ON DEGENERATIONS OF ALGEBRAIC SURFACES 7
In the context of a blow up at a point p, we have two mappings from
the divisors of X to the ones of B (X) .
P
We have the total transform D - TT~ (D) , and TT"" gives the natural
embedding of N(X) into N(B(X)); for that reason we will employ the con-
vention of writing D = TT"" (D)
The more interesting concept is that of proper transform. Naively
we define D = TT (D - p) . The relationship between the total and proper
transform is given in N(B(X)) by D = D - m E, where m = mult (D) . Hence
we get the formula (D«D') = (D»D') -mm*.
K behaves well under blow ups?we have K = K + E. Using this we
m (m -1)
get p (D) = p (D) - P P . For more details the reader is referred
a a 2
to [18].
These formulas have obvious generalizations to any number of points,
including infinitely near ones, as we will see below.
If TT: X - X is a map onto a minimal model, then T T is gotten by
blowing up points (including infinitely near points) on X ; for each point
x (in the sense of infinitely near) we let E = TT (x ) —the total
v v v
transform. Note E »E = -$ , hence we get a natural isomorphism
N (X) + N (XJ ® TL E .
0 v
Note: In general E is not irreducible. E.g. blow up X. and then
v 1
blow up x infinitely near to x . Then E is not irreducible, we have
the decomposition E. = (E - E ) + E2 into irreducible components. (Note
that E. - E is the proper transform of E under the blow up of x .)
(See figure 1.)
We also get a partial order on the E by inclusion. Thus x is
infinitely close to x iff E c E .
Note: As was indicated above we have the following simple general-
ization of formula (p )
Pa (D) = p^ (D) - V 1/2(m (m - 1))
a a {_j v v
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