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

S° 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: