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.