On the slope of bielliptic fibrations
Abstract
Let $\pi :S\longrightarrow B$ be a bielliptic fibration. We prove $S$ is, up to base change, a rational double cover of an elliptic fibration and that $\pi $ is isotrivial provided it is smooth. Finally, we prove that the slope of $\pi $ is at least four provided the genus of the fibre is at least six.
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1 ON THE SLOPE OF BIELLIPTIC FIBRATIONS Miguel A. BARJA 1 Abstract Let : S ;! B b e a bielliptic bration. We prove S is,uptobasechange, a rational double cover of an elliptic bration and that is isotrivial provided it is smo oth. Finally,we prove that the slop e of is at least four provided the genus of the bre is at least six 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. 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). Using the glueing results of 2] weknow that, 1 Partially supp orted by CICYT PS93-0790 and HCM pro ject n.ERBCHRXCT-940557. 1991 Mathematics Sub ject Classication: Primary 14H10, Secundary 14J29
2 MIGUEL A. BARJA if g 6 a bielliptic bration is a (generically) double cover of an elliptic bration. We prove (example 1.2) that this is not true in general if g 5 due to the existence of several bielliptic maps in the general bre. Using this we get the following sharp b ound for the slop e of bielliptic brations. Theorem 2.1 Let : S ;! B bearelatively minimal biel liptic bration of genus g 6 . Let V be the relative minimal model of the el liptic bration obtainedinsection 1. 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 encouragement and interesting comments. 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. All throughout this pap er wework over the eld of complex numb ers C . 1 Bielliptic brations Let F b e a smo oth curve of genus g . F is called bielliptic if F admits a 2-to-1 map onto an elliptic smo oth curve E .Sucha mapis always given by the quotientbyaninvolution i 2 Aut( F ), called a bielliptic involution on F .It is a well known fact that suchan involution is unique if g 6 (see 1]). Let : S ;! B b e a bration of genus g .Wesaythat is bielliptic if so it is the general bre F of . The following result claries the structure of such brations. Recall that the bration is said to b e smo oth if every bre is smo oth and it is said to b e isotrivial if all the smo oth bres are mutually isomorphic. Prop osition 1.1 Let : S ;! B be a biel liptic bration of genus g . Then (a) Up to base change, S is a rational double cover of an el liptic surface over the base curve. (b) If g 6 the same is true without base change. (c) If is smooth, then is isotrivial.
ON THE SLOPE OF BIELLIPTIC FIBRATIONS 3 Pro of. (a) and (b) are consequence of general results given in 2]. Wegive here a sketch of pro of and refer there for details. Given : S ;! B we can consider :Aut 2 2 g ; 2 S=B ;! B the scheme of relative automorphisms of S over B of order 2 having 2 g ; 2 base p oints (which corresp onds brewise to double covers of elliptic curves) which is a quasi-pro jective B -scheme. After a base change B 0 ;! B such a map has always a section dened over a non-empty Zariski op en subset U 0 B 0 which corresp onds to a rational automorphism of the minimal desingularization S 0 of S B B 0 suchthat j F t is a bielliptic involution for t 2 B 0 general. If V is a desingularization of S 0 . < > wehave a rational dobule cover S 0 / / ___ V over B 0 . If g 6 then is clearly 1-to-1 and then base change is not needed in order to have a section. (c) Isotrivially can b e checked after base change. Following 2] section 2 we can consider after base change S ! ! C C C C C C C C C / / i J ( ) / / f ___ J ( ) { { w w w w w w w w w B where J ( ) is the relative Jacobian varietyof S over B and f is a rational relative endomorphism of J ( )such that f i pro duces a bielliptic map on the general bre of .Let V = ( f i )( S ). Note that V is an elliptic surface over B (p ossibly singular). Nevertheless classication of singular bres of a smo oth elliptic surface shows, since J ( ) t is an ab elian variey for every t 2 B ,that V is smo oth and the map : V ;! B is also smo oth. Moreover, the map g = f i : S / / ___ V can b e solved after some blow-ups but then exceptional curves must b e contracted since V J ( ). So wehavethat S is a double cover of a smo oth elliptic bration (p erhaps after base change). In particular every bre of is bielliptic. Consider now the double cover g : S ;! V . Since g has degree two the branching divisor of g must b e smo oth and hence it is etale over B . After new base changes we can assume that the irreducible comp onents of the branching divisor D are sections of . Moreover, since : V ;! B is a smo oth elliptic bration it is isotrivial and then, after base change, we can assume V = B E ( E : elliptic smo oth curve). Let D 1 be an irreducible comp onentof D .If D 1 is a trivial section of then so must b e the other comp onents and then is clearly isotrivial. Assume D 1 is not a trivial section. Then D 1 = f ( b ( b )) 2 B E j : B ;! E non constantmap g . Consider a xed structure of group on E =( E 0) and consider the automorphism of V over B : ( b x )= ( b x + ( b )). Note that ; 1 ( D 1 )= B f 0 g and, hence, ; 1 ( D ) is comp osed of trivial horizontal sections. If wechange the base S / / ' e g S g B E / / B E the branching divisor of e g is just ; 1 ( D )which is trivial. Hence is isotrivial. 2
4 MIGUEL A. BARJA A bielliptic curveofgenus g 5 can have more than one bielliptic involution the number of suchinvolutions are in corresp ondence with the elliptic comp onents of W 1 4 ( F ), the Brill-No ether lo cus of linear series on F of typ e g 1 4 .Wegive an example which shows that these involutions do not glue indep endently for a general bration. Example 1.2 Take a genus ve curve F with exactly two bielliptic involutions i : F ;! E i suchthat E 1 6 = E 2 , with E i having no exceptional automorphisms (a count of constants shows that suchan F can b e chosen). Then wehavethat 1 2 : F ;! E 1 E 2 embeds F as a smo oth curve, F 2j ` 1 (2 p 1 ) ` 2 (2 p 2 ) j , b eing ` i : E 1 E 2 ;! E i the pro jections and ( p 1 p 2 ) 2 E 1 E 2 . Since Aut ( E 1 E 2 ) acts transitively on E 1 E 2 wehave that for every ( q 1 q 2 ) 2 E 1 E 2 there exists e F 2j ` 1 (2 q 1 ) ` 2 (2 q 2 ) j , e F = F . Let B be any smo oth curvehaving an involution and let g : B ;! B = B =<> . Consider a morphism : B ;! P 1 with no factorization through B .Takeaxed t 2 B such that if g ; 1 ( t )= f t 1 t 2 g then ( t 1 ) 6 = ( t 2 ). After an automorphism of P 1 wecan supp ose that ( t i ) is the mo dular invariantof E i in C P 1 . Then, by 3] p.160, there exists an elliptic bration : V ;! B with a section, such that ; 1 ( t i ) = E i . Let B 0 b e the image in V of the section of . Consider the following pull-back Z := V B V 1 / / 2 $ $ J J J J J J J J J J J J J J J J J V V / / B Then, for t 2 B wehave Z t = ; 1 ( t )= E ( t ) E t ,where E m = ; 1 ( m ). The natural involution on V C V induces a commutative diagramm Z / / Z B / / B and then Z / / g Z := Z =<> B / / g B Note that Z is a threefold bred over B and the bre over g ( t ) 2 B general is E ( t ) E t . We can assume Z is already smo oth. Let B 00 = g ( ; 1 2 ( B 0 )) and L = O Z (2 B 00 ). Wehave that L j Z t = ` 1 (2 q 1 ) ` 2 (2 q 2 ) for some ( q 1 q 2 ) 2 E 1 E 2 .Notethatif a 2 Pic B is ample enough wehave an epimorphism H 0 ( Z L ( a )) ;! H 0 ( E 1 E 2 L j Z t ) : Since byhyp othesis there exists F 2jL j Z t j weget S 2jL ( a ) j a surface bred over B , smo oth at a general bre and suchthat S t = F . Again, we can supp ose S is already
ON THE SLOPE OF BIELLIPTIC FIBRATIONS 5 smo oth. Let : S ;! B and F m = ; 1 ( m ). For m 2 B general wehavethat F m is an smo oth curve of genus 5 having at least two bielliptic involutions given by the inclusion F m E ( m ) E m (if g ( m )= m )asa (2 2)-divisor. We claim that for general m 2 B , F m has exactly two bielliptic involutions. Since this is the case for F = F t we only have to prove that having at most two of them is an op en condition. Consider W 1 4 ( ) ;! B , the relative Brill-No ether lo cus of (at least over an op en set of B , see 16]), after a base change if necessary. The numb er of bielliptic involutions of F m is given by the number of elliptic comp onents of W 1 4 ( F m ) = W 1 4 ( ) m . Then, having at most twoof such comp onents is obviously an op en condition. We claim that S is not a (birational) double cover of any elliptic bration : V ;! B . Indeed, assume wehave a double cover f : S ;! V (we can supp ose f everywhere dened after some blow-ups). Consider the base change diagram Z / / Z S ? O O / / e f S ? O O f e V / / e V B / / B For S wehave three double covers of elliptic brations over B : e f : S ;! e V f i : S ;! V f i = i j S i =1 2 Set U = f m 2 B j E m 6 = E ( m ) E m E ( m ) and e E m are smo oth and F m has exactly two bielliptic involutions g (where e E m = e ; 1 ( m )). Wehavethat U is a non-empty op en set of B . Since f 1 j F m , f 2 j F m , e f j F m are double covers of E ( m ) , E m and e E m resp ectively wehave that for every m 2 U , e E m = E ( m ) or e E m = E m . If g 1 = g j U : U ;! P 1 , g 2 = g j U : U ;! P 1 and e g : U ;! P 1 are the mo dular morphisms induced by , and e over U resp ectively wehavethat e g = g 1 or e g = g 2 . Assume e g = g 2 . As wehave t 1 t 2 2 U and ( t 1 )= t 2 weget E t 1 = ; 1 ( t 1 )= e ; 1 ( t 1 ) = e ; 1 ( t 2 )= ; 1 ( t 2 )= E t 2 since e is induced by : V ;! B and then e ; 1 ( m ) = e ; 1 ( ( m )) for all m 2 B . But this is imp osible since byhyp othesis E t 1 = E 1 6 = E 2 = E t 2 . 2
6 MIGUEL A. BARJA 2 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 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 1 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
ON THE SLOPE OF BIELLIPTIC FIBRATIONS 7 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 2.1 Let : S ;! B bearelatively minimal biel liptic bration of genus g 6 . Let V be the relative minimal model of the el liptic bration obtainedinsection 1. Then (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) : (1) 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 : (2) 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 : (3)
8 MIGUEL A. BARJA 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 +(4) +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 : Then (a) follows from (1), (2), (3) and (4) 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 (1), (2), (3) and (4). 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 2.2 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. Griths, J. Harris. Geometry of algebraic curves , vol I. Grund. Math. Wiss. 267 . Springer-Verlag 1985. 2. M.A. Barja, J.C. Naranjo. Glueing structures on a bration . Preprint. 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 . Springer- Verlag (1977). 6. E. Horikawa. On deformation of Quintic Surfaces .Inventiones math. 31 (1975), 43{85.
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