RATIONAL CURVES ON QUASI-PROJECTIVE SURFACES

9

of this in §22. Note Shokurov's result, and (1.15) imply there is a uniform TV 0 such that

| — NKs\ is non-empty for all S. This result is interesting in view of the fact that there is no

bound on the index of Ks-

Complements in dimension n give information on extremal neighbourhoods in dimension n-h2

(see [27]). They play an important role in Shokurov's program for log flips. These observations

were pointed out to us by Alessio Corti.

Tigers have the following relation to the notion of affine-ruled (as defined by Miyanishi). The

proof is given in (21.4).

L e m m a . // S is affine-ruled (that is contains a product neighbourhood U x A1) then S has a

tiger.

There are S with simply connected smooth locus (cf. §21). but no tiger. These give counter-

examples to Miyanishi's conjecture (see [15] ) that the smooth locus of any rank one log del

Pezzo has a finite etale cover, that is affine-ruled.

It is a fairly simple matter to reduce the proof of (1.3) to the case when 7r" g(S°) is trivial

(cf. §7). We will assume this for the rest of the introduction. We will sometimes abuse notation

and say that S° (or even S) is simply connected. Of course, a posteriori, by (1.6), the two are

even equivalent.

As remarked above, not every S has a tiger. In §15-19 we generate a finite set of families of

surfaces 5 which includes all (simply connected) S without tigers. For each S € 5 we have an

explicit description of the minimal desingularisation, S. in terms of blow ups of P 2 , and we check

in each case that 5° is uniruled, by explicitly exhibiting a dominating family of rational curves.

We note here that it is quite possible that 5 is too big, in the sense that some of the surfaces

in 5 actually have tigers, or non-simply connected smooth locus. Given an explicit surface it is

usually easier for us to check that the smooth locus is uniruled, than to check either that the

smooth locus is simply connected, or that the surface has no tiger, as we have rather robust

tools for the first, and only rather ad-hoc techniques for the second and third. Of course for

(1.3) this approach is sufficient.

It is perhaps surprising, that even with an explicit description of 5, it can still be very difficult

to show S° is dominated by rational curves. Typically the strict transform of these covering

curves have rather high degree in P 2 , often over a hundred.

To prove uniruledness in a specific case, we try to find rational curves, Z C 5, with an

endomorphism lifting to Sb. In practical situations this is only possible when Z meets the

singular locus at most twice, see (4.9.4). Then we try to deform the lifted endomorphism. We

have the following sufficient condition (6.5):

Suppose Z meets the singular locus twice, and the local analytic index of its two branches

(meeting the singular locus) are u and v. Then there is a surjection / : P1 — Z of degree