THE PICARD GROUP OF A QUASI-BUNDLE

A quasi-bundle is defined to be a morphism from an algebraic surface onto a curve having all smooth fibres connected and isomorphic, and allowing as only singular fibres multiples of smooth curves. When no multiple fibre occurs it is called a fibre bundle. Toe general fibre F of a quasi-bundle is said to be divisible by an integer k if (I/k)F is still the numerical class of an integral divisor. This paper focuses on the relationship between the divisibility properties of F and the torsion of H 2 ( S, Z). For fibre bundles, the link between those two notions is established by means of Serre spectral sequence. As for general quasi-bundles, a suitable base change leads back to the fibre bundle case. Toe results become most explicit for elliptic quasibundles, where the action of the monodromy can be fully computed. For any prime number p, the paper contains examples of fibre bundles whose fibre is divisible by p. THE PICARD GROUP OF A QUASI-BUNDLE


THE PICARD GROUP OF A QUASI-BUNDLE
The simplest kind of morphisms from an algebraic surface onto a curve are those having all smooth fibres connected and ísomorphic to each other, and allowing as only singular fibres multiples of smooth curves. These fibrations will be called quasi-bundles, the fibre bundles being the specíal case where no singular fibre occurs. The aim of this paper is to study the Picard group of surfaces S endowed with a quasi-bundle fibration.
In particular, we are interested in linking two seemingly unrelated notions, namely the divisibility of a fibre in H 2 (S, Z)/ (torsion) and the torsion of H 2 (S, Z).
If <p : S _. C is a quasi-bundle with general fibre F, we say that F is divisible by an integer k if there exists L E H 2 (S, l) such that F -kL is a torsion element of H 2 ( S, l) ( or zero). One sees easily that the cocycle L must be algebraic as well. Every integer which occurs as multiplicity of sorne fibre of <p obviously divides F in the abovementioned sense. However, not all the divisibility properties of F are accounted for by the existence of multiple fibres. The action of the monodromy is playing a role, too. As a matter of fact, for each prime number p we are giving examples in § 2 of fibre bundles ' with fibre divisible by p ( despite the absence of singular fibres ). Once the possibility of such a phenomenon has been shown it is a question of searching for implications. For a fibre bundle S _. C this is done by means of Serre spectral sequence. In this case we derive a close connection between the divisibility of F and the torsion of H 1 (S, Z) ( non-canonically isomorphic to the torsion of H 2 ( S, l) ). This is the content of § 2. The results for fibre bundles will be extended in § 3 to general quasi-bundles. The idea now will be to perform a suitable base change, so that the multiple fibres disappear, and the information about the fibre bunclle 80 obtained cm1 be appliecl to our fibration. r-.Ioreover, sorne results of the author ( [12]) about the behaviour of multiple fibres in homology are required. Here, of course, the results will be less precise since one loses information along the process of base-change. Finally, § 4 contains a more detailed study of elliptic quasibundles (i.e. g(F) = 1). In this context, the results ofthe prececling sections will become most explicit, inasmuch as the action of the monodromy can be fully described.
The main results of this paper are stated as Theorems 2.2 and 3.6 for general fibre genus, and Theorems 1.10, 4.1, 4.3 and 4.4 for elliptic fibrations.
The research leading to this paper was carried out while the author was visiting the University of Utah. I want to thank H. Clemens for his invitation, and J. Kollar for severa! helpful remarks. § l. NOTATION AND PRELIMINARIES All varieties will be defined over the field of complex numbers. A surface ( respectively, a curve) is a projective, irreducible, non-singular scheme of dimension 2 (resp. 1). vVe shall employ the following terminology: The irregularity and geometric genus of a surface S are denoted q(S) : = h 1 Os, p 9 (S): = h 2 0s respectively. For a curve or divisor C, g(C) stands for the arithmetic genus of C. The symbol ~ (respectively -) represents linear (resp. numerical) equivalence of divisors. If D is a divisor on a surface S, often we will write also D to mean its class in the distinct groups Pie S, H 2 (S, Z), etc., as the context will indicate.
A fibration <p : S -+ C is a morphism with connected fibres from a surface onto a curve. \Vhen the fibre genus is equal to one it is called an elliptic fibration. In general, r.p is said to be relatively minimal if no fibre contains a (-1 )-curve. \Ve say that r.p 1s the Albanese fibration if the image of the Albanese map a : S --+ Alb(S) is a curve isomorphic to C, and a : S--+ a(S) coincides with r.p. From the universal property of the Albanese variety ( [2]) i t follows that q( S) = g( C) if and only if ei ther q( S) = O or r.p is the Albanese fibration.
Let F = ¿i n¡B¡ be a fibre of r.p, where the B~s are the irreducible reduced components, and the nis their multiplicities. If m denotes the greatest common divisor of the nis, then we will say that m is the multiplicity of F, and will write F = m D, where D = ¿¡(n¡/m)B¡. Whenever we use the expression "let m D be a multiple fibre" we always mean that m is the multiplicity of m D, and m 2' .: 2.
This paper deals with the simplest types of fibrations, to be defined now: Definition 1.1. A fibration r.p : S --+ C is called a quasi-bundle if all smooth fibres are isomorphic, and the only singular fibres are multiples of smooth curves. If moreover r.p has no singular fibres, then r.p is said to be a fibre bundle. For economy of notation, a surface S will also be called a quasi-bundle (respectively, a fibre bundle) if it admits a quasi-bundle (resp., a fibre bundle) fibration.

Remark 1.2.
A quasi-bundle surface always admits two distinct quasi-bundle fibrations.
In fact, it is proved in [13] that every quasi-bundle surface is the quotient of a product of two curves by the action of a finite group.
• Proposition 1.4. With the preceding notation one has h 1 0s(-F) = g(C) for any fibre F. In particular, the image of Picº(S) -+ Picº(F) (induced by the inclusion Proof: Let F be the fibre over p E C. Leray spectral sequence yields In the next two Lemmas we will restrict our attention to elliptic fibrations.  A central theme of this paper is the divisibility of the fibre of a morphism, where divisibility is understood in the following sense: Deflnition l. 7. Let D be a divisor on a surface S. We will say that D is divisible by an integer k if (1/k)D E H 2 (S,Z)/(torsion), that is, if there exists a cocycle E E H 2 (S, Z) such that D -kE is either a torsion element of H 2 (S, Z) or zero.
We claim that the cocycle E of Definition l. 7 is also algebraic. The exponential sequence yields the exact piece Obviously the torsion of H 2 (S, Z) lies in the kernel of T, and so it is algebraic. Now, if D is a divisor and D -kE E tor H 2 (S, Z) then kE E Im(u). But H 2 (5,l)/Irn(a) is embe<lded m the C-vector space H 2 0 5 , which has no torsion.
Let N um( 5) denote the group of numerical equivalence classes of clivisors on 5.
By the sequence (*) above ancl the Algebraic lndex Theorem we see that Num (5)

•
The following result gives a converse of Proposition 1.9 for "most" elliptic fibrations, namely, the ones with x0s > O. The remaining ones fall into the class of quasi-bundles (Lemma 1.5) and will be the object of § 4.
Thus q divides all the e~s, so that q divides µ as well.

O § 2. FIBRE BUNDLES
As explained in the Introduction, the main object of this paper is to establish a relationship between the divisibility of the fibre of a qua.si-bundle and the torsion of the integral homology of the surface. In this section we will restrict ourselves to fibre bundles.
Given a fibre bundle r..p : S -+ C with fibre F, we can consider Serre's spectral sequence E;,q = Hp(C,Hq(F,l)) ===} Hp+q(S,l) where H P ( C, H q( F, Z)) denotes the p!h homology group of C with coefficients in the bundle of abelian groups {Hq(r..p-1 (t),Z)Lec· (see e.g. [15] or [10]). H¡(F,Z) is a 1r 1 (C)-module by the action of the monodromy, where 1r1(C) denotes the fundamental group of C. Ho(C,H 1 (F,l)) will be denoted H1(F)1r 1 (c), and by ( [15], VI 3.2) it is computed as Inasmuch as the map r..p is analytically locally trivial ([1], I 10.1), it follows that the action of 1r 1 (C) over H 1 (F,Z) factors through the action of the group of analytic automorphisms of F, denoted Aut(F). In general, if G is a finite group of order IGI, and M a G-module with invariants Mª, then IGI annihilates the kernel of the norm map N : If moreover M is a torsion-free abelian group then IGI annihilates the Z-torsion of Ma as well. In our situation we way conclude that the torsion of H 1 (F)1ri(C) is killed by the order of Aut(F).
The first terms of Serre's spectral sequence yield an exact sequence where the homomorphisms i.p. are induced by i.p ([10], Thm. 5.8).
The divisibility of F m H 2 (S, Z)/(torsion) can be read off from this sequence as follows.   Let F be any smooth fibre. With this generality, the following has been proved in (12) In this Section we are going to develop the main theme of this paper, namely divisibility of fibres versus torsion of homology, for general quasi-bundles. For this purpose we are going to perform a suitable base change on our fibration which will lead us to a fibre bundle. This will allow us to apply the results of the preceding Section. The main tool, then, is the following The exponential sequences far S and R give rise to a commutative diagram with exact rows: H 2 (S,Z))º Let F 0 the fibre of r.p over Pt+t. By construction, 1r 1s totally ramified over F 0 , so that 1r-1 (F 0 ) ~ F 0 • Identifying 1r-1 (F 0 ) and Fo we may consider h : F 0 <---+ R, j : F 0 <---+ S to be the corresponding inclusions, which determine a commu- j\ f·
\Vithout this precaution, only the inequality > holds in general.
\Ve shall apply the results of § 2 by means of the follmving As a consequence, the torsion oí J is also a quotient of the torsion of I. This latter fact is one of the crucial points of our construction. O With the above notation, we now want to compare the divisibility of F with that of F. • \Ve reach at last the conclusion we were seeking:   A translation is homotopic to the identity, and thus induces the identity map m homology. Consequently we may restrict our attention to automorphisms of F defined as multiplication by sorne e E C. ldentifying H1(F, Z) ~ Z · w EB Z one sees that the induced automorphism on H 1 (F,Z) is also multiplication by e. We have é·w=aw+b , é·l=cw+d with a,b,c,dEl. Hence w=(aw+b)/(cw+d), so that (*)

cw +(d-a)w-b=O
In the ordered ha.sis {w, 1} of Z · w EB Z, the morphism is given by the matrix (: ~) with determinant ± l. The discriminant of (*) must be strictly negative be- •