6

SEAN KEEL AND JAMES McRERNAN

Note when B is empty then (1.8.1) implies that log del Pezzo surface can have at most 4-f 2-p

singularities. As far as we know, no such bound has been previously observed.

For p = 1, we prove a stronger version of (1.8.1) (it applies to any log terminal surface

5), following an argument of [20], see (9.2). The proof is independent of (1.1) (and indeed,

anything else in the paper). We call this intermediate result the Bogomolov Bound and use

it repeatedly in the proof of (1.3). The Bogomolov Bound has an interesting corollary, see (9.3).

In [2], using different methods, Alexeev obtains results related to (but much deeper than) (9.3).

In the light of (1.0) and (1.1), we propose the following:

1.9 Conjecture. Let U be a smooth quasi-projective variety. Then, either

(1) k{U) 07 or

(2) U is dominated by images of A1.

Note that (1.9) is equivalent to the following:

1.10 Conjecture. Let the pair (X, D) consist of a smooth variety X, and reduced normal

crossings divisor D. Then, either

(1) \rn(Kx -f D)\ is non-empty for some m 0, or

(2) there is a dominating family of rational curves meeting D in at most one point.

By (5.11), (1) and (2) of (1.10) are mutually exclusive, and one can view the conjecture as

saying either X has a log pluricanonical section, or there is a clear geometric reason why none

can exist.

There are natural analogues of (1.4-7) in higher dimensions, and similar implications will

hold between them, if one has the MMP. We wish however to note one difference; in dimension

two (1.1) implies (1.6). This follows from deformation theory, and the fact that log terminal

surface singularities are quotient singularities, see Appendix L and §5. Three-fold log terminal

singularities need not be quotient singularities.

In higher dimensions there is not much evidence for (1.9-10). We can prove (1.9) in the case

— {Kx + D) is ample, see (5.4), and a version of (1.3) for a threefold X in the case D has two

components, see (6.7).

Under the assumptions of (1.9), if X has dimension at most three, then the MMP and log

Abundance [22] imply that X is covered by (not necessarily rational) curves C with {Kx 4- D) -

C 0. Thus one is moved to consider:

1.11 Conjecture: Log Bend and Break, (notation as in 1.10). If {Kx + D) • C 0, and

C £ D then through a general point of C there is a rational curve meeting D at most once.

Mori's famous bend and break argument proves (1.11) in the case D empty. We don't see any

way to extend the proof to the case when D is not empty. In Mori's argument, Kx • C 0 is