RATIONAL CURVES ON QUASI-PROJECTIVE SURFACES
3
contraction, the Picard number goes down by one, and from this point of view, Y is simpler than
X. The complication is that Y can have quotient singularities (if E has self-intersection at most
- 2 ) . We replace (X, D) by (V, DY 7r*(D)), and continue, by looking for a (AV + 22y)-negative
contraction. Such a series of contactions is called the (Kx + D)-minimal model program (MMP
for short).
If at the start \m(Kx + D)\ = 0 for all m 0, then the Log Abundance Theorem implies
that either X has a fibration X C whose general fibre is IP1 and meets D at most once, or,
we have a birational morphism IT : X S, a composition of contractions as above, with S of
Picard number one, and K$ + &s anti-ample and log terminal (discussed below). In the first
case of course we take the general fibre. In the second case, X is dominated by rational curves
meeting D at most once, iff S is dominated by rational curves meeting ir(D) at most once. The
one dimensional part of 7r(D) is Ds, the zero dimensional part, V, is a union of components of
D contracted by 7r. In this way, (1.1) is reduced (and in fact equivalent) to:
1.3 Theorem. Let S be a normal projective surface of Picard number one, with quotient singu-
larities. Suppose D C S is a reduced curve, such that K$ + D is log terminal, and (Ks + D) is
ample. Let V C S be any finite set of points. Then S is dominated by rational curves, meeting
D U V at most once.
An elementary discussion of the notion of log terminal is given in Appendix L. We note here
(so that the reader may have some idea of the term's meaning) that if D is irreducible then
K$ -f 22 is log terminal iff the pair (S. D) has quotient singularities, that is locally analytically
the quotient of a smooth pair (Sf, D1) by a finite group. For the (Ks + D)-MMP, log terminal is
the correct generalisation of normal crossings, in the sense that it is preserved by the birational
contractions that occur in the (Ks 4- D)-minimal model program.
In fact in (1.3) it is enough to consider the case V = Sing(S). The main point behind this
reduction is Mori's observation that on a smooth space, the general member of a dominating-
family of images of A1 deforms freely (as an image of A1) and so in particular misses any fixed
codimension two subset, see (5.5) and (5.8). Thus (1.3) can be equivalently (and somewhat
more aesthetically) stated as:
1.3.1 Theorem. Let S be a normal projective surface of Picard number one, with quotient
singularities. Suppose D is a reduced curve, such that Ks 4- D is log terminal, and —(Ks 4- D)
is ample. Then is dominated by rational curves which meet D at most once.
As we will explain shortly, the main work is proving (1.3) in the case when D is empty, and
is thus to show that the smooth locus of a log del Pezzo surface is uniruled (that is dominated
by complete rational curves).
Our proof of (1.3) proceeds roughly as follows: For one class of rank one log del Pezzo surfaces,
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