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§10. Enriques Classification for Surfaces 377

(9.7) Relation Between Numerical Invariants. From Poincar´ e duality the topo-
logical Euler number e(X) = · on X × X is
+
−
c 2 = e(X) = b 0 − b 1 + b 2 − b 3 + b 4 = 2 − 2b 1 + b 2 = 2 − 2b 1 + b + b .
Combining the topological Euler number calculation with the Riemann–Roch
2
formula 12χ(O X ) = c + c 2 we obtain the relation
1
2
12 − 12q + 12p g = K + 2 − 2b 1 + b 2
which we rewrite in the form
2
10 − 8q + 12p g = K + b 2 + 2
where = 2q − b 1 .

(9.8) Remark. For complex surfaces either b 1 is even when = 0or b 1 is odd
0
1
when = 1. Furthermore, dimH (& ) = q − and b + = 2p g + 1 − .he
following two assertions are used to prove this statement.
Firstly, every holomorphic 1-form is closed. This is false in dimensions strictly
greater than two and in characteristic p > 0. Secondly, if ω 1 ,...,ω r is a basis
0
1
for H (& ), then the set of forms ω 1 ,...,ω r , ω 1 ,..., ω r is linearly independent in
1
H (C) and thus we have the inequality 2h 1,0 ≤ b 1 .
For algebraic surfaces there is a quantity in characteristic p > 0 with proper-
ties as in the previous paragraph.

§10. Enriques Classification for Surfaces

(10.1) Exceptional Curves and Minimal Surfaces. The Enriques classification the-
orem is for surfaces without exceptional curves, that is, without rational curves E
with E, E =−1. These curves on a surface X are the result of blowing up a smooth
point P on a another surface Y. This blowing up is a map X → Y such that the
restriction X − E → Y − P is an isomorphism. A surface is called minimal provided
there are no exceptional curves. Every surface can be mapped onto a minimal surface
with only exceptional curves mapping to points, and the process is unique with one
type of exception related to P 2 and P 1 × P 1 .

(10.2) Nature of the Classification of Minimal Surfaces. We can classify very
roughly curves into three classes by the genus g where g = 0, g = 1, and g ≥ 2.
In this classification, the subject of the book is in the middle g = 1. The Enriques
classification comes in four parts where the canonical divisor K with O(K) = & 2
X
again plays a basic role.
Step 1. For curves the first step is to identify genus g = 0as P 1 with a global
meromorphic function with one simple pole. For surfaces, the corresponding class
of minimal surfaces X consists of those with a curve C satisfying K · C < 0. These
are further classified by the irregularity q: