RATIONAL CURVES ON QUASI-PROJECTIVE SURFACES 5

ample. Every curve meets (Ks 4- A')-negatively. However, if the curve meets the exceptional

locus of / , its strict transform may be (Ks -f A)-positive. Thus in order to prove (1.5) along

these lines some result such as (1.3) is required. Such an attempt (in dimension three !) was

the original impetus for this paper.

1.6 Corollary. If (£, A) is a log Fano surface then S° is rationally connected, and U — 5 \ L A J

is connected by images ofA\. In particular, TCI(S°) is finite, and ir\(U) is almost Abelian.

(Almost abelian is defined in §7.) The main issue in (1.6) is the uniruledness of the smooth

locus of a rank one log del Pezzo. This was conjectured by Miyanishi and Tsunoda, [30]. The

finiteness of the fundamental group 7Ti(5°) has been established previously in [14]-[15] and

separately in [8]. The smooth locus of a log Fano threefold (or surface) has finite algebraic

fundamental group by a boundedness result of [6].

1.7 Corollary. Suppose Ks + D -f A is log terminal for some effective Q-divisor A. Then

every co-extremal ray (cf. [5]) is spanned by classes [C], for C the general member of a covering

family of rational curves, contained in the smooth locus, and meeting D in at most one point.

In particular if —(Ks + D + A) is ample, then the nef cone of S is polyhedral and spanned by

such classes.

The fact that the nef cone in (1.7) is polyhedral was proved in [5], our contribution is the

description of the generators. (1.7) should be compared to the cone theorem, which describes

generators for the cone of effective curves.

(1.7) in turn gives:

1.8 Corollary. Suppose (5, D + A) is log Fano. Then Ts(— logD) is generically semi-positive,

in other words for a general member C of a sufficiently ample linear series, the vector bundle

Ts(—\ogD)\c has no quotient line bundles of negative degree.

The fact that (1.8) is implied by (1.7) was pointed out to us by Mori.

We note that for terminal Fano threefolds, the analogue of (1.6) implies the analogue of (1.8).

Thus if one could prove terminal Fano threefolds have uniruled smooth locus, it would follow as

in [20] that the set of terminal Fano threefolds is bounded.

(1.8) itself has some interesting corollaries:

1.8.1 Corollary. Let (S,B) be a log Fano pair (with S a projective surface, and B C S a

reduced curve).

£

r-2-ll2

+ p(S)-etop(B)

peSing(S\B)

Tp

where rp is the order of the local fundamental group, etop(B) is the topological Euler characteristic

of B, and p is the Picard number.