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 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).
+ p(S)-etop(B)
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.
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