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§4. Generalities on Elliptic Fibrations of Surfaces Over Curves 393

(4.5) Theorem. Let f : X → B be a relatively minimal elliptic or quasi-elliptic
fibration of a surface over a curve B satisfying f ∗ (O X ) = O B . There is an enumera-
tion of the multiple fibres f −1 (b i ) = m i P i such that the P i are indecomposable. With
1
the notation R f ∗ (O X ) = L ⊕ T we have the following formula for the canonical
line bundle on X

ω X = f ∗ L (−1)⊗ ⊗ ω B ⊗ O a i P i
i
where
(i) 0 ≤ a i ≤ m i − 1, and
(ii) a i = m i − 1 if P i is not exceptional.
Hence the length of T , denoted l(T ) is greater than or equal to the number of a i <
m i − 1. Moreover,

(−1)
deg L ⊗ ω B = 2g(B) − 2χ(O X ) + l(T )
or
deg(L) =−χ(O X ) − l(T ).

Proof (Sketch of parts of the proof).

∗
(1) We show that the line bundle O(K X ) equals f (M) ⊗ O a i P i where
i
0 ≤ a i < m i for some line bundle M on B. In the language of divisors, we show
that K X is linearly equivalent to f −1 (M) + i a i P i for some divisor M on B.For
this, consider r nonsingular fibres C(i) of f . Then the structure sheaves O C(i) on
C(i) are isomorphic to O C(i) ⊗ O(K X + C(i)) because C(i) is an indecomposable
divisor of canonical type, see (4.5), and in fact they are curves with p(C(i)) = 1.

(4.6) Definition. An effective divisor D = i n i C i on a surface X is of canonical
type provided K · C i = D · C i = 0 for all i. In addition, D is indecomposable of
canonical type provided D is connectedandthe greatest common divisor of the n i
equals 1.
Tensoring the following exact sequence with O(K X )

0 → O → O i C(i) → i O C(i) ⊗ O(C(i)) → 0,
we obtain the exact sequence

0 → O(K X ) = ω X → O K + i C(i) → i O C(i) → 0.
The cohomology sequence is the following exact sequence
0 0
0 → H (O(K X )) −−−−→ H (O K X + C(i)
i



0 1
i H (C(i), O C(i) ) −−−−→ H (O(K X )) → ...