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390 20. Families of Elliptic Curves

(2.13) Remark. Let X be a scheme over a field k, and consider a line bundle L
such that (X, L) is a k-vector space of dimension n + 1. For any basis s 0 ,..., s n ∈
(X, L) we consider the open set U of x ∈ X where at least one s i (x) = 0. Ap-
n
plying (2.2), we have a morphism X ⊃ U → P (k) denoted φ L . This morphism is
independent of basis, and we can denote it simply as

φ L : U → P (X, L)
Here P(V ) denotes the projective space on the vector space V ,and if L = O X (D),
then the projective space P (X, L) is also denoted |D|, and the mapping is denoted
n
φ D : U →|D|= P . The projective space |D| is called the (complete) linear
system associated with the divisor in the classical terminology, and n = dim|D| is
its dimension.
The aim in the next sections is to construct morphisms φ D : X → P a projective
space with image B, and then further the morphism φ : X → B whose fibres are
identified from properties of L or D in the total space X.

§3. Fibrations EspeciallySurfaces Over Curves

(3.1) Definition. A fibering or fibration p : X → B is a proper, flat morphism of
finite presentation. A fibration of curves of genus g (resp. K3-surfaces) is a fibration
p : X → B such that every fibre X b is a curve of genus g (resp. a K3 surface).
For definitions of properties of morphisms, see Mumford, Red Book, SLN 1358,
pp. 121 and 215.
This brings up the first question related to the smoothness of a fibration p, for
where p is not smooth the fibre can have a singular point. In the case of genus 1
curves we understand these singular curves, but for K3 surfaces the situation is more
complicated. In these two cases, we can “enrich” the fibration with additional struc-
tures.

(3.2) Definition. An elliptic fibration p : X → B is a fibration of genus one curves
together with a section e : B → X of p such that the geometric fibres are irreducible
reduced curves. A polarized fibration of K3 surfaces is K3 fibration p : X → B is a
fibration of surfaces such that the geometric fibres of p are K3 surfaces together with
aclass ξ ∈ Pic(X) such that its restriction to each fibre ξ b ∈ Pic(X b ) is the class of
an ample line bundle.
It is possible that all fibres over geometric points are singular rational curves of
genus 1 in a genus one fibration over a curve. This happens only in characteristic
p = 2 and 3, but in general, only a finite number of fibres are singular.

(3.3) General Fibre. If f : X → B is a fibration with general fibre F,thenexcept
for a finite set of b ∈ B, the fibre X b of f over b is an irreducible curve. We can
apply the genus formula (8.4) to obtain the genus q(F) as 2q(F) − 2 = K x · F since
F · F = 0. We have an elliptic fibration if and only if K X · F = 0.