Tohoku Math. J. I 43 (1991), 141{148. 8. E. Horikawa. Algebraic surfaces of general type with smal l c 2 1 ,V .J.Fac. Sci. Univ. Tokyo. Sect. A 28 (1981), 745{755. 9. K. Konno. Non-hyperel liptic brations of smal l genus and certain irregular canonical surfaces . Ann. Sc. Norm. Pisa Ser. IV vol XX (1993), 575{595.
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