4 ULF PERSSON

B (X). See, e.g., Shafarevicz [18, p.9].

A blowup is given by a regular map rr: B (X) • X, such that:

P

a) TT:B (X) - E - X - p is an isomorphism,

b) TT(E) = p, where E is a smooth divisor.

E is called an exceptional divisor, and turns out to be a rational

2

curve with E = -1.

Conversely a theorem of Grauert says that given a rational curve E,

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with E = -1, then E is exceptional, and hence "comes from" a blow up.

(For a proof see e.g. Kodaira [10 a].)

Observation: h2(B (X)) = h (X) +1.

We call a surface X minimal, iff it contains no exceptional curves.

We conclude that any surface can, by performing a sequence of blow downs,

be made minimal.

For classification purposes it is convenient to just consider minimal

surfaces; the case of the nonminimal ones can easily be referred to the

former.

We note that p ,q,x are invariant under blow ups, but we of course

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have e(B (X)) = e(X) + 1, and hence K goes down by one.

Let us now chiefly focus on the algebraic case. In this case w(X)

is rich in information due to:

Classical fact: If X, Y are two algebraic surfaces, then we have

$t(X) = $t(Y) iff X, Y are birational, i.e., there exists a birational

correspondence $: X - Y.

X,Y are said to be models for the function field 5BN

We say X dominates Y, or X Y, iff there is a regular birational

map ^: X - * Y.