Isotrivial fibred surfaces

Annali di Matematica pura ed applicata (IV), Vol. CLXXI (1996), pp. 63-81

Isotrivial Fibred Surfaces (*).

FERNANDO SERRAN0

Abstrac t . - An isotrivial surface is a smooth projective surface endowed with a morphism onto a curve such that all smooth fibres are isomorphic to each other. Such a surface is birationaUy isomorphic to a quotient of a product of curves by the action of a finite group. Starting with this birational description, several biregular features are analysed~ In particular, the canonical bundle of a particular model is explicitely computed.

1. - I n t r o d u c t i o n .

This paper deals with particular kinds of fibrations 9: S - ) C from a smooth projective surface onto a smooth curve. We are interested in the case when all smooth fibres are mutually isomorphic (and non-rational). One such fibration, and the surface supporting it, are said to be isotrivial. The birational aspect of the situation is known to be rather simple, namely.

(1.1) If 9 is isotrivial, with general fibre isomorphic to A, then there exists a smooth curve B and a finite group G acting algebraically on A and B, such that S is birational to (A x B)/G, C ~-B/G, and the diagram

S - - -9 (A x B) /G

C > B /G

(*) Entrata in Redazione il 25 ottobre 1994. Indirizzo dell'A.: Departament d'/~lgebra i Geometria, Facultat de Matem~tiques, Universi-

tat de Barcelona, Gran Via 585, 08007 Barcelona (Spain). Partially supported by the European Science Program, project (,Geometry of Algebraic Va-

rieties~,, contract no. SCI-0398-C(A), and by CICYT research project no. PS90-0669 (Ministerio de EducaciSn y Ciencia; Spain).

64 FERNANDO SERRANO: Isotrivial fibred surfaces

commutes. Here, G is acting on A x B componentwise (i.e.V. (a, b) = (~. a, 7" b) for E G) and the right vertical arrow is the natural projection.

Our aim is to apply this birational information in order to obtain results of a biregular nature. These are best described on a particular birational model of S, called standard isotrivial in the sequel, defined to be the minimal desingularization R of (A x B)/G. The obvious maps a: R ~ (A/G), fl: R ---) (B/G) will be referred to as the natural projections. After describing the singular fibres that can occur in a or fl (The- orem 2.1), the main result of the paper is stated as Theorem 4.1, where the canonical bundle of R is computed in terms of the natural projections.

Some required preliminaries are developed in Section 3. They also lead, under some restrictions, to a characterization of the quotients (A x B)/G which are smooth as being precisely those surfaces with splitting tangent bundle (Proposition 3.3).

In the last Section we gather together some independent remarks. The remaining implication of the criterion for isotriviality in [16] is proven here. It is also shown that isotrivial surfaces have a non-positive topological index. We give at the end an alternative proof of a result first stated by KATSURA and UENO [10].

2. - S t r u c t u r e o f i s o t r i v i a l m a p s .

We work, for simplicity, over the field of complex numbers. The name fibration is here employed to mean any morphism ~0: S --, C with connected fibres from a smooth projective surface onto a smooth projective curve. The arithmetic genus of the fibres is always assumed to be > 0. The irregularity of S (respectively, the genus of C) is de- noted q(S) := dimHl(S , Os) (resp. g(C)). As customary, we speak of a ( - 1) or a ( - 2)-curve B c S to refer to a smooth rational curve with B 2 = - 1 or - 2 respectively. Relatively minimal fibrations are those whose fibres are free from ( - D-curves. The multiplicity of a fibre is the greatest common divisor of the multiplicities of its irreducible components.

An isotrivial fibration is called a quasi-bundle if every singular fibre is a multiple of a smooth curve. An isotrivial and smooth fibration is afibre bundle. For economy of notation, a surface is also called isotrivial (resp. quasi-bundle, fibre bundle) if it admits an isotrivial (resp. quasi-bundle, fibre bundle) fibration.

No confusion should arise if we sometimes denote with the same symbol a divisor D and its associated invertible sheaf r and employ additive or multiplicative notation interchangeably.

Statement (1.1) from the Introduction works in higher dimensions as well (see [12]; also [11], 2.11). A sketchy proof for surfaces goes as follows:

(2.0.1) Let C denote the open set in C over which ~ is smooth (notation as in (1.1)); : = ~ - 1 ( ~ ) . The monodromy r Z l ( C ) - ) A u t H l ( A, Z) is derived from an analytic

action on the general fibre A.

FERNANDO SERRANO: Isotrivial fibred surfaces 65

We claim that a finite morphism C1 ~ C exists such that S x C, is birational to c

A x C1 over C1. For one thing, if g(A) ~> 2 then Aut (A) (so also Im Q) is finite, and the surjection z l (C) --+ ImQ gives rise to an unramified cover / ) --* C such that the funda- mental group z l ( / ) ) equals KerQ. Since Q(Zl( /)))= 0, S x / ) ~ / ) has trivial mon-

odromy, and the claim follows readily by taking as C~ any smooth compactification o f / ) .

When A is elliptic, Im Q is also finite because translation by the grouplaw of A be- come trivial in cohomology. We do as before and get a map S x D --+ D with trivial

monodromy, from which we derive a relatively minimal morphism ~2: T--+E with trivial monodromy, T and E being projective manifolds. Consider the exact sequence

0 - - )HI (E , Q)-- .H1 (T, Q)-~i H1 (A, Q).

If E denotes the open set on E on which ~ is smooth, by [3] we get I m f = = H ~ (A, Q)~IcV) = H ~ (A, Q). Then q(T) = g(E) + 1, so )~(OT) = 0 and tP is a quasi-bundle ([15]). A suitable base change on ~ yields a smooth fibration, from which the above claim follows ([2], VI 7 and VI 8).

The next step is to consider the Galois closure C2 of C1 over C, with Galois group H. Then S • C2 is birational to A • C2 over C2. By acting trivially on S, H acts on S x C2

C C (so also on A • C2), and (S x c C2)/H -- S. Finally, by ([2], VI. 10), there is an ~tale cov-

ering B-~ C2, and a finite group G acting on A and B which satisfy the conditions stated in (1.1). Note that the actions of G can be assumed to befaithfid (i~e. no non- zero element acts trivially), since the subgroup N _c G of element acting trivially on A is normal, and we have (A x B)/G -- (A x B' )/G ' with B' := B /N , G' := GIN.

(2.0.2) Unless otherwise mentioned, we shall fix a i'mite group G which acts faith- fully on two smooth projective curves A and B, and set V := (A x B)/G. The stabilizer H c G of a point b e B is a cyclic group ([4], p. 106). I f H acts freely onA (i.e. no point is fixed by a non-zero element of G) then V is smooth along the scheme-theoretic fibre of a: V--~ (B/G) over b e B/G, and this fibre consists of the curve A / H counted with multiplicity [H I := order of H. Then, smooth fibres of o are 'all isomorphic to A. On the contrary, if a e A is fixed by some non-zero element of H, then V has a cyclic quotient singularity over the point (a, b) e (A x B)/G. In this case, the fibre of (a, b) on the minimal desingularization )l: R --) V is an H-J string (abbreviation of Hirzebruch- Jung string), that is to say, a connected union of smooth rational curves Z1, ..., Z~ with self-intersection numbers ~< - 2, and ordered linearly so that Z~Zi +, = 1 for all i, and ZiZj = 0 if l i - J l >~ 2 ([1], III 5.4). These observations lead to the following statement, which describes the singular fibres that can arise in a standard (cf. Intro- duction) isotrivial fibration:


TI-IEOREYi 2.1. - Set V := (.4 x B)/G as above. Denote by 4: R -+ V the minimal desingularization and let q~: R--+ (BIG) be the associated isotrivial fibration. Take any point b ~ B, and let F denote the fibre of ~ over b ~ BIG. Then:

(i) The reduced structure of F is the union of an irreducible smooth curve Y, called the central component of F, and either none or at least two mutually disjoint H-J strings, each one meeting Y at one point. These strings are in one-to-one corre- spondence with the branch points of A--) (A/H), H c G being the stabilizer of b.

(ii) The intersection of a string with Y is transversa~ and it takes place at only one of the end component2 of the string.

(iii) Y is isomorphic to A/H, and has multiplicity equal to I HI in F.

(iv) Denote by ~f: R --+ (A/G) the other natural projection. Let L = 2 Zi be an 4=1

H-J string in F as in (i), ordered linearly, and consider the central component X of ~-10p(L)). Then L meets X and Y at opposite ends, i.e. either Z IX = Z~Y--- 1 or Z , X = Z I Y = 1.

PROOF. - Let Y be the closure of Fre d after removing the curves collapsed by )t. Consider the diagram

A x B

B

-----> (A x B)/H ----> (A x B)/G

- - - . B/H B/G.

For one thing, the set-theoretic image ofA • {b} in (A • B) /H is birationally isomorphic to A/H, so actually isomorphic in view of the projection (A • B)/H-+ (A/H). Also, (B/H) --+ (B/G) is unramified at the image b' of b in B/H. Since smoothness is preserved by base-change, the reduced fibre of b' in (A • B) /H maps isomorphically onto the reduced fibre of b in (.4 • B)/G, which is 2(Y). This shows that ~(Y) is irreducible, smooth and isomorphic to A/H, so the same holds for Y. To complete (iii) it suffices to note that a general fibre of V-+ (A/G) meets the general fibre of V--+ -+(B/G) at ]GI points, while it meets +~(Y) at IGI/IHI points.

The fact that not a unique (maximal) H-J string can exist in F follows either by noting that A --+ (A/H) cannot have a unique branch point (Lemma 5.7), or by the fol-

lowing direct computation. Indeed, if we write Zo := Y, and let F = ~ miZ~, with n i = 0

the string ~ Z~ ordered linearly, then we get the absurdity i = l [ t 2 n n - 1

Next we look at the local situation. Consider the cyclic group Er as acting on C 2 by


(ul, u 2 ) ~ (~1ul , ~ u 2 ) , where a~, a2 are positive integers and ~ is a primitive r-th root of unity. Without loss of generality we may assume (al, r) = (a2, r) = 1 ([6], p. 295). The singularity of C 2/Z~ at the origin is resolved by replacing the origin with

n

an H-J string U 1= zi (linearly ordered). If Zo, Zn+~ denote the proper transforms of

the coordinate axis Ul = 0, u2 = 0 in the desingularization, then it follows from [8] (see also [18], p. 43) that Z0, Z1, ..., Z~, Z . + 1 form a linear chain. This completes the proof of the Theorem. []

In the sequel, when we speak of an H-J string contained in a fibre F of a standard isotrivial fibration, we mean a connected component of the closure of F - Y, Y being the central component.

FREITAG ( [ 5 ] , p. 99) proves that if a finite group G acts analytically on a compact complex manifold V, and W denotes any non-singular model of V/G, then

k k H~ A ~2w)= H~ A ~ v ) G for any k >~ 0. From this it readily follows:

PROPOSITION 2.2. - Let S be a smooth surface birationally isomorphic to (A x B)/G. Then

q(S) = g(A/G) + g(B/G).

PROOF. - If Zl and z2 denote the projection of A x B onto its factors, we have QA • B = z~' (QA) ~ ~ ' (QB), and thus

q(S) = dimH~ x B, ~2AxB) 6 = dimH~ ~A) 6 +

+dimH~ ~B) G = g(A/G) + g(B/G). []

REMARK 2.3. - In view of (2.1), the structure of the fibres of arbitrary isotrivial maps is somehow explained by the following observation, If ~: S---) C is a relatively minimal isotrivial fibration, birationally equivalent to (A x B)/G ~ (B/G) -- C as in (1.1), and R is the minimal desingularization of (A • B)/G, then the natural birational map a: R---~S is a morphism. Moreover, f is an isomorphism provided that q(S) > g(C). This last assertion is proved as follows. If Y denotes the central component of some fibre of 9 ~ a: R--> C, then Y maps surjectively onto A/G. Therefore g(Y) ~ g ( A / G ) = q ( S ) - g ( C ) by (2.2), so in case q(S)>g(C), ~ o a is relatively minimal.

FURTHER REMARKS:

(2.4) The map a in (2.3) is not always an isomorphism. A relatively minimal isotrivial but non-standard map can be easily constructed by considering the family of plane cubics given in affine coordinates by y2= X a + )~, 2 e C. The smooth curves in the


family are all isomorphic, since the change of variables Y = Y0 V~, X = Xo ~/~ leads to the equation Y02 = X0 ~ + i. The family degenerates to the cuspidal cubic y2 = X 8" Now take the surface T in F s defined by X2X~ - XSo - XaX~ = O. The pencil of planes aXe - - bX8 = 0, (a : b) e p1, cuts T along the above isotrivial family. Suitable blowing-up and desingularization leads to a smooth surface which admits an elliptic fibration onto F 1 with the cuspidal cubic as one of its singular fibres. This is the example sought.

(2.5) A quasi-bundle surface can never lie as an ample divisor in a smooth threefold, the reason being that a quasi-bundle is isomorphic to a smooth quotient (A x B) /G, and we can here apply Fujita's criterion ([7], 2.3 and 2.7).

(2.6) Every smooth fibration with hyperelliptic fibres is isotrivial ([19], 2.10).

3. - A c h a r a c t e r i z a t i o n o f quas i -bundle s .

We start with some generalities. Let us fix an arbitrary fibration ~: S --, C from a smooth surface onto a curve. The cotangent sheaves fit into an exact sequence

Dualizing one gets

(3.0.1)

0 --> ~v* (~2c) ~ P~s ~ Qs/c "--> O.

1 ~2 where Ts, Tc are the corresponding tangent sheaves, ~ := 8xtos( s/c, Os), and Ts/c := ~2 s/c is the so-called relative tangent sheaf of ~. Here * denotes dual of a module. Define

Js/c := image of Ts ---) cf * (Tc) .

Both Ts/c and J~/c can be written as second syzygies, so they are invertible sheaves. Since Js/c is torsion-free, the quotient J ~ c / J s / c is the structure sheaf of a zero-dimen- sional subscheme F c S.

Consider the exact sequence

(8.0.2) O.

The sheaves det (Ts) = K [ 1 and Ts/c | J~*c are isomorphic outside the codimension 2 subset F. Hence

(3 .o .3) 1 = T s / c | .


The following equality holds ([16], 1.1):

c2(S) + Cl(Ts/c)'cl(J~/c)= ~ length(Or, x). x ~ S

In the following statement we consider the line bundles Ks, Ts/c and J~/c as divisors, and apply additive notation throughout. Accordingly, products and squares are always understood as intersection numbers.

PROPOSITION 3.1. - Let { v iEi }i ~ x stand for the set of components of all singular fibres of q~. Here v i denotes the multiplicity of Ei within its fibre. Then:

(i) J$c = ~ * ( g c ) + ~ ( v i - 1)Ei. i E I

(fO T~/C = Ks - q~*(gc) - ~ (v~ - 1)E~. i e I

(ffi) The suppor~ of s is the set of points x e S such that ( ~ -1 (~0(X)))red/S singular at x.

(iv) K~ <~ T~/c + 2c2(S), with equality i f and only i f aU singular fibres are multiples of smooth curves.

(v) (J~c) 2 ~< T2/c .

PROOF. - Parts (i) and (ii) are proven in ([16], 1.1). To see (iv), write G = J ~ for simplicity, and note that G 2 ~< 0 because of (i). Squaring (3.0.3) and combining with (3.0.4) one gets

- - T 2 G 2 K~ = T~/c + G2 + 2Ts/c'G - s/c + + 2 c ~ ( S ) - 2 ~ length ( or,~) <~ T21c + 2C2(S ). x E S

If the preceding inequality is an equality then/~ is the empty set, and so all fibres have a smooth reduced structure in view of (iii), thus proving (iv). Let F be a general fibre. Statement (v) is equivalent to

K ~ - 8 ( g ( C ) - 1 ) ( g ( F ) - 1 ) - 2Ks ~ ( v ~ - 1 ) E ~ i> 0, i e l

which is an immediate consequence of (3.5.1) in [16]. Another proof of (v) goes by applying the Ks-semistability of ~2s ([19]) to J ~ c c Q s , and combining with (3.0.3); for

J~/c'Ks <~ (1 /2) (Ks .de t~s ) = K2/2 .

Finally, (iii) will be derived from results of Iversen [9]. Take x e S, and pick a parameter r for C at ~(x), and a parameter system s, t for x at S. Set f : = r o r and write f~, ft instead of af/as, af/at for simplicity. The completion of ~ in (3.0.1) at x is ~x = = o/(fs , Ji), where O := Os, x. Let now d denote the greatest common divisor off~,f~. Note that the following sequence is exact:

(3.1.1) 0 ----> � 9 ~ 0 / (~ ,~) --> O/(d) ---> O.


Set A := ~ ( ~ - 1)Ei. The snake lemma applied to the diagram i ~ I

yields

( 3 . 1 . 2 )

o - - - . J s / c - - - > ~ * ( T c ) - - - > ~ >

0 > J ~ -----> ~v*(Tc) ----> OA >

0

0

0 ~ Or--* 8 -* (9A-* O.

Locally at x, A is defined by the equation d = 0. Consequently, (3.1.1) is the local ver- sion at x of (3.1.2). In particular

length ((Or,,) = length O/(f~/d, f t /d) .

Let us denote length(Or, z) by/tx(cp). After ([9], 6.1) we know

~(~o) = ~ (25~(Ek) - v~(Ek)) + ~ (Ek'E1)x + 1 k e L (k, l ) e L x L

k ~ l

where the following symbols are involved:

- - (Ek}k~L is the set of components of ~o-l(~o(x)) passing through x.

- - 5~ (E) := codimension of OE. x in its normalization (as a vector space over the ground field).

- - v~(E) := number of analytic branches of E at x.

- - (Ek'El)~ := local intersection number of Ek, Ez at x.

Obviously (E~ El)~ >I v~ (Ek)" u~ (E~). Therefore tt x (~v) ~> 1, unless only one component E of cp -1 (~0(x)) in passing through x. In this case, tt~(r = 25~(E) - % ( E ) + 1. If 5~(E) = 0 then %(E) = i and ~(~0) = 0. But 5x(E) I> v~(E) - 1, as is readily seen. Summing up, we conclude that tt~ (~0) = 0 if and only if (~0-1 (~v(x)))r~ d is non-singular at x. []

LEMMA 3.2. - Let el: S - - )C be any fibration with fbre of genus >I 2. Then cf is isotrivial i f and only i f q). (Ts) is not the zero sheaf.

PROOF. - Set C := {P ~ CI~-1 (p) is smooth}, S : = ~ - 1 (~), ~: ~__~ ~ the restriction of ~0. Applying ~ . to the exact sequence 0-~ T~/5--~ T~--~ ~ * (T S)--~ 0 yields

0 --) ~). (T~/~) --, ~ . (T~) --> TS~-~ R 1 q). (T~/5).

For any fibre F of ~, (T~/~)IF = TF has no global sections, so ~ , (T~/~) = 0. It is well-known that ~ is isotrivial if and only if the Kodaira-Spencer map 5 vanishes. Inasmuch as T5 is invertible and RI~, (T~/5) is locally free, one sees that 5 being zero


is equivalent to failing to be one-to-one, and this is equivalent to 0 r ~, (T~) (= =~o.(Ts)tS). Moreover 9 . ( T s ) is torsion-free over a smooth curve, so locally free. []

The following result yields a characterization of quasi-bundles in terms of the splitting behaviour of the tangent bundle:

PROPOSITION 3.3. - Let S be a smooth surface of general type, with irregularity either >- 3 or = 1. Then S admits a quasi-bundle structure i f and only i f the tangent bundle of S splits as a direct sum of line bundles. More precisely, i f S = (A • B)/G is a quasi-bundle, then

Ts = TS/(A/e) ~ Ts/(8/o) �9

PROOF. - First suppose that Ts splits, mid write f2s = M ~ N, where M, N are invertible sheaves. Let us deal with the case q(S) I> 3 first. We may assume h ~ (S, M) ~> I>2. Take any two independant sections f l l , f l 2 eH~176 f2s). Then fll Aft2 -= 0, and by Castelnuovo-de Franchis Theorem ([1], IV. 4.1), there exists a curve C of genus at least 2, a fibration ~: S--* C, and two 1-forms a l , a2 e H~ f21) such that fli = ~*(a~), i = 1, 2.

Let L denote the invertible subsheaf of 9"(f2 c) generated by f f*(a l ) . The composite

L c-->q~*($-2c)C--) D~s = M ~ N---> N

turns out to be the zero map. Since cp*(~c)/L is torsion, its image in N vanishes. Hence e f * ( f 2 c ) ~ N and its dual N*- - )9* (Tc ) are the zero maps. Consider the sequence

(3.3.1) II M * ~ N * .

Since N*--)Js/c vanishes, M*-->Js/c is onto. We thus have the isomorphism M* -- Js/c, which gives the splitting of (3.3.1). Js/c being an invertible sheaf, it follows that all singular fibres of 9 are multiples of smooth curves (3.1 (ill)). Note that 9 , (Js/c) is locally free of rank 1, because (Js/c)It" ~- OF for a general fibre F. Therefore cp,(Ts)=Cp,(Ts/c)(~cp,(Js /c)~O. The preceding Lemma implies that r is isotrivial.

72 FERNANDO SERRANO: Isotrivial f ibred surfaces

Next assume q(S) = 1. Let ~v: S --~ C denote the Albanese map of S, where C = = AIb(S) is an elliptic curve. Suppose H~ N ) = 0, and consider the sequence

(3.3.2) 0 --> cp * ( Q c ) --> Q s ---> ~ s / c "-> O .

Here ~v * (Qc) = COs, and the composite ~v * (~9c) ---) ~gs --* N vanishes because N has no sections. If D denotes the zero set of ~* ( tQc ) - - .M then M = Os(D), and t~s/c = = OD (D) @ N. Therefore

V s ) = , (3.3.3) 8xtos ( s/c, Os) (OD (D), OD

the last equality coming from

O --* Os--* Os(D) --~ OD(D) --> O .

Dualizing (3.3.2) yields

O Ts/c Ts * ( Tc ) --, O ,

from which Js/c = O s ( - D ) . This implies that D is contained in fibres of % so that 9 , ( M * ) - - 9 , ( O s ( - D ) ) ~ O. We get as above that q ~ , ( T s ) ~ 0 and ~0 is isotrivial. Moreover Js/c is locally free, so every singular fibre of ~o is a multiple of a smooth curve (3.1).

Conversely, let S = (A x B ) / G be a quasi-bundle. The sheaves JS/(A/G), JS/(B/G) are invertible. Theorem 4.1 yields K[ l=Js / (n /a ) | (3.0.3) gives K [ ~= = TS/(A/G)| JS/(A/G). Then TS/(A/O)~--JS/(B/G), and the inclusion TS/(A/G) r Ts provides the splitting of (3.3.1), with C = B/G. �9

4. - T he c a n o n i c a l bundle .

Theorem 4.1 below provides an expression for the canonical bundle of a standard isotrivial surface in terms of the fibres of the two natural projections. First we shall fLX the notation employed throughout.

Let S be a standard isotrivial surface, given as the minimal desingularization of V := (A x B ) / G , with morphism ;t: S --~ V. Set C := B /G , D := A / G . Write Q: V--~ C, a: V-->D for the natural projections. The data are summarized in the commutative diagram

(4.0.1)

S

B/G = C < V ~ D = A/G l(P, C x D


THEOREM 4.1. - Let S be a standard isotrivial surface as above, with associated fibrations qg: S--~C, v2: S---) D. Let {SiEi}i~i (respectively {yjFj}j~j) denote the components of all singular fibres of qJ (resp. of ~), with their multiplicities attached. Finally, let {Zh}h~H stand for the set of curves contracted to points by (~, ?): S ~ C • D (the exceptional locus). Then we have

Ks = q~*(Kc) + ~fl*(KD) + E (~i - 1)E~ + E (~j - 1)Fj + E Zh. i e l j ~ J h e l l

PROOF. - As at the beginning of Section 3, we define

Js/c := image of Ts --* ~ * (Tc),

JS/D :=image of Ts--*~P*(TD).

In view of (3.1), the formula to be proven is equivalent to

Note that {Zh}h~H is a subset both of {Ei}i~l and {Fj}j~z. We shall use the notation of diagram (4.0.1). To begin with, let {Xl, ..., x~ } c C

stand for the set of points on which ~ has a singular fibre. Write ~ - l ( x i) = miAi + + Mi, where Ai denotes the central component (see (2.1)), mi ~> 2 is the multiplicity of

Ai, and Mi is contained in the exceptional locus of 4: S--* V. Likewise, every smooth fibre of ~ is isomorphic to B. We take {YI, ..-, Y~ } c D to be the points on which ~ has a singular fibre, and write ~ - l ( y j ) = njBj + Nj, the central component being Bj, and with Nj in the exceptional locus of 4. The scheme-theoretic fibre ~-1 (xi) is of the form miAi, where Ai is a Weft divisor. Likewise a- l (y j ) = njBj.

Let Vo denote the smooth locus of V = (A • B)/G, with natural inclusion j: V0 ~ V. Set z := (Q, a): V--. C • D. The restriction V0 --* z(Vo ) of z is finite of degree equal to I GI.

d

Set L := ~ (mi - 1).4i + ~ (nj - 1)/~j. By ([1], I, 16.1), the ramification locus of i = l j = l

V0---> z(Vo) is Liv o . Since Kv = j . (Kyo), the canonical bundle formula for a finite map ([1], p. 41) yields Kv = z * ( K c • Equivalently,

(4.1.1) d e

Kv= e*(Kc) + o*(Ko) + E + E - i=1 j=l

Working with Q-Weil divisors we have

(4.1.2) 4" (Ai) = (1/m~) ~* Oc(xi) = Ai + (1/mi)Mi ,

(4.1.3) 4* (Bj ) = (1/nj ) ~* r (Yj ) = Bj + (1/nj ) Nj .

The support of W := 4" (Kv) - Ks is contained in [J Zh, i.e. the exceptional set of 4, hGH

74 FERNANDO SERRANO: Isotrivial f ibred surfaces

and Z h ' ~ * ( K v ) = 0 for all h e H . The intersection pairing is negative-definite on the Q-vector space spanned by {Zh}h~H. AlSO WZh = - K s ' Z k <<-0 for all h. It follows that W is an effective Q-divisor ([20], 7.1).

Combining Ks = ) . * ( K v ) - W with (4.1.1), (4.1.2) and (4.1.3) we have

(4.1.4) d

Ks = cf*(Kc) + ~g*(KD) + E (mi - 1)Ai + E (nj - 1)Bj + i = l j = l

d + ~, ((mi - 1 ) /m~)Mi + E ((nj - 1 ) / n j ) N j - W .

i=1 j = l

On the other hand, the pull back of ~ C • by (~, ~ g ) : S - - ~ C x D equals ~* (t9 c) ~ ~* (t2D). Dualizing the exact sequence

( ~ , ~1):~ (~'~C x D ) "'> ~r ~ ~'~S/C x D "-"> 0

we get an injection Ts ~ ~ * (T c) @ ~g*(TD) whose kernel is Ts/c A TS/D = O. The natural map Ts/c--> Js/n derived from the sequences

0 --* Ts/c--* T s " ~ J s / c O O,

o ~ Ts/o ~ Ts--~ Js/o ~ O ,

turns out to be one-to-one. Then, by symmetry, both Ts/c-~ J~*/*D and Ts/D--* J~*/~ are one-to-one. Taking (3.0.3) into account, we have a monomorphism

-1 = Vs/c | |

and so

(4.1.5) Ks = Jhc | JhD | v s ( ~

for some effective divisor F. Denote by/~a (respectively, vh) the multiplicity of Zh within its fibre by ~ (resp.

by kY). In view of (3.1), we can write (4.1.5) as

(4.1.6) d e

Ks = qg*(Kc) + ~u* (KD) + ~ (mi - 1)Ai + E (nj - 1)Bj + i = 1 j = l

+ E ( (~h - 1) + (vh - 1 ) ) z~ + r . h e l l

By (4.1.4), the support of F is contained in the exceptional locus of )k Take any Zh, and write Z = Zh, ~ =t th, v = vh for simplicity. Z is contained in some fibre ~o-l(x~) = = m~A~ + Mi, so that Z is a component of Mi with multiplicity it. Also, Z is contained in ~ - l ( y j ) = njBj + Nj , and has multiplici~ , as a component of Nj. Write m = mi , n = nj. Finally, denote by a I> 0 (respectively, fl >I 0) the multiplicity of Z as a component of F (resp. W). While a is an integer, fl is just a rational number. By equating the

FERNANDO SERRANO: I so t r iv ia l f i b red sur faces 75

coefficients of Z in (4.1.4) and (4.1.6) we derive

( m - 1 ) # / m + (n - 1 ) v / n - fl = # + v - 2 + a

and

(4.1.7) - 2 < /~ /m + v / n - 2 = - a - fl <~ O .

We thus conclude a = 0 or 1. This holding for all Zh, it implies that F is reduced. We momentarily interrupt the proof, of (4.1) in order to show:

REMARK 4 .2 . - With the previous notation we have:

(i) # / m + v / n <~ 1.

(ii) If )~(Z) is not a rational double point of V, then # / m + v / n < 1.

For one thing, (i) follows from (4.1.7) together with the fact, to be proven later, that a = 1 for all Z. Now, assume there is equality in (i). Then fl = 0 because of (4.1.7), so W Z >I O. On the other hand,

W Z = (~ * ( K v ) - K s ) Z = - K s Z <<. O .

So 0 = W Z = K s Z , and Z 2 = - 2 since Z = t )1 . We recall that all connected components of the exceptional locus of )~ are H-J strings. In the one containing Z, the irreducible components meeting Z are not part of W, because W Z = O. By the previous ar- gument, these are also ( - 2)-curves. Moving along the H-J string we conclude by in-

duction that the whole connected component of U zh containing z is an H-J string of h e l l

( - 2)-curves, and none of these is a component of W. Such a string is the exceptional locus of a rational double point on V. �9

The proof of (4.1) continues. We have

(4.1.8) J~/c = q ) * ( K c ) + E ( 5 i - 1)Ei , i e I

(4.1.9) J~. = ~*(K.) + Z (r~ - 1)F~. j~J

Let A0 denote the central component of ~-1 (~(Z)), and B0 the one of ~y-1 (~P(Z)). Note that Z and Ao are two of the Ei' s. We have Ao ~ 5iEi = 0, Aoq~* (Kc ) = 0. In

i e l

the sequel we use additive notation with T~ c and J~c- One has

T ~ c ' A o = (Ks - ~* ( K c ) - i~, ~ (5~- i)Ei)Ao =

=(Ks+ E , ) m = 2 g ( m ) - 2 +.4o .

i e l E i ~ A o


From (4.1.5), we get

SO

J~/D + F)Ao = 2g(Ao) - 2 + Ao ~ Ei . Ei ~ -4o

Also /'Ao ~< A0 Ei ~Ao

(4.1.10)

Ei, because F is reduced. Then

J~*/DAo >I 2g(Ao) - 2.

On the other hand, the degree of the ramification locus of ~ IAo : Ao --* D is not smaller than Ao" ~ ( y j - 1)Fj. This fact, combined with (4.1.9) and Hurwitz formula for

j e J

~YlAo:Ao--*D yields J~D'Ao <~ 2g(Ao) - 2. We conclude

(4.1.11) J~D'Ao = 2g(Ao) - 2,

(4.1.12) FAo = Ao ~ Ei �9 E~ ~Ao

By symmetry we have

(4.1.13) FBo = Bo ~, Fj . Fj ;~ B 0

Equality (4.1.11) is equivalent t o T~/D'A o = - A ~ . In the sequel we assume that Z is not in the support of F, so we shall reach a contradiction. From (4.1.12) and (4.1.13) we get

(4.1.14) ZAo = ZBo = O.

Then

(4.1.15) Z F = Z(Ks - J ~ c - J~/D) = ZKs + Z ~, E~ + Z Z Fj = " i e I j ~ J

= - 2 + Z 2 + Z ~ E i + Z ~ Fj . E~ ~ Z Fj ~ Z

But Z ~ Ei ~< 2 because ZAo = 0, and the Ei ~ s distinct from A0 which appear in E ~ Z

-I(~(Z)) are organized as disjoint H-J strings. In the same fashion Z ~ F i <~ 2, F~Z

which combined with Z 2 ~< - 2 and (4.1.15) yield ZF <~ O. Hence ZF = 0, and Z E ~ Z

E i = Z ~ Fj = 2. We deduce that the components of X-1 (X(Z)) which intersect Z are Fizz

not inside F. Apply the same reasoning to them. After finitely many steps we must reach a component of X-1 (X(Z)) which does not belong to Supp (F) but meets Ao. This contradicts (4.1.14) when applied to this component. The absurdity comes from as- suming that Z is not inside Supp (/3.

FERNAND0 SERRAN0: Isotrivial f ibred surfaces 77

As a conclusion, we can improve (4.1.6) as

d

(4.1.16) Ks = r + W*(KD) + ~ (mi - 1)Ai + i = 1

+ ~ (nj-1)Bj+ Z (tth+Vh--1)Zh. j = l h ~ H

REMARK 4.3. - The proof of (4.1) has not fully used Theorem 2.1, but only parts (i) and (ifi). We can indeed derive parts (ii) and (iv) of (2.1) from (4.1) above. To this pur-

n

pose, let S, ~ and yJ be as in (4.1), and take any connected component L = ~ Zi of the i = l

exceptional locus of (~, y~): S - ) C x D. L is an H-J string which we assume to be linearly ordered. Denote by Ao (respectively, B0) the central component of ~-1 (~(L)) (resp. V -1 (V(L))). Our aim is to show that L meets Ao and B0 at distinct end components, and transversally. Let Z be any of the Zi' s. From (4.1.5), and (4.1.9) we get

Z F = Z(Ks - J ~ / c - J~D) = - 2 + z2 + z ~'. Ei + Z E Fi . Ei~Z Fj~Z

From the structure of F it follows

Z F = Z2 + Z ~, E i - ZAo . E i ~ Z

We thus get

(4.3.1) O >I - Z A o = - 2 + Z ~ Fj . F~.~Z

By symmetry, Z ~ Ei ~< 2. It readily follows that if Z meets A0 then ZAo = 1, and Z Ei~Z

is either ZI or Zn. The same holds for Bo. Finally, suppose ZnBo = 1 and n I> 2. Then Zn ~ Fj= Z n(Bo + Zn_1)= 2,so(4.3.1)yields -Z~A o= -2 + Zn ~ =0. There-

F~Z~ Fj~Z~ fore Z1Ao = 1.

5. - N u m e r i c a l p r o p e r t i e s .

We gather in this Section some independant results about isotrivial surfaces. ((5.1), (5.3), (5.5), (5.6)).

For D a divisor on a surface S, denote by K(S, D) the D-dimension of S, defined as the maximum dimension of ~ inDi ( S ) , for all n I> 1, where ~b inDi is the rational map de-

** termined by the linear system I nDI . In [16] it is shown that K(S, ~s / c ) = 2 for a non- isotrivial fibration ~: S o C with fibre genus I> 2. The converse is essentially true, but not always. More precisely, if ~ is isotrivial but not standard, then it may happen

* * that K ( S , ~ 2 s / c ) = 2 , but the standard associated fibration R--->C satisfies *(R, * * QR/C) = 1 in view of:

78 FERNkaNDO SERRANO: Isotrivial fibred surfaces

PROPOSITION 5.1. - Let qo: S -+ C be a standard isotrivial fibration. Then (Ts/c) 2 <<. ~< 0, and K~ <<. 2c~(S). Moreover, ~:(S, ~2~/* c) = 1.

PROOF. - Suppose S is the minimal desingularization of (A x B)/G, with ~ being the natural map S -+ (B/G) = C. Set D := A /G, and write g*: S -+ D for the other natural projection. In view of (3.1) (ii) and (4.1) we have

* * - T~c = ~ ' * ( K ~ ) + A s / C -

where A is a divisor contained in fibres of g*. Thus T~2/c ~ 0, and K(S, ~2s/c) = 1. The inequality Ks 2 ~< 2 c2 (S) follows from (3.1) (iv). �9

Let v(S) := (1/3)(K~ - 2c2(S)) denote the topological index of S. We aim at proving that v(S) <<. 0 when S -+ C is isotrivial. In view of (5.1), this holds true if S is standard. When q(S) = g(C), S may not be standard, but a suitable sequence of blow-ups will eventually lead to a standard surface S. Since v decreases as we blow-up, the fact that r(S) ~< 0 does not guarantee v(S) <~ O, so we must find an alternative approach. To this purpose we shall recall a notion due to MIYAOKA [13]. Let 8 be a vector bundle over a projective variety X. The associated projective bundle P(8) is naturally endowed with a morphism z: P(8)--+X and a line bundle O(1) satisfying z . (O(1)) = 8. Take any ample divisor H on X. In [13], Miyaoka calls 8 almost everywhere ample (a.e. ample, for short) whenever there exists a proper Zarisld closed subset F _c X, and a real number /t > 0, such that, for any irreducible curve D in P(8), O(1).D ~> t> ,~z* (H)D as long as z(D) ~ F. The definition is actually independant of the choice of H. In the above situation one says that 8 is ample modulo F. It follows readily that in case 8 is ample modulo F, and Y is an irreducible subvariety of X not contained in F, then 8 IF is ample modulo s N Y.

LEMMA 5.2. - I f 8 is a.e. ample, then H~ 8*) = 0.

PROOF. - 2my global section of 8* yields a non-trivial morphism C9 x--+ 8". Dualizing we get a non-trivial map 8 --+ Ox whose image we denote by ~. One can find a smooth irreducible curve B in X such that B ~g F and aB := a ~ OB is a non-zero ideal of r hence invertible. Consider the cartesian diagram

v X

?

P(Sl8 ) ----> p(g)

B ~ > X.

The surjection 81B-+j~ defines a section a:B-+P(81B) of 0~' such that ~B = = (7oa)* O(1). Write D = ( 7 o a ) ( B ) . Then O>--deg(gs)=D.O(1)>O, absurd. �9

FERNAND0 SERRAN0: Isotrivial fibred surfaces 79

In [13] it is shown that t~ s is almost everywhere ample for surfaces with positive index. From this we derive:

PROPOSITION 5.3. - I f S admits an isotrivial fibration then K~ <~ 2c2 (S).

PROOF. - Otherwise ~9 s is a.e. ample. If F stands for a general fibre of the isotrivial map ~: S -o C, then (t9 s )iF is a.e. ample, so that H ~ (F, (~ ~ )IF) ~ - - 0 by (5.2). It follows that ~ . ( T s ) = 0, against (3.2). �9

We deal next with the numerical divisibility of fibres. Let ~: S -o C be any fibration. Suppose ~ has exactly t multiple fibres, whose multiplicities we denote ml, ..., mr. Set/~ for the least common multiple of ml, ..., mr. A general fibre F is always divisible by tt in H ~ (S, Z)/(torsion), or, equivalently in the group Num (S) of numerical equivalence classes (cf. [15], p. 68). But it may well happen that F is also divisible by a larger number. As a matter of fact, for each prime number p there exist fibre bundles with fibre divisible by p ([15]).

PROPOSITION 5.4. - With notation as above, i f at least one f b r e is simply connected, then tt is the largest integer dividing F.

PROOF. - Suppose F is divisible by r#, for some integer r > 1. We may assume that F is linearly equivalent to rttL + M, where L and M are divisors, and M is a torsion element in Pic (S). If sM is linearly equivalent to zero, then sF = srttL in Pic (S). Write n = srtt, and choose smooth fibres F1, . . . ,F~. The isomorphism Os(nL)~- = r + ... +F~) gives rise to an n-cyclic covering z: R - o S , totally ramified at F1, ..., F~ ([1], p. 42). Let R -o B -o C denote the Stein factorization of ~ o z.

CLAIM: deg (B -o C) < n. - Once this is proven, the contradiction arises as follows. The commutative diagram

B ~- C

shows that for any fibre F of ~, distinct from F1, ..., Fs, and for some connected component E of z -1 (F), Z lE: E - o F is unramified of degree > 1, which is imposible as long as F is simply connected.

As for the Claim, first assume r (L) = 0. Then H~ LIE) = 0 for a general fibre F, and so Ljr -~ r Since OF(F1 + ... + F~) = OF, the covering ~: R -o S restricts on F to an unramified n-cyclic covering, the one given by (LIE) | (9 F. B u t ~ - l ( F ) does not have n connected components, because LiE ~ r So d e g ( B - o C ) < n.

Finally we show that cp, (L) is zero, Suppose otherwise. Pick any fibre F =


= go -1 ( p ) , p e C. Then

N~ L | Os(mF)) = H ~ (C, go, (L | go* Oc(mP))) =

= H~ go,L | Oc(mP)) ~ 0 for m >> O.

We can thus assume that L + mF is effective. Since (L + m F ) F = 0, L + mF is contained in fibres. Let A = ~ diAi denote any connected component of L + mF, con-

i rained in the fibre Fo = ~ e~Ai. If we manage to show that/~A is always an integral

multiple of F0, we reach the desired contradiction, because rttL - F, and r > 1. Here - denotes numerical equivalence. From L 2= F 2= 0 we get A 2= O. This implies

A - ( p / q ) F o ([2], VIII 4), with p, q relatively prime integers. Combining ~ ( q d i - p e i ) A i - O with the fact that the Ai's are linearly independent in

Q @z Num (S), one gets qdi = pei for all i. Hence q divides all the ei' s, so that q divides/~

as well. []

COROLLARY 5.5. - Suppose go: S--~ C is isotrivial. I f there exists a fibre whose components are all rational, even i f singular, then tt is the largest integer dividing F. This applies, in particular, for a minimal isotrivial non-standard surface.

PROOF. - The multiplicity of a fibre coincides with the multiplicity of its total transform under the blow-up of S at a point. So, we can restrict ourselves to the situation where go is standard isotrivial, and one of the central components is a p l . It is now obvious that the corresponding fibre is simply connected, so that (5.4) applies. []

The following result is stated by KATSURA and UENO [10] for a rational base curve, but their proof works for an arbitrary base as well. Here l.c.m, stands for least common multiple.

PROPOSITION 5.6 ([10], 4.1). - Let go: S ---> C be an elliptic fibration with Z(Os) = 0, and q(S) = g( C) + 1. By ml , ..., mt we denote the multiplicities of the singular fibres of go. Then either t = 0 or t >i 2 and, for all i ~ { 1, ..., t }, m~ divides

1.c.m. (ink). [] k ~ i

k = l , . . . , t

We give below an alternative, more structural proof of this statement. It relies on the observation that the surfaces involved are elliptic quasi-bundles (see e.g. [15]), so they can be described as quotients (A • B)/G, where A is elliptic and go is the projection onto B/G = C. Since q(S) = g(A/G) + g(B/G), the assumptions imply that A / G is elliptic. Then G is acting on A by translations, so it is an abelian group of the form Z / (a ) (3 Z/(b). The multiplicities {ml, ..., mt} coincide with the branching orders of B---) (B/G). Then, (5.6) is just a consequence of


LEMMA 5.7. - Let G be any finite abelian group acting on a smooth projective curve B, and let ml, ..., mt denote the branching orders of the projection B ---) (B/G). Then either t = 0 or t >I 2 and~ for all i = 1, ..., t, m~ divides 1.c.m. (ink), where k runs in {1, . . . , t } but k ~ i .

A proof of (5.7) is given in ([14], 2.3.29).

REFERENCES

[1] W. BARTtt - C. PETERS - A. VAN DE VEN, Compact Complex Surfaces, Ergeb. Math. Grenzgeb. (3) 4, Springer-Verlag, Berlin (1984).

[2] A. BEAUVILLE, Surfaces alggbriques complexes, Ast~risque, 54 (1978). [3] P. DELIGNE, Th~orie de Hodge H, Publ. Math. I.H.E.S., 40 (1971), pp. 5-57. [4] H. M. F ~ I ~ S - I. KaA, Riemann Surfaces, Grad. Texts Math., 71, 2 ~d edition, Springer-Ver-

lag., Berlin (1992). [5] E. FREITAG, Uber die Straktu~" der FunktionenkSrper zu hyperabelschen Gruppen I, J.

Reine Angew. Math., 247 (1971), pp. 97-117. [6] A. FUJIKI, On resolutions of cyclic quotient singularities, Pub1. RIMS, Kyoto Univ., 10

(1974), pp. 293-328. [7] T. FUJITA, Impossibility criterion of being an ample divisor., J. Math. Soc. Japan, 34 (1982),

pp. 355-363. [8] F. HIRZEBRUCI-I, Uber vierdimensionale Riemannsche Fldchen mehrdeutiger analytischer

Funktionen yon zwei Vergnderlichen, Math. Ann., 126 (1953), pp. 1-22. [9] B. IVERSEN, Critical points of an algebraic function, Invent. Math., 12 (1971), pp.

210-224. [10] T. KATSURA - K. UENO, On elliptic surfaces in characteristic p, Math. Ann., 272 (1985), pp.

291-330. [11] J. KOLLMr Subadditivity of the Kodaira dimension: fibers of general type, Alg. Geom.

Sendal, 1985, Adv. Studies Pure Math., 10 (1987), pp. 361-398. [12] M. LEVINE, Deformations of Irregular Threefolds, Lect. Notes in Math., 947, Springer

(1982), pp. 269-286. [13] Y. MIYAOKA, Algebraic surfaces with positive indices, in Class of Alg. and Anal. Manifolds

(K. Ueno ed.). Progress in Math., 39, Birkhafiser (1983). [14] M. NAMBA, Branched Coverings and Algebraic Functions., Pitman Research Notes in Math.

Series, vol. 161, Longman (1987). [15] F. SERRANO, The Picard group of a quasi-bundle, Manuscripta Math., 73 (1991), pp.

63-82. [16] F. SERRANO, Fibred surfaces and moduli, Duke Math. J., 67(1992), pp. 407-421. [17] H. TsuJI, Stability of tangent bundle of minimal algebraic varieties, Topology, 27 (1989), pp.

429-442. [18] G. VAN DER GEER, Hilbert Modular Surfaces, Ergeb. Math. Grenzgeb, 16, Springer-Verlag,

Berlin (1988). [19] G. XIAO, Surfacesfibrdes en courbes de genre deux, Lecture Notes in Math., 1137, Springer-

Verlag, Berlin (1985). [20] O. Z~ISKI, The theorem of Riemann-Roch for high multiples of an effective divisor on an

algebraic surface, Ann. Math., 76 (1962), pp. 560-615.

Isotrivial fibred surfaces

Documents

Transcript of Isotrivial fibred surfaces