12 SEAN KEEL AND JAMES

MCKERNAN

In §23 we classify rank one log del Pezzos S = So (no assumption on fundamental group)

such that Ei (of the first hunt step) is a tiger. As we remarked above, this includes all S with

sufficiently large coefficient e(5), which in turn includes all but a bounded collection of S.

Our classification, which is independent of the hunt analysis §14-19, is of the following sort:

First we classify abstract pairs (S\:Ai) of a rank one log del Pezzo surface containing an

integral rational curve A\ such that Ksx + A\ is anti-nef, and log terminal at singular points

of S\. They fall into a short number of series. We note that a similar classification of pairs is

obtained in [31] and [32]. Our argument is based on quite different ideas, and is considerably

simpler.

We apply this classification to the first hunt step. Let A\ C S\ be the image of Ei (assuming

ITi is birational). It is easy to show that if E\ is a tiger, then either 1\ is a F^fibration, or

(S\,Ai) is as in the preceding paragraph. The first case is easy to classify. To classify 5, it

remains to classify possibilities for the transformation Si —- S. This amounts to classifying

possibilities for n\ : T\ — Si such that Ei (the strict transform of A\) is contractible, and

contracts to a log terminal singularity. We indicate how this can be done in §23. It is easy and

elementary, but we do not actually list the possibilities, as it would be notationally too involved.

Observe that the existence of such a classification is at least to some degree counter-intuitive.

One might have expected a simple classification of S with mild singularities, with a progressively

less tractable list of possibilities as more complicated singularities are allowed. Indeed, P 2 is

the only smooth S, Gorenstein S are classified in [11], and log del Pezzos with index (of K) at

most two are classified in [1]. However, comparison of our two main classification results -the

collection 5, and our classification of S with large coefficient-gives an indication in the opposite

direction: Surfaces of small coefficient, though bounded, appear (at least to us) rather sporadic,

while surfaces with sufficiently large coefficient exhibit relatively uniform behaviour.

Given the effectiveness of our techniques at either singularity extreme, we believe that re-

peated application of the same methods would eventually yield a complete classification of rank

one log del Pezzo surfaces.

Several people have asked the following question; is it true that {K^} (over all log del Pezzos)

satisfies ACC for bounded subsequences ? The question is suggested by analogy with a result of

[2] that the corresponding set over minimal log terminal surfaces of general type satisfies DCC.

The answer to this question is a resounding no, see (22.5).

Thanks: We would like to thank A. Baragar, A. Corti, D. Morrison, S. Mori, P. Deligne,

D. Huybrechts, T. Hsu, M. Lustig, R. Morelli, Y. Petridis and V. Shokurov for advice and

suggestions. We would like to particularly thank Janos Kollar, who suggested (1.5) as a natural

companion to Log Abundance, and who has assisted us considerably throughout our research.