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.
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