RATIONAL CURVES ON QUASI-PROJECTIVE SURFACES
11
the following equivalent formulation of the definition of tiger (which follows readily from the
definitions):
Lemma. Let E be an exceptional divisor over S. Let T S be the extraction (of relative
Picard number one) of E. E is a tiger iff —{KT + E) has non-negative Kodaira dimension.
Since a tiger is a divisor of coefficient at least one, in hunting for a tiger, it is natural to extract
the exceptional divisor from the minimal desingularisation which has maximal coefficient. This
is the idea behind the hunt. We will use the tiger/hunt metaphor in various notations throughout
the paper, occasionally to a tiresome degree.
Beginning with (So, Ao) = (S, 0), we inductively construct a sequence of pairs (S*,A.;) of a
rank one log del Pezzo with a boundary, such that:
(1) ~(KSi + A,) is ample.
(2) If (Sj, Aj) has a tiger, then so does S.
The construction is by a sequence of a ^-positive extraction (that is a blow up) ft : rJ\+i
Si followed by a /f-negative contraction, 7^+], each of relative Picard number one. Ki+\ is either
a P1-fibration, or a blow down -Ki+\ : T!t+i Si+i- In the first case we say Ti+i is a ne t and the
process stops. We give the details in §8. Such sequences are frequently studied in the MMP, see
for example [34]. The only choice in the sequence is which divisor is extracted by / j . The hunt
is a sequence given by extracting an exceptional divisor Ei+i, of the minimal desingularisation
of Si, for which the coefficient e(E{+i, Kst + Aj) is maximal.
To generate the collection # ( and complete the proof of (1.3) ) we classify all possibilities for
the hunt for which we are unable to find a tiger. Let us give a few remarks to explain why such
a classification is possible:
If the coefficient e(S) is sufficiently close to one (cf. (21.1)), then by (5.4) of [25], E\ is a tiger.
By (9.3) the collection of S with e(S) 1 e is bounded. Thus in all but a bounded number
of cases, the hunt finds a tiger at the very first step, and what is needed is an efficient means
of dealing with the exceptions. Our choice for the hunt, that is always extracting a divisor of
maximal coefficient (which is a natural choice, from the point of view of tigers) turns out to
have remarkably strong geometric consequences. We will explain this in considerable detail in
the introduction to §8. It is these consequences which make feasible an explicit classification of
the exceptional cases. In (8.4.7) we give a detailed breakdown describing possibilities for the
hunt. We then complete the proof by analysing each of the possibilities.
1.17 Classification of all but a bounded collection of S.
As discussed above, a detailed analysis of the hunt yields a collection # containing all S (with
simply connected) without tiger. We introduced the hunt for exactly this purpose. Somewhat
surprisingly, the hunt is also a useful tool for classification at the other extreme:
Previous Page Next Page