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§2. Morphisms Into Projective Spaces 389

(2.7) Remark. The group of divisors (X, D X ) is written additively so that if f ,

f are local equations of divisors D , D ,at x ∈ X respectively, then the prod-

uct f · f ±1 is a local equation of D ± D .Also, if D = D , then f /f ∈
x x
∗
∗
im( (U, O ) → (U, K )) in some open neighborhood U of x.
X X
The short exact sequence
∗ ∗
1 −−−−→ O −−−−→ K −−−−→ D X −−−−→ 1
X X
leads to a long exact sequence of cohomology starting with the terms
1
1
∗
∗
0 → (X, O ) → (X, K ) → (X, D ) → H (X, O ) → H (X, K ) → ... .
∗
∗
∗
X X X X X
∗
(2.8) Definition. Associated to each rational function f ∈ (X, K ) is a divisor
X
∗
( f ) whose local equation at any x ∈ X is f . The divisor class [D]of D ∈ (X, K )
X
1
∗
is its image in H (X, O ).
X

Hence in terms of the above additive notation we have [D] = [D ] if and only if
∗

D = D + ( f ) for some f ∈ (X, K ).
X
(2.9) Definition. To a divisor D ∈ (X, D X ), we associate an O X -subsheaf O X (D)
⊂ K X by the condition that O X (D) x = O x [ f −1 ] x where f is any local equation of
D on an open set U with x ∈ U.

Observe that O X (D) ⊗ O X (D ) and O X (D + D ) are isomorphic, and in terms

of the homomorphism sheaf we have an isomorphism
ˇ
O X (−D) = HomO X (D) = Hom(O X (D), O X ).

(2.10) Proposition. For two divisors D and D , the divisor classes [D] = [D ] are

equal if and only if O X (D) and O X (D ) are isomorphic as O X -sheaves.

This is another way of looking at the second morphism in the following exact
1
∗
sequence (X, K ) → (X, D ) → H (X, O ), namely, to the divisor D we
∗
∗
X X X
assign the isomorphism class of the line bundle O X (D).
Adivisor D is positive (or effective) provided O X ⊂ O X (D) ⊂ K X , and this is
equivalent to O X (−D) is an ideal sheaf in O X .
(2.11) Remark. For an effective divisor D with structure sheaf O D for the related
closed subscheme D we have the exact sequence 0 → O X (−D) → O X → O D →
0 of sheaves. The closed subscheme D has O X (−D) as defining sheaf of ideals,
and the local equation f of D at x has the property that O X (−D) x = f · O x .In
particular, the divisor D is determined by a codimension 1 closed subscheme D.For
an effective divisor D, the image s of 1 under the natural morphism (X, O X ) →
(X, O X (D)) ⊂ (X, K X ) is like a global equation for D and Z(s) = D.
(2.12) Proposition. Let L be a line bundle. The sections s ∈ (X, L), which are
nonzero divisors, up to multiplication by elements of (X, O ) are in bijective cor-
∗
X
respondence with effective divisors D such that O D is isomorphic to L.If O X (D)

and O X (D ) are isomorphic, then D = D + ( f ) where f ∈ (X, K ).
∗
X