The basic facts are:
a) Given X, Y birational, then there is Z,
Z dominates X, Y.
b) Any regular, birational map is a sequence of blow ups. (Hence,
in particular, p ,q,x are birational invariants.)
X is said to be a minimal model iff it is minimal among the models.
(Note that the minimal model coincides with minimal as defined above.)
A birational class of surfaces is said to be positive, if in addition to
a) we have:
+ X Y
a ) Given X, Y, then we can find W, \ ^ X, Y dominates W.
Obviously positive surfaces are the ones having unique minimal models.
These will then furnish the canonical representatives for the function
fields. And the structure of all models will then be easily described in
terms of the minimal one. (Note that any birational automorphism of a
positive surface is automatically biregular.)
Intimately related to this is the useful language of infinitely near
points. If X Y the points of X are said to be infinitely near
points on Y, the points on Y are said to be ordinary points of Y.
Using b) it is clear what we will mean by an infinitely near point of
given order.
The Hodge numbers are crude invariants in describing surfaces. In
order to appreciate the finer details we have to know the group of
divisors more closely.
This is a huge group and in order to make it less unwieldly, we
introduce linear equivalence. Then we get a 1-1 correspondence between
divisors D, and invertible sheaves (or equivalently line bundles) via
D » 3(D), which will enable us to define intersection via
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