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
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