8

SEAN KEEL AND JAMES M^KERNAN

the Bug-Eyed cover gives a simple, transparent proof, as well as some generalisations to higher

dimensions.

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