10

SEAN KEEL AND JAMES McRERNAN

lcm(u, v) lifting to Sb, and if

-KS'Z l/u + l/v

then / deforms to cover S°. We usually use the criterion when Z has two branches meeting at one

point, and u — v. In this case uZ is Cartier, and so the criterion only fails when —Ks • Z = 1/u.

Example (6.8) indicates that the criterion is essentially sharp.

In order to apply the criterion to a particular case, we have to find rational curves meeting the

singular locus twice. The Z we use occasionally come from interesting geometric configurations

(see for example 17.5.1-3). For concrete example applications of the criterion, see (6.10) and

(8.1).

There are around sixty surfaces in J. The generation of 5 is one main focus of the paper,

and accounts for most of its volume.

Th e hunt: Our construction of 5 is based on a simple idea. To explain it (and the notion

of tiger) we use the following:

1.16 Definition. Let X be a normal surface, and A an effective Q-Weil divisor (that is A =

Y^GL%Di with di € Q, di 0). Let IT : Y — X be any birational morphism. Let A = ^aiDi

be the strict transform. There is a unique Q-Weil divisor F supported on the exceptional locus

such that

KY + A + F = 7i*(Kx + A).

r = A 4- F is called the log pullback of A. The coefficient e(E, Kx 4- A) of any irreducible

divisor E on Y is just the coefficient as it appears in T, it depends only on (X, A) and the

discrete valuation associated to E. The coefficient e(X, A) of the pair (X, A) is the largest

coefficient of any divisor (or discrete valuation). We will write e(X) for the coefficient of (X, 0).

Remark-Definition. e(E, Kx 4- A) is just the negative of its discrepancy. In particular, Kx 4- A

is log canonical iff the coefficient of (X, A) is at most one, and log terminal iff in addition

the coefficient of every exceptional divisor is less than one. It is Kawamata log terminal iff its

coefficient is less than one.

We note one trivial, but useful, fact: coefficients are invariant under log pullback.

Now let us explain the motivation behind definition (1.13), specifically the definition of a

special tiger for S. In view of case I, when D in (1.3) is empty, one naturally wonders if there is

some curve C on S with Ks 4- C anti-ample and log terminal. It turns out this fails for infinitely

many families of 61, so one looks for weaker conditions. Note that if Ks 4- C is anti-ample, then

we can add on some effective Q-Weil divisor fi so that Ks + C+(3 is trivial. A general philosophy

of the log category is to treat exceptional divisors and divisors on S uniformly. Applying the

philosophy to a — C 4- /?, leads to the notion of a special tiger (for Ks) which can be defined

as an effective a with Ks 4- ex numerically trivial, with coefficient at least one. We note also