6 ULF PERSSON
£_(D*.D) = ( V $ G^D')* defined up to linear equivalence on each curve.
u u (y
Line bundles (or divisors mod linear equivalence) are classified
by Pic X = H ($*) . The exact sequence
0
•-
Z —*r
$
-J? Q*
~^
0
gives
H1^,*) ~ H1(X,0) -^ H1(X,(3*) -^H2(X,jE)
Hence Pic X is factored into a continuous part
Pic X = H (X,$)/H (X,^ describing the divisors
algebraically equivalent to zero, and a discrete
part N « Im c, the Neron-Severi group describing
divisors mod algebraic equivalence.
2
We have N which is finite dimensional and in fact N = H (X,Z) n
H ' . The intersection defined above gives a nondegenerate pairing on
N. The basic fact is the
2
Hodge Index Theorem; If we have C 0 and CD = 0, then either
2
D 0 or D ~ 0 ( ~ denotes numerical or homological equivalence, both
coinciding in the case of surfaces).
As a consequence, we have that the (light) cone
D =
2
C = {C|C £ 0, CD £ 0) is self-adjoint
2
Furthermore, we have the adjunction formula KC+ C = 2p (C) - 2 for
2
irreducible curves; hence, in particular, KC + C = 0(2) for all C.
(p denotes the arithmetic genus.)
An inane observation of extreme usefulness is the fact that for two
distinct irreducible curves C, D we have CD ^ 0.
The Neron-Severi group behaves simply under blow ups. Indeed we
have N(B(X)) a N(X) © 2ZE, with the intersection pairing decomposing and
with E2 - -1.
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