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:
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
curve with E = -1.
Conversely a theorem of Grauert says that given a rational curve E,
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
We note that p ,q,x are invariant under blow ups, but we of course
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.