SEAN KEEL AND JAMES M^KERNAN
the Bug-Eyed cover gives a simple, transparent proof, as well as some generalisations to higher
Case two: When D is empty, the idea is to replace S by S\, birational to 5, which contains
a non-empty D, in such a way that (1.3) for (S\,D) implies S° is uniruled. (S\,D) will come
from extracting a divisor E via a blow up / : T — S, and then blowing down in a different
direction n : T — S\, D = n(E). To find (Si, D) essentially reduces to finding E. For this we
introduce the following, the most important technical definition of the paper:
1.13 Definition. Let X be proper, normal variety. Let A be an effective Q-Weil divisor on X.
A special tiger for Kx + A is an effective Q-divisor a such that Kx + A 4- a is numerically
trivial, but not Kawamata log terminal (kit).
For some m, ma is a very singular element of | — m(Kx + A)| and so a sort of antithesis
to Reid's general elephant. Hence the terminology. By a special tiger for X, or a special tiger
(without further reference), we mean a special tiger for Kx-
Note that if we have a special tiger there is at least one divisor E of discrepancy (with respect
to Ks + a) at most —1. We will call any such E a tiger. In fact we are much more interested
in E (and the resulting new log Mori fibre space structure) than in a. Sometimes we will be
a little sloppy in our notation and use the word tiger to mean either E or a. This should not
cause any harm, because normally we are only interested in when X does or does not have a
tiger, E, which is equivalent to the existence of a special tiger, a.
We will define kit, and explain the motivation behind (1.13) in (1.15) below.
The next two results indicate the usefulness of tigers for (1.3):
1.14 Proposition. If S is a projective surface with quotient singularities and S has a tiger,
then S° is uniruled.
1.15 Proposition. The collection of rank one log del Pezzo surfaces which do not have a special
tiger is bounded.
Note (1.14) is implied by (6.1).
(1.15) follows from (9.3) and quite general boundedness principals. We prove a stronger
statement in §23. Of course, together Case I and (1.14-15) imply (1.3) in all but a bounded
collection of cases. We will turn to the question of classifying these cases (that is surfaces without
tiger) in a moment. But first some general remarks on tigers.
For the rest of the introduction, unless otherwise noted, S will indicate a log del Pezzo surface
of rank one.
Shokurov has independently considered tigers, in relation to complements. He has observed
that if S has a tiger, then Ks is 1, 2, 3, 4 or 6-complemented (Ks is n-complemented if there is
a member M G | — nKs\ such that K + 1/nM is log canonical). We include Shokurov's proof