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384 20. Families of Elliptic Curves

ditions. In the classification of surfaces, the role of the existence of elliptic fibrations
is central. The problem of whether or not an elliptic curve on a surface X over k can
be the fibre of some fibration π : X → B by curves of genus one or elliptic curves
is an important starting point in the theory, and we examine this problem further for
K3 surfaces.
Finally passing to three dimensional varieties X,wecan,byextension of the
study of elliptic fibrations on surfaces, consider fibrations by K3 surfaces and abelian
surfaces. This we do only for three dimensional Calabi–Yau varieties where criteri-
ons for the existence of fibrations by elliptic curves and/or by K3 surfaces are given.
There are many questions in this direction that were raised by physicists working
on string theory. For this, see the appendix by S. Theisen.

§1. Algebraic and Analytic Geometry

In the previous chapters on elliptic curves we saw that there is an algebraic theory
and a closely related analytic theory. The concept used to include both theories is the
following.
(1.1) Definition. A local ringed space X is a pair (X, O X ) consisting of a topologi-
cal space X together with a sheaf of rings O = O X on X such that the stalks O x are
local rings for each x ∈ X.
For each x ∈ X, we have the unique maximal ideal m x ⊂ O x and the residue
class field K(x) = O x /m x . For a cross-section s ∈ (U, O) over an open set U ⊂ X,
the value of s at x ∈ U is a germ denoted s x ∈ O x and the value of s at x ∈ U in
the residue class field is the image s(x) ∈ K(x) of s x under the natural quotient
morphism O x → K(x).

(1.2) Remark. Since the projection of a sheaf F to its base space is a local home-

omorphism, the set x ∈ U where two cross sections s , s ∈ (U, F) are s (x) =

s (x) is a closed set. For a section s ∈ (U, O), the set of points where s x = 0isa
closed set, while on the other hand, the set of points where s(x) = 0 is an open set.
This is a special property of local ringed spaces for s(x) = 0 means that s x is a unit
in O x . Thus there exists an open set U(x) containing x and t ∈ (U(x), O X ) with

s x t x = 1. Since s and t are sections of a sheaf, there exist an open set U (x) ⊂ U(x)

still containing x such that s y t y = 1 for all y ∈ U (x). This implies that s(y) = 0 for

y ∈ U (x).
(1.3) Definition. A morphism f : (X, O X ) → (Y, O Y ) of local ringed spaces is a
pair consisting of a continuous function f : X → Y together with a morphism of
sheaves of local rings f ∗ : (O Y → O X ), so it preserves the maximal ideals in the
stalks.

The category of local ringed spaces is denoted (l/rg/sp), and its objects are those
defined in (1.1) and its morphisms are those defined in (1.2) with composition being