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392 20. Families of Elliptic Curves

(3.8) Application II. Let X be a surface with K X · K X = 0and K X · D ≥ 0 for
all divisors D ≥ 0. In the linear system of mK X for m > 0 consider D ∈|mK X |

with D = 0. Decompose D = n i E i with n i > 0, and hence, we have 0 =
i

(mK X ) · K X = D · K X = n i (E i · K X ). Since each E i · K X ≥ 0, we see that
i
E i · K X = 0and E i · D = 0 for all i. Apply (3.3) this time to the Q-vector space

generated by these E i ’s, and we deduce that E i · E i ≤ 0 for D = n i E i ∈|mK X |
i
on such a surface. This is a useful property of surfaces whose canonical divisor has
zero self intersection.
A general reference for this section is Barth, Peters, and van de Ven [1980].

§4. Generalities on Elliptic Fibrations of Surfaces Over Curves

In the carrying out of the Enriques classification of surfaces, an understanding of the
canonical divisor on an elliptic surface plays a basic role.
(4.1) Remark. For many considerations concerning a family f : X → B we can
reduce to the case of f ∗ (O X ) = O B by Stein factorization. Any proper morphism
f : X → Y between Noetherian schemes has a factorization

f q
X −−−−→ B −−−−→ Y

where f is proper and q is finite with B Spec( f ∗ (O X )). In this factorization,
f (O X ) = O B , and under a separability hypothesis on f , the morphism q is ´ etale. In

∗
general, a standard modification of a family of curves or surfaces over B is to induce

it by an ´ etale map B → B to a family over B .
(4.2) Remark. Let f : X → B be an elliptic fibration with f ∗ (O X ) = O B . Then
1
R f ∗ (O X ) = L ⊕ T where L is a line bundle and T is a torsion sheaf. This in-
cludes quasi-elliptic fibrations which are those nonsmooth fibrations arising only in
characteristic 2 and 3 mentioned in (3.2).
(4.3) Definition. With the previous notation for an elliptic fibration f : X → B
with f ∗ (O X ) = O B the exceptional points of f are b ∈ Supp(T ). A fibre D =
f −1 (b) is exceptional provided it satisfies either of the following two equivalent
conditions:

(1) b is an exceptional point, i.e. b ∈ Supp(T ).
0
(2) dimH (O D ) ≥ 2.
Recall that a curve C is of the first kind provided C is rational and C · C =−1,
and a surface is minimal provided there are no curves of the first kind.

(4.4) Definition. A fibration f : X → B of a surface over a curve B is a relatively
minimal fibration provided there are no curves of the first kind contained in any fibre.