2 ULF PERSSON
The rest of the conditions, however, do not imply 2) in the cases
of higher dimensions.
Note that condition 6 illustrates the fact that algebraic surfaces
are characterized by their wealth of curves.
To describe the nonalgebraic ones with respect to this property we
have the following, also due to Kodaira [10 a, Thm. 4.1, 4.2, 5.1].
Theorem B: Let X be a surface. Then we have:
1. tr.deg !IR(X) = 1, iff we have a map g:X - * C, with elliptic
curves as fibers. (X is a so called elliptic surface) and
such that X has no transversal divisors with respect to the
fibration.
If this is the case then q#: ©(C) - * w(X) induces an isomor-
phism of fields.
2. tr.deg ^(X) = 0, iff there are only a finite number of curves
on X.
Given any surface X we can of course look at the Hodge spectral
sequence and the Hodge numbers h = dim H (X,n )•
We define p = h ' « dim H (X,fl ) by Serre duality
h
2

=
h
0
'
2
,
= h
0A , . .. 1
= dim H (X,0) and we have
p is referred to as the geometric genus of X, and q is called the
irregularity.
We define x ~ P - Q + 1* an* ^et e denote the Euler character-
istic.
These invariants are related by Noether's formula
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