Page 413 -

388 20. Families of Elliptic Curves

(2.2) Proposition. For a local ringed space, let (X) denote the set of pairs
(L; s 0 ,..., s n ) where L is a line bundle on X and s 0 ,..., s n is a set of sections
n
generating L, all up to isomorphism. The function Hom (Sh/R) (X/R, P ) → (X)
R
∗
which assigns to a morphism f : X/R → P n the pair of the line bundle f (O(1))
R ∗
∗
and the generating sections f (y 0 ), ..., f (y n ) is a bijection.
∗
Proof. We prove the result by constructingan inverse to this function by startingwith
L a line bundle on X with generating sections s 0 ,..., s n . Denote by X i the open set
of x ∈ X where s i (x) = 0. The related f is the result of gluing the compatible
∗
morphism f i : X i → W i = Spec(R[y 0 /y i ,..., y n /y i ]) defined by f (y k /y i ) =
i
s k /s i usingthe adjunction in (1.4). The rest is easily checked and left to the reader.
(2.3) Remark. For a section s ∈ (X, L) of a line bundle on X the closed subset
Z(s) of zeros of s. It is defined by x ∈ Z(s) provided s(x) = 0 or equivalently
the germ s x ∈ m x L x . In fact, Z(s) is a closed subscheme of X with ideal sheaf J
generated by one element. Locally at each x ∈ X the line bundle L|U is isomorphic
to O X |U and the section s of L corresponds to f ∈ (U, O X ) where the germ f x
generates the ideal J x ⊂ O x definingthe closed subscheme Z(s).
For a nonzero section s ∈ (X, L) the closed subscheme Z(s) has codimension
one and is called a divisor. Now we recall some elements of the theory of divisors
using the sheaf of germs of total rings of fractions K X on a scheme (X, O X ) which
is the general setting for zero and poles.

(2.4) Definition. Let (X, O X ) denote a local ringed space. The sheaf of rational
functions K X on X is the sheafification of the presheaf on X which assigns to each
open set U the total ringof fractions of the ring (U, O X ).
Observe that K X is a sheaf of O X -algebras.

(2.5) Remark. For x ∈ X,the O x -algebra K x is the total ringof fractions of the
ring O x since the operation of formingthe total ringof fractions commutes with
direct limits. For a Noetherian scheme X and an affine open subscheme U the ring
(U, K U ) is the total ringof fractions of the ring (U, O U ).

∗
∗
(2.6) Notation. Let O denote the sheaf of invertible elements in O X , K the sheaf
X X
∗
∗
of invertible elements in K X ,and D X = K /O . We have a short exact sequence of
X X
sheaves of abelian groups containing a short exact sequence of sheaves of monoids
∗ ∗
1 −−−−→ O −−−−→ K −−−−→ D X −−−−→ 1
X X
∪ ∪
∗ ∗ ∗
1 −−−−→ O −−−−→ K ∩ O X −−−−→ D −−−−→ 1.
X X X
The sheaf D X of abelian groups is called the sheaf of germs of divisors and the
∗
subsheaf D is called the sheaf of germs of positive divisors. For a germ D x ∈ D x
X
a local equation is f ∈ (U, K x ) where U is a neighborhood of x and D x = f x
∗
mod O in the quotient. Compare with 19(8.1).
X,x