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372 19. Higher Dimensional Analogs of Elliptic Curves: Calabi–Yau Varieties

∗ ∗ ∗ ∗
1 −−−−→ O −−−−→ M −−−−→ D X = M /O −−−−→ 1
X X X X
  
 id ord


id
ord
∗ ∗ -
1 −−−−→ O −−−−→ M −−−−→ Z x −−−−→ 0
X X codim(x)=1
and for positive divisors
∗ +
1 −−−−→ O −−−−→ O X −{0}−−−−→ D −−−−→ 1
X X
  
  
id id ord
ord
∗ ∗ -
1 −−−−→ O −−−−→ O X −{0} −−−−→ N x −−−−→ 0
X codim(x)=1
where ord is the function which assigns to a germ the order of zero or pole.
For the first diagram we extract the following exact sequence of low dimensional
cohomology groups for X over k
0
0
∗
∗
∗
1 → H (O ) = k → H (M ) = k(X) ∗
X
X
0 1
∗
→ H (D X ) = Div(X) → H (O ) = Pic(X) → ...
X
This leads to the exact sequence
1
1
∗
∗
∗
∗
1 → k(X) /k = Div p (X) → Div(X) → H (O ) = Pic(X) → H (M ) → ...
x
X
with the first arrow mapping a germ of the nonzero function f to the principal divisor
( f ).
(8.2) Groups of Divisors and of Line Bundles. In terms of sheaf cohomology we
0
define divisors as elements of Div(X) = H (D X ). Line bundles up to isomorphism
are described by elements of
1
∗
H (O ) = Pic(X)
X
Finally the group of divisor classes is the quotient group Div(X)/Div p (X) which
maps by an injection Div(X)/Div p (X) → Pic(X) into the Picard group Pic(X).
1
∗
In the algebraic case H (M ) = 0 and we have an isomorphism
X
Div(X)/Div p (X) → Pic(X)
from the group of divisors classes to the group of isomorphism classes of line bun-
dles.
(8.3) The Line Bundle of a Positive Divisor. For a divisor D ≥ 0viewedasa
closed subscheme D → X, we have the exact sequence
0 → J D = O(−D) → O X → O D → 0
of structure sheaves on X and D together with the ideal sheaf J D of the locus D in
X.