SEAN KEEL AND JAMES
We note that the reverse direction of the implication in (1.1) is fairly straightforward, and
holds in all dimensions, see (5.11). Thus (1.1) can be viewed as saying, either U has a log
pluricanonical section, or there is a clear geometric reason why it cannot.
The other main results of this paper are very strong partial classifications of log del Pezzo
surfaces, that is projective surfaces with quotient singularities and —K$ ample. This includes
a classification of all but a bounded family of log del Pezzo surfaces of Picard number one. We
will explain this classification at the end of this introduction. We believe that with sufficient
effort the methods of the paper would yield a complete classification.
Log del Pezzo surfaces are of interest for several reasons, independently of (1.1). They are
obviously important for the study of open surfaces. Beyond this, the log category is important
even for the study of projective varieties with, for example, log surfaces (surfaces with bound-
ary) playing an intermediary role between surfaces and threefolds. Good examples of this are
Kawamata's proof of the Abundance conjecture, and Shokurov's program for Flips. Log del
Pezzos also occur as the centres of 4-fold log flips.
In addition log del Pezzos play an essential role in the compactification of the moduli space of
surfaces of general type. They occur at the boundary of the moduli space of surfaces of general
type (just as rational curves occur at the boundary of the moduli space of curves), and are the
main object of study in the proofs of Alexeev's boundedness theorems , used to show the
moduli space is projective.
We will use the following terminology. Given a variety X, let Ar° denote the smooth locus of
1.2 Definition. Let X be a variety, C a curve in X and D a subset of X'. We say that C meets
D k times if the inverse image of D on the normalisation of C is a set of A: points.
For example a smooth point or a unibranch singularity of C counts once, and more generally
a singularity with r branches counts r times, regardless of the order of contact.
We note also that throughout the paper, by a rational curve, we mean a complete rational
curve, that is an image of P 1 . By uniruled, we mean dominated by rational curves.
1.2.1 Remarks. If a complete variety is dominated by rational curves, then it is in fact covered
by rational curves (that is through every point there is a rational curve), since the degeneration
of a rational curve is again rational.
In the definition of dominating, one can equivalently require that the maps / : C — U form
a flat family (see (IV.1.3.5) of ).
Connection betwee n (1.1) and log del Pezzo surfaces. In proving (1.1) one tries to
simplify the situation by a birational contraction ir: X — Y of an irreducible curve E C X. In
order to preserve the dimension of \m(Kx + D)\, it is sufficient that (Kx + D)-E 0. Under the